&Digraphs\ ;

\n[BM06] Boyer, J. M.\ and Myrvold, W. J.,\n On the Cutting Edge: Simplified\ O(n) Planarity by Edge\n Addition,\n in Gr\ aph Algorithms and Applications 5,\n {WORLD} {S\ CIENTIFIC}\n (2006),\n 241--273.\n

\n[CK86] Calder\ bank, R. and Kantor, W. M.,\n The geometry of two-weig\ ht codes,\n Bull. London Math. Soc.,\n 18 (2)\n (1986),\n 97--122.\n

\n[Gab00] Gabow, H. N.,\n Path-based depth-first search for strong and biconnected\n \ components,\n Information Processing Letters,\n 74 (34)\n (\ 2000),\n 107 - 114.\ \n

\n[JK07] Junttila, T. and\ Kaski, P. (Applegate, D., Brodal, G. S., Panario,\ D. and Sedgewick, R., Eds.),\n Engineering an effi\ cient canonical labeling tool for large and\n sparse graphs,\n \ in Proceedings of the Ninth Workshop on Algorithm Eng\ ineering and\n Experiments and the Fourth Workshop on Analytic Algorithms and\ \n Combinatorics,\n SIAM\n\ (2007),\n 135--149\ .\n

\n[MP14] McKay, B. D. an\ d Piperno, A.,\n Practical graph isomorphism, II,\ \n Journal of Symbolic Computation,\n 60 (0)\n (2014),\n 94 - 112.\n

\n[vLS81\ ] van Lint, J. H. and Schrijver, A.,\n <\ i class='BibTitle'>Construction of strongly regular graphs, two-weight codes a\ nd\n partial geometries by finite fields,\n Combinatorica,\n 1 (1)\n (1981),\n 63--73.\n

1 \n The Digraphs\ package\n \n

1.1 \ Introduction\n\n

\n
1.1-1 Def\ initions\n\n

2 Installing Digraphs\n
2.1 For those in a hurry\n\n
\n
2.2 Optional package\ dependencies\n\n
\n
  2.2-1 The Grape package\n\n
\n
2.\ 3 Compiling the kernel module\n\n\n
&nb\ sp;2.4 Rebuilding the documentation\n\n
\n
  2.4-1 DigraphsMakeDoc\n
\n
2.5 Testing your installation\n\n
\n
  \ 2.5-1 DigraphsTestInstall\n
  2.5-2 DigraphsTestStandard\ \n
  2.5-3 DigraphsTestExtreme\n
\n
\n", 3 := "
3 Creating digraphs\n
3.1 Creating digraphs\n\n
\n
<\ span class=\"nocss\">  3.1-1 IsDigraph\n
  3.1-2 Is\ MutableDigraph\n
\   3.1-3 IsImmutab\ leDigraph\n
\ ; 3.1-4 IsCayleyDigrap\ h\n
  <\ /span>3.1-5 IsDigraphWithAdjacency\ Function\n
\ 3.1-6 DigraphByOutNei\ ghboursType\n
&nb\ sp; 3.1-7 Digraph<\ /span>\n
  \ 3.1-8 DigraphByAdjacencyMatrix\ \n
  3.1-9 DigraphByEdges\n\
  3.1-10 EdgeOrbitsDigraph\n
  3.1-11 DigraphByInNeighbours\n
  3.1-12 CayleyDigraph\n
\n<\ div class=\"ContSect\"> 3.2 Ch\ anging representations\n\n
\n
  3.2-1 AsBinaryRelation\n
  3.2-2 AsDigraph\n  3.2-3 Graph\n
  3.2-4 As\ Graph\n
&nb\ sp;3.2-5 AsTransformation\n
\n
3.3 \ New digraphs from old\n\n
\n
&nb\ sp;3.3-1 DigraphImmutableCo\ py\n
  \ 3.3-2 DigraphImmutableCopyI\ fImmutable\n
&nbs\ p; 3.3-3 InducedSubdig\ raph\n
&nbs\ p;3.3-4 ReducedDigraph<\ /span>\n
  \ 3.3-5 MaximalSymmetricSubdigraph\n
  3.3-6 MaximalAntiSymmetricSubdi\ graph\n
&nb\ sp;3.3-7 UndirectedSpanning\ Forest\n
&n\ bsp;3.3-8 QuotientDigraph\n
  3.3-9 DigraphReverse\ \n
  3.3-10 DigraphDual\n
  3.3-11 DigraphSymmetricClosure\n
  3.3-12 DigraphTransitiveClosure\n
  3.3-13 DigraphTransitiveReduction\n
  3.3-14 DigraphAddVertex\n
  3.3-15 DigraphAddVertices\n
  3.3-16 DigraphAddEdge\n
  3.3-17 DigraphAddEdgeOrbit\n\
  3.3-18 DigraphAddEdges\n
<\ span class=\"nocss\">  3.3-19 DigraphRemoveVertex\n
  3.3-20 DigraphRemoveVertices\n
  3.3-21 DigraphRemoveEdge\n
  3.\ 3-22 DigraphRemoveEdgeOrbit\n
  3.\ 3-23 DigraphRemoveEdges\n
  3.3-2\ 4 DigraphRemoveLoops\n
  3.3-25 Di\ graphRemoveAllMultipleEdges\n
  3.\ 3-26 DigraphReverseEdges\n
  3.3-2\ 7 DigraphDisjointUnion\n
  3.3-28 \ DigraphEdgeUnion\n
  3.3-29 Digra\ phJoin\n
&n\ bsp;3.3-30 DigraphCartesian\ Product\n
&\ nbsp;3.3-31 DigraphDirectPr\ oduct\n
&nb\ sp;3.3-32 DigraphCartesianP\ roductProjections\n
  3.3-33 Digra\ phDirectProductProjections\n
  3.3\ -34 LineDigraph\n
  3.3-35 LineUnd\ irectedDigraph\n
\   3.3-36 DoubleDi\ graph\n
&nb\ sp;3.3-37 BipartiteDoubleDi\ graph\n
&nb\ sp;3.3-38 DigraphAddAllLoop\ s\n
  <\ /span>3.3-39 DistanceDigraph\n
  <\ a href=\"chap3_mj.html#X86F9CCEA839ABC48\">3.3-40 DigraphClosure\n<\ span class=\"ContSS\">
  3.3-41 DigraphMycielskian\n
\n
3.4 Random digraphs\n\n
\n
  3.4-1 RandomDigraph\n
  3.4-2 RandomMultiDigraph\n
  3.4-3 RandomTournament\n<\ br />  3.4-4 RandomLattice\n
\n
3.5 Standard examples\n\n
\n
  3.5-1 ChainDigraph\n
  3.5-2\ CompleteDigraph\n
  3.5-3 Comple\ teBipartiteDigraph\n
  3.5-4 Compl\ eteMultipartiteDigraph\n
  3.5-5 C\ ycleDigraph\n
&nb\ sp; 3.5-6 EmptyDigraph\ \n
  3.5-7 JohnsonDigraph\n
  3.5-8 PetersenGraph\n
  3.5-9 GeneralisedPetersenGraph\n
\n
\n", 4 := "
4 Operators\n
4.1 Operators for digraphs\n\n
\n
  4.1-1 IsSubdigraph\n
  4.1-2 Is\ UndirectedSpanningTree\n
\n
\n", 5 := "
5 Attributes and operations\n
5.1 Vertices and\ edges\n\n
\n
  5.1-1 DigraphVertices\n
  5.1-2 DigraphNrVertices\n
  5.1-3 DigraphEdges\n
  5.1-4 Di\ graphNrEdges\n
&n\ bsp; 5.1-5 DigraphNrLo\ ops\n
  \ ;5.1-6 DigraphSinks\n
  5.1-7 DigraphSources\n
  5.1-8 DigraphTopologicalSort\n
  5.1-9 DigraphVertexLabel\n
  5.1-10 DigraphVertexLabels\n
  5.1-11 DigraphEdgeLabel\n
  5.1-12 DigraphEdgeLabels\n\
  5.1-13 DigraphInEdges\n
  5.1-14 DigraphOutEdges\n
  5.\ 1-15 IsDigraphEdge\n
  5.1-16 IsMa\ tching\n
&n\ bsp;5.1-17 DigraphMaximalMa\ tching\n
&n\ bsp;5.1-18 DigraphMaximumMa\ tching\n
\n
5.2 Neighbours and degree\n\n\n
&\ nbsp; 5.2-1 AdjacencyM\ atrix\n
&nb\ sp;5.2-2 CharacteristicPoly\ nomial\n
&n\ bsp;5.2-3 BooleanAdjacencyM\ atrix\n
&nb\ sp;5.2-4 DigraphAdjacencyFu\ nction\n
&n\ bsp;5.2-5 DigraphRange<\ /span>\n
  \ 5.2-6 OutNeighbours\n
  5.2-7 InNeighbours\n
  5.2-8 OutDegrees\n  5.2-9 InDegrees\n
  5.2\ -10 OutDegreeOfVertex\n
  5.2-11 O\ utNeighboursOfVertex\n
  5.2-12 In\ DegreeOfVertex\n
\   5.2-13 InNeighb\ oursOfVertex\n
&n\ bsp; 5.2-14 DigraphLoo\ ps\n
  \ 5.2-15 PartialOrderDigraphM\ eetOfVertices\n
&\ nbsp; 5.2-16 DegreeMat\ rix\n
  \ ;5.2-17 LaplacianMatrix\ \n
\n
5.3 Reachability and connectivity\n\n
\n
&nb\ sp; 5.3-1 DigraphDiame\ ter\n
  \ ;5.3-2 DigraphShortestDista\ nce\n
  \ ;5.3-3 DigraphShortestDista\ nces\n
&nbs\ p;5.3-4 DigraphLongestDista\ nceFromVertex\n
&\ nbsp; 5.3-5 DigraphDis\ tanceSet\n
\ 5.3-6 DigraphGirth\n
  5.3-7 DigraphOddGirth\ \n
  5.3-8 DigraphUndirectedGirth\ \n
  5.3-9 DigraphConnectedComponents\n
  5.3-10 DigraphConnectedComponent<\ /span>\n
  \ 5.3-11 DigraphStronglyConnectedCom\ ponents\n
&\ nbsp;5.3-12 DigraphStrongly\ ConnectedComponent\n
  5.3-13 Digr\ aphBicomponents\n
  5.3-14 Articul\ ationPoints\n
&nb\ sp; 5.3-15 Bridges\ \n
  5.3-16 StrongOrientation\n
  5.3-17 DigraphPeriod\n
  5.3-18 DigraphFloydWarshall\n
  5.3-19 IsReachable\n
  5.3-20 VerticesReachableFrom\n
  5.3-21 DigraphPath\n
<\ span class=\"nocss\">  5.3-22 DigraphShortestPath\n
  5.3-23 IteratorOfPaths\n
  5.3-\ 24 DigraphAllSimpleCircuits\n
  5.\ 3-25 DigraphLongestSimpleCircuit\n
  5.3-26 DigraphLayers\n
  5.3-2\ 7 DigraphDegeneracy\n
  5.3-28 Dig\ raphDegeneracyOrdering\n
  5.3-29 \ HamiltonianPath\n
  5.3-30 NrSpann\ ingTrees\n
\ 5.3-31 DigraphDijkstr\ a\n
\n
5\ .4 Cayley graphs of groups\n\n
\n
&nbs\ p; 5.4-1 GroupOfCayley\ Digraph\n
&\ nbsp;5.4-2 GeneratorsOfCayl\ eyDigraph\n
\n
5.5 Associated semigroups\n\n\
\n
  5.5-1 AsSemi\ group\n
&nb\ sp;5.5-2 AsSemigroup\n
\n
5.6 Planarity\n\n
\ \n
  5.6-1 KuratowskiPlanarSubdigraph\n
  5.6-2 KuratowskiOuterPlanarSubdigraph\ \n
  5.6-3 PlanarEmbedding\n
  5.6-4 OuterPlanarEmbedding\ \n
  5.6-5 SubdigraphHomeomorphicToK23\n
\n
\n", 6 := "
6 Properties of digraphs\n
6.1 Edge propertie\ s\n\n
\n  6.1-1 DigraphHasLoops\n
  6.1-2 IsAntiSymmetricDigraph\n
  6.1-3 IsBipartiteDigraph\n
  6.\ 1-4 IsCompleteBipartiteDigraph\n
  6.1-5 IsCompleteDigraph\n
  6.1-6\ IsCompleteMultipartiteDigraph\n
  6.1-7 IsEmptyDigraph\n
  6.1-8 Is\ EquivalenceDigraph\n
  6.1-9 IsFun\ ctionalDigraph\n
\   6.1-10 IsMultiD\ igraph\n
&n\ bsp;6.1-11 IsReflexiveDigra\ ph\n
  \ 6.1-12 IsSymmetricDigraph\n
  6.1-13 IsTournament\ \n
  6.1-14 IsTransitiveDigraph\n<\ span class=\"ContSS\">
  6.1-15 IsPreorderDigraph\n
  6.1-16 IsPartialOrderDigraph\n
  6.1-17 IsMeetSemilatticeDigraph\n
\n
6.2 Regularity\n\n
\n
  6.2-1 IsInRegularDigraph\n
  6.2-2 IsOutRegularDigraph\n
  6.2-3 IsRegularDigraph\n\
  6.2-4 IsDistanceRegularDigraph\n
\n
6.3 Connectiv\ ity and cycles\n\n
\n
  6.3-1 IsAcyclicDigraph\n
  6.3-2 IsChainDigraph\n
  6.3-3 IsConnectedDigraph\n
  6.3-4 IsBiconnectedDigraph\n
  \ 6.3-5 IsBridgelessDigraph\n
  6.3-\ 6 IsStronglyConnectedDigraph\n
  6\ .3-7 IsAperiodicDigraph\n
  6.3-8\ IsDirectedTree\n
  6.3-9 IsUndire\ ctedTree\n
\ 6.3-10 IsEulerianDigr\ aph\n
  \ ;6.3-11 IsHamiltonianDigrap\ h\n
  <\ /span>6.3-12 IsCycleDigraph\n
\n
6.4 Planarity\n\n
\ \n
  6.4-1 IsPlanarDigraph\n
  6.4-2 IsOuterPlanarDigraph\n
\n
&nb\ sp;6.5 Homomorphisms and transformations\n\n
\n
  6.5-1 IsDigraphCore\n\
  6.5-2 IsEdgeTransitive\n
  6.5-3 IsVertexTransitive\n
\ \n
\n", 7 := "
7 Homomorphisms\n
7.1 Acting on digraphs\n\n
\n
  7.1-1 OnDigraphs\n
  7.1-2 OnMu\ ltiDigraphs\n
\n
7.2 Isomorphisms and canonical labellings\n\n
\n
  7.2-1 DigraphsUseNauty\n
  7.\ 2-2 AutomorphismGroup\n
  7.2-3 Bl\ issAutomorphismGroup\n
  7.2-4 Nau\ tyAutomorphismGroup\n
  7.2-5 Auto\ morphismGroup\n
&\ nbsp; 7.2-6 Automorphi\ smGroup\n
&\ nbsp;7.2-7 BlissCanonicalLa\ belling\n
&\ nbsp;7.2-8 BlissCanonicalLa\ belling\n
&\ nbsp;7.2-9 BlissCanonicalDi\ graph\n
&nb\ sp;7.2-10 DigraphGroup<\ /span>\n
  \ 7.2-11 DigraphOrbits\n<\ span class=\"ContSS\">
  7.2-12 DigraphOrbitReps\n
  7.2-13 DigraphSchreierVector\n
  7.2-14 DigraphStabilizer\n
  7.2-15 IsIsomorphicDigraph\n
  7.2-16 IsIsomorphicDigraph\n
  7.2-17 IsomorphismDigraphs\n
  7.2-18 IsomorphismDigraphs\n\
  7.2-19 RepresentativeOutNeighbours\n
  7.2-20 IsDigraphIsomorphism\n
  7.2-21 IsDigraphColouring\n
\n
7.3 Homomorphis\ ms of digraphs\n\n
\n
  7.3-1 HomomorphismDigraphsFinder\n
  7.3-2 DigraphHomomorphism\n
  7.3-3 HomomorphismsDigraphs\n
  7.3-4 DigraphMonomorphism\n
  7.3-5 MonomorphismsDigraphs\n
  7.3-6 DigraphEpimorphism\n  7.3-7 EpimorphismsDigraphs\n
  7.3-8 DigraphEmbedding\n
  7.3-9 EmbeddingsDigraphs\n
  7.3-\ 10 IsDigraphHomomorphism\n
  7.3-1\ 1 IsDigraphEmbedding\n
  7.3-12 Di\ graphsRespectsColouring\n
  7.3-1\ 3 GeneratorsOfEndomorphismMonoid\n
  7.3-14 DigraphColouring\n
  7.3\ -15 DigraphGreedyColouring\n
  7.3\ -16 DigraphWelshPowellOrder\n
  7.\ 3-17 ChromaticNumber\n
  7.3-18 Di\ graphCore\n
\n
\n", 8 := "
8 Finding cliques and independent sets\n\
8.1 F\ inding cliques\n\n
\n
  8.1-1 IsClique\n
  8.1-2 CliquesFinder\n
  8\ .1-3 DigraphClique\n
  8.1-4 Digra\ phMaximalCliques\n
  8.1-5 Clique\ Number\n
\n
8.2 Finding independent sets\n\n\
\n
  8.2-1 IsInde\ pendentSet\n
&nbs\ p; 8.2-2 DigraphIndepe\ ndentSet\n
\ 8.2-3 DigraphMaximalI\ ndependentSets\n
\n
\n", 9 := "
9 Visualising and IO\n
9.1 Visualising a digra\ ph\n\n
\n  9.1-1 Splash\n
  9.1-2\ DotDigraph\n
&nb\ sp; 9.1-3 DotSymmetric\ Digraph\n
&\ nbsp;9.1-4 DotPartialOrderD\ igraph\n
&n\ bsp;9.1-5 DotPreorderDigrap\ h\n
  <\ /span>9.1-6 DotHighlightedDigraph<\ /a>\n
\n
9.2\ Reading and writing digraphs to a file\n\n
\n
  9.2-1\ String\n
&\ nbsp;9.2-2 DigraphFromGraph\ 6String\n
&\ nbsp;9.2-3 Graph6String\ \n
  9.2-4 DigraphFile\n
  9.2-5 ReadDigraphs\n
  9.2-6 WriteDigraphs\n<\ br />  9.2-7 IteratorFromDigraphFile\n<\ br />  9.2-8 DigraphPlainTextLineEncoder\n
  9.2-9 TournamentLineDecoder\n
  9.2-10 AdjacencyMatrixUpperTriangleLineDecoder\n
  9.2-11 TCodeDecoder\n
  9.2-12 PlainTextString\n
  9.2-13 WritePlainTextDigraph\n\
  9.2-14 WriteDIMACSDigraph\n
\n
\n", A := "
A \n Grape to Digraphs Command Map\n \n
\ ;A.1 \n Functions to construct and modify graphs\n \n\ \n
\n
A.2 \n Functions to inspect graphs, vertices and edges\n \ \n\n
\n
A.3 \n Functions to determine regularity properties of \ graphs\n \n\n
\n
A.4 \n Some special vertex subsets o\ f a graph\n \n\n
\n
A.5 \n Functions to construct new\ graphs from old\n \n\n
\n
A.6 \n Vertex-Colouring an\ d Complete Subgraphs\n \n\n
\n
A.7 \n Automorphism g\ roups and isomorphism testing for graphs\n \n\n
\n\n" ), chapsectlinkstext := rec( 1 := "
1 \n The Digraphs\ package\n \n
1.1 \ Introduction\n\n
\n
  1.1-1 Def\ initions\n\n
\n
\n", 2 := "
2 Installing Digraphs\n
2.1 For those in a hurry\n\n
\n
2.2 Optional package\ dependencies\n\n
\n
  2.2-1 The Grape package\n\n
\n
2.\ 3 Compiling the kernel module\n\n\n
&nb\ sp;2.4 Rebuilding the documentation\n\n
\n
  2.4-1 DigraphsMakeDoc\n
\n
2.5 Testing your installation\n\n
\n
  \ 2.5-1 DigraphsTestInstall\n
  2.5-2 DigraphsTestStandard\ \n
  2.5-3 DigraphsTestExtreme\n
\n
\n", 3 := "
3 Creating digraphs\n
3.1 Creating digraphs\n\n
\n
<\ span class=\"nocss\">  3.1-1 IsDigraph\n
  3.1-2 Is\ MutableDigraph\n
\   3.1-3 IsImmutab\ leDigraph\n
\ ; 3.1-4 IsCayleyDigrap\ h\n
  <\ /span>3.1-5 IsDigraphWithAdjacency\ Function\n
\ 3.1-6 DigraphByOutNei\ ghboursType\n
&nb\ sp; 3.1-7 Digraph<\ /span>\n
  \ 3.1-8 DigraphByAdjacencyMatrix\ \n
  3.1-9 DigraphByEdges\n\
  3.1-10 EdgeOrbitsDigraph\n
  3.1-11 DigraphByInNeighbours\n
  3.1-12 CayleyDigraph\n
\n<\ div class=\"ContSect\"> 3.2 Ch\ anging representations\n\n
\n
  3.2-1 AsBinaryRelation\n
  3.2-2 AsDigraph\n  3.2-3 Graph\n
  3.2-4 As\ Graph\n
&nb\ sp;3.2-5 AsTransformation\n
\n
3.3 \ New digraphs from old\n\n
\n
&nb\ sp;3.3-1 DigraphImmutableCo\ py\n
  \ 3.3-2 DigraphImmutableCopyI\ fImmutable\n
&nbs\ p; 3.3-3 InducedSubdig\ raph\n
&nbs\ p;3.3-4 ReducedDigraph<\ /span>\n
  \ 3.3-5 MaximalSymmetricSubdigraph\n
  3.3-6 MaximalAntiSymmetricSubdi\ graph\n
&nb\ sp;3.3-7 UndirectedSpanning\ Forest\n
&n\ bsp;3.3-8 QuotientDigraph\n
  3.3-9 DigraphReverse\ \n
  3.3-10 DigraphDual\n
  3.3-11 DigraphSymmetricClosure\n
  3.3-12 DigraphTransitiveClosure\n
  3.3-13 DigraphTransitiveReduction\n
  3.3-14 DigraphAddVertex\n
  3.3-15 DigraphAddVertices\n
  3.3-16 DigraphAddEdge\n
  3.3-17 DigraphAddEdgeOrbit\n\
  3.3-18 DigraphAddEdges\n
<\ span class=\"nocss\">  3.3-19 DigraphRemoveVertex\n
  3.3-20 DigraphRemoveVertices\n
  3.3-21 DigraphRemoveEdge\n
  3.\ 3-22 DigraphRemoveEdgeOrbit\n
  3.\ 3-23 DigraphRemoveEdges\n
  3.3-2\ 4 DigraphRemoveLoops\n
  3.3-25 Di\ graphRemoveAllMultipleEdges\n
  3.\ 3-26 DigraphReverseEdges\n
  3.3-2\ 7 DigraphDisjointUnion\n
  3.3-28 \ DigraphEdgeUnion\n
  3.3-29 Digra\ phJoin\n
&n\ bsp;3.3-30 DigraphCartesian\ Product\n
&\ nbsp;3.3-31 DigraphDirectPr\ oduct\n
&nb\ sp;3.3-32 DigraphCartesianP\ roductProjections\n
  3.3-33 Digra\ phDirectProductProjections\n
  3.3\ -34 LineDigraph\n
  3.3-35 LineUnd\ irectedDigraph\n
\   3.3-36 DoubleDi\ graph\n
&nb\ sp;3.3-37 BipartiteDoubleDi\ graph\n
&nb\ sp;3.3-38 DigraphAddAllLoop\ s\n
  <\ /span>3.3-39 DistanceDigraph\n
  <\ a href=\"chap3_mj.html#X86F9CCEA839ABC48\">3.3-40 DigraphClosure\n<\ span class=\"ContSS\">
  3.3-41 DigraphMycielskian\n
\n
3.4 Random digraphs\n\n
\n
  3.4-1 RandomDigraph\n
  3.4-2 RandomMultiDigraph\n
  3.4-3 RandomTournament\n<\ br />  3.4-4 RandomLattice\n
\n
3.5 Standard examples\n\n
\n
  3.5-1 ChainDigraph\n
  3.5-2\ CompleteDigraph\n
  3.5-3 Comple\ teBipartiteDigraph\n
  3.5-4 Compl\ eteMultipartiteDigraph\n
  3.5-5 C\ ycleDigraph\n
&nb\ sp; 3.5-6 EmptyDigraph\ \n
  3.5-7 JohnsonDigraph\n
  3.5-8 PetersenGraph\n
  3.5-9 GeneralisedPetersenGraph\n
\n
\n", 4 := "
4 Operators\n
4.1 Operators for digraphs\n\n
\n
  4.1-1 IsSubdigraph\n
  4.1-2 Is\ UndirectedSpanningTree\n
\n
\n", 5 := "
5 Attributes and operations\n
5.1 Vertices and\ edges\n\n
\n
  5.1-1 DigraphVertices\n
  5.1-2 DigraphNrVertices\n
  5.1-3 DigraphEdges\n
  5.1-4 Di\ graphNrEdges\n
&n\ bsp; 5.1-5 DigraphNrLo\ ops\n
  \ ;5.1-6 DigraphSinks\n
  5.1-7 DigraphSources\n
  5.1-8 DigraphTopologicalSort\n
  5.1-9 DigraphVertexLabel\n
  5.1-10 DigraphVertexLabels\n
  5.1-11 DigraphEdgeLabel\n
  5.1-12 DigraphEdgeLabels\n\
  5.1-13 DigraphInEdges\n
  5.1-14 DigraphOutEdges\n
  5.\ 1-15 IsDigraphEdge\n
  5.1-16 IsMa\ tching\n
&n\ bsp;5.1-17 DigraphMaximalMa\ tching\n
&n\ bsp;5.1-18 DigraphMaximumMa\ tching\n
\n
5.2 Neighbours and degree\n\n\n
&\ nbsp; 5.2-1 AdjacencyM\ atrix\n
&nb\ sp;5.2-2 CharacteristicPoly\ nomial\n
&n\ bsp;5.2-3 BooleanAdjacencyM\ atrix\n
&nb\ sp;5.2-4 DigraphAdjacencyFu\ nction\n
&n\ bsp;5.2-5 DigraphRange<\ /span>\n
  \ 5.2-6 OutNeighbours\n
  5.2-7 InNeighbours\n
  5.2-8 OutDegrees\n  5.2-9 InDegrees\n
  5.2\ -10 OutDegreeOfVertex\n
  5.2-11 O\ utNeighboursOfVertex\n
  5.2-12 In\ DegreeOfVertex\n
\   5.2-13 InNeighb\ oursOfVertex\n
&n\ bsp; 5.2-14 DigraphLoo\ ps\n
  \ 5.2-15 PartialOrderDigraphM\ eetOfVertices\n
&\ nbsp; 5.2-16 DegreeMat\ rix\n
  \ ;5.2-17 LaplacianMatrix\ \n
\n
5.3 Reachability and connectivity\n\n
\n
&nb\ sp; 5.3-1 DigraphDiame\ ter\n
  \ ;5.3-2 DigraphShortestDista\ nce\n
  \ ;5.3-3 DigraphShortestDista\ nces\n
&nbs\ p;5.3-4 DigraphLongestDista\ nceFromVertex\n
&\ nbsp; 5.3-5 DigraphDis\ tanceSet\n
\ 5.3-6 DigraphGirth\n
  5.3-7 DigraphOddGirth\ \n
  5.3-8 DigraphUndirectedGirth\ \n
  5.3-9 DigraphConnectedComponents\n
  5.3-10 DigraphConnectedComponent<\ /span>\n
  \ 5.3-11 DigraphStronglyConnectedCom\ ponents\n
&\ nbsp;5.3-12 DigraphStrongly\ ConnectedComponent\n
  5.3-13 Digr\ aphBicomponents\n
  5.3-14 Articul\ ationPoints\n
&nb\ sp; 5.3-15 Bridges\ \n
  5.3-16 StrongOrientation\n
  5.3-17 DigraphPeriod\n
  5.3-18 DigraphFloydWarshall\n
  5.3-19 IsReachable\n
  5.3-20 VerticesReachableFrom\n
  5.3-21 DigraphPath\n
<\ span class=\"nocss\">  5.3-22 DigraphShortestPath\n
  5.3-23 IteratorOfPaths\n
  5.3-\ 24 DigraphAllSimpleCircuits\n
  5.\ 3-25 DigraphLongestSimpleCircuit\n
  5.3-26 DigraphLayers\n
  5.3-2\ 7 DigraphDegeneracy\n
  5.3-28 Dig\ raphDegeneracyOrdering\n
  5.3-29 \ HamiltonianPath\n
  5.3-30 NrSpann\ ingTrees\n
\ 5.3-31 DigraphDijkstr\ a\n
\n
5\ .4 Cayley graphs of groups\n\n
\n
&nbs\ p; 5.4-1 GroupOfCayley\ Digraph\n
&\ nbsp;5.4-2 GeneratorsOfCayl\ eyDigraph\n
\n
5.5 Associated semigroups\n\n\
\n
  5.5-1 AsSemi\ group\n
&nb\ sp;5.5-2 AsSemigroup\n
\n
5.6 Planarity\n\n
\ \n
  5.6-1 KuratowskiPlanarSubdigraph\n
  5.6-2 KuratowskiOuterPlanarSubdigraph\ \n
  5.6-3 PlanarEmbedding\n
  5.6-4 OuterPlanarEmbedding\ \n
  5.6-5 SubdigraphHomeomorphicToK23\n
\n
\n", 6 := "
6 Properties of digraphs\n
6.1 Edge propertie\ s\n\n
\n  6.1-1 DigraphHasLoops\n
  6.1-2 IsAntiSymmetricDigraph\n
  6.1-3 IsBipartiteDigraph\n
  6.\ 1-4 IsCompleteBipartiteDigraph\n
  6.1-5 IsCompleteDigraph\n
  6.1-6\ IsCompleteMultipartiteDigraph\n
  6.1-7 IsEmptyDigraph\n
  6.1-8 Is\ EquivalenceDigraph\n
  6.1-9 IsFun\ ctionalDigraph\n
\   6.1-10 IsMultiD\ igraph\n
&n\ bsp;6.1-11 IsReflexiveDigra\ ph\n
  \ 6.1-12 IsSymmetricDigraph\n
  6.1-13 IsTournament\ \n
  6.1-14 IsTransitiveDigraph\n<\ span class=\"ContSS\">
  6.1-15 IsPreorderDigraph\n
  6.1-16 IsPartialOrderDigraph\n
  6.1-17 IsMeetSemilatticeDigraph\n
\n
6.2 Regularity\n\n
\n
  6.2-1 IsInRegularDigraph\n
  6.2-2 IsOutRegularDigraph\n
  6.2-3 IsRegularDigraph\n\
  6.2-4 IsDistanceRegularDigraph\n
\n
6.3 Connectiv\ ity and cycles\n\n
\n
  6.3-1 IsAcyclicDigraph\n
  6.3-2 IsChainDigraph\n
  6.3-3 IsConnectedDigraph\n
  6.3-4 IsBiconnectedDigraph\n
  \ 6.3-5 IsBridgelessDigraph\n
  6.3-\ 6 IsStronglyConnectedDigraph\n
  6\ .3-7 IsAperiodicDigraph\n
  6.3-8\ IsDirectedTree\n
  6.3-9 IsUndire\ ctedTree\n
\ 6.3-10 IsEulerianDigr\ aph\n
  \ ;6.3-11 IsHamiltonianDigrap\ h\n
  <\ /span>6.3-12 IsCycleDigraph\n
\n
6.4 Planarity\n\n
\ \n
  6.4-1 IsPlanarDigraph\n
  6.4-2 IsOuterPlanarDigraph\n
\n
&nb\ sp;6.5 Homomorphisms and transformations\n\n
\n
  6.5-1 IsDigraphCore\n\
  6.5-2 IsEdgeTransitive\n
  6.5-3 IsVertexTransitive\n
\ \n
\n", 7 := "
7 Homomorphisms\n
7.1 Acting on digraphs\n\n
\n
  7.1-1 OnDigraphs\n
  7.1-2 OnMu\ ltiDigraphs\n
\n
7.2 Isomorphisms and canonical labellings\n\n
\n
  7.2-1 DigraphsUseNauty\n
  7.\ 2-2 AutomorphismGroup\n
  7.2-3 Bl\ issAutomorphismGroup\n
  7.2-4 Nau\ tyAutomorphismGroup\n
  7.2-5 Auto\ morphismGroup\n
&\ nbsp; 7.2-6 Automorphi\ smGroup\n
&\ nbsp;7.2-7 BlissCanonicalLa\ belling\n
&\ nbsp;7.2-8 BlissCanonicalLa\ belling\n
&\ nbsp;7.2-9 BlissCanonicalDi\ graph\n
&nb\ sp;7.2-10 DigraphGroup<\ /span>\n
  \ 7.2-11 DigraphOrbits\n<\ span class=\"ContSS\">
  7.2-12 DigraphOrbitReps\n
  7.2-13 DigraphSchreierVector\n
  7.2-14 DigraphStabilizer\n
  7.2-15 IsIsomorphicDigraph\n
  7.2-16 IsIsomorphicDigraph\n
  7.2-17 IsomorphismDigraphs\n
  7.2-18 IsomorphismDigraphs\n\
  7.2-19 RepresentativeOutNeighbours\n
  7.2-20 IsDigraphIsomorphism\n
  7.2-21 IsDigraphColouring\n
\n
7.3 Homomorphis\ ms of digraphs\n\n
\n
  7.3-1 HomomorphismDigraphsFinder\n
  7.3-2 DigraphHomomorphism\n
  7.3-3 HomomorphismsDigraphs\n
  7.3-4 DigraphMonomorphism\n
  7.3-5 MonomorphismsDigraphs\n
  7.3-6 DigraphEpimorphism\n  7.3-7 EpimorphismsDigraphs\n
  7.3-8 DigraphEmbedding\n
  7.3-9 EmbeddingsDigraphs\n
  7.3-\ 10 IsDigraphHomomorphism\n
  7.3-1\ 1 IsDigraphEmbedding\n
  7.3-12 Di\ graphsRespectsColouring\n
  7.3-1\ 3 GeneratorsOfEndomorphismMonoid\n
  7.3-14 DigraphColouring\n
  7.3\ -15 DigraphGreedyColouring\n
  7.3\ -16 DigraphWelshPowellOrder\n
  7.\ 3-17 ChromaticNumber\n
  7.3-18 Di\ graphCore\n
\n
\n", 8 := "
8 Finding cliques and independent sets\n\
8.1 F\ inding cliques\n\n
\n
  8.1-1 IsClique\n
  8.1-2 CliquesFinder\n
  8\ .1-3 DigraphClique\n
  8.1-4 Digra\ phMaximalCliques\n
  8.1-5 Clique\ Number\n
\n
8.2 Finding independent sets\n\n\
\n
  8.2-1 IsInde\ pendentSet\n
&nbs\ p; 8.2-2 DigraphIndepe\ ndentSet\n
\ 8.2-3 DigraphMaximalI\ ndependentSets\n
\n
\n", 9 := "
9 Visualising and IO\n
9.1 Visualising a digra\ ph\n\n
\n  9.1-1 Splash\n
  9.1-2\ DotDigraph\n
&nb\ sp; 9.1-3 DotSymmetric\ Digraph\n
&\ nbsp;9.1-4 DotPartialOrderD\ igraph\n
&n\ bsp;9.1-5 DotPreorderDigrap\ h\n
  <\ /span>9.1-6 DotHighlightedDigraph<\ /a>\n
\n
9.2\ Reading and writing digraphs to a file\n\n
\n
  9.2-1\ String\n
&\ nbsp;9.2-2 DigraphFromGraph\ 6String\n
&\ nbsp;9.2-3 Graph6String\ \n
  9.2-4 DigraphFile\n
  9.2-5 ReadDigraphs\n
  9.2-6 WriteDigraphs\n<\ br />  9.2-7 IteratorFromDigraphFile\n<\ br />  9.2-8 DigraphPlainTextLineEncoder\n
  9.2-9 TournamentLineDecoder\n
  9.2-10 AdjacencyMatrixUpperTriangleLineDecoder\n
  9.2-11 TCodeDecoder\n
  9.2-12 PlainTextString\n
  9.2-13 WritePlainTextDigraph\n\
  9.2-14 WriteDIMACSDigraph\n
\n
\n", A := "
A \n Grape to Digraphs Command Map\n \n
\ ;A.1 \n Functions to construct and modify graphs\n \n\ \n
\n
A.2 \n Functions to inspect graphs, vertices and edges\n \ \n\n
\n
A.3 \n Functions to determine regularity properties of \ graphs\n \n\n
\n
A.4 \n Some special vertex subsets o\ f a graph\n \n\n
\n
A.5 \n Functions to construct new\ graphs from old\n \n\n
\n
A.6 \n Vertex-Colouring an\ d Complete Subgraphs\n \n\n
\n
A.7 \n Automorphism g\ roups and isomorphism testing for graphs\n \n\n
\n\n" ), chaptitle := rec( 1 := "\n The Digraphs package\n ", 2 := "Installing Digraphs<\ /strong>", 3 := "Creating digraphs", 4 := "Operators", 5 := "Attributes and operations", 6 := "Properties of digraphs", 7 := "Homomorphisms", 8 := "Finding cliques and independent sets", 9 := "Visualising and IO", A := "\n Grape to Digraphs Command Map\n " ), content := [ rec( content := "Book SYSTEM \"gapdoc.dtd\"\n [ Digraphs\">\n http://www.maths.qmul.ac.uk/~leonard/grape/Grape\">\n http://www.tcs.tkk.fi/Software/bliss/bliss\">\n https://github.com/graph-algorithms/edge-addition-planarity-suiteedge-addition-planarity-suite\">\n ht\ tp://pallini.di.uniroma1.it/nauty\">\n https://github.com/gap-packages/NautyTracesInter\ faceNautyTracesInterface\">\n \n http://gap-p\ ackages.github.io/io/\nIO\">\n \n http://http:/\ /gap-packages.github.io/orb/\n Orb\">\n https://github.com/gap-packages/datastructuresdatastructures\">\n \n\n\n\n\n\n\n\n\n\n ]", count := [ 0, 0, 0, 1 ], name := "XMLDOCTYPE", root := ~ ), rec( attributes := rec( Name := "digraphs" ), content := [ rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ) ], count := [ 0, 0, 0, 1 ], name := "Title", next := 2024, root := ~, start := 1999, stop := 2023 ), rec( attributes := rec( ), content := [ rec( content := "Version 1.4.0", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Version", next := 2063, root := ~, start := 2027, stop := 2062 ), rec( attributes := rec( ), content := [ rec( content := "Jan De Beule\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "jdebeule@cage.ugent.be", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2129, root := ~, start := 2092, stop := 2128 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "http://homepages.vub.ac.b\ e/~jdbeule", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2190, root := ~, start := 2134, stop := 2189 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2202, root := ~, start := 2067, stop := 2201 ), rec( attributes := rec( ), content := [ rec( content := "Julius Jonušas\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "julius.jonusas@tuwien.ac.\ at", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2280, root := ~, start := 2238, stop := 2279 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "http://julius.jonusas.wor\ k", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2332, root := ~, start := 2285, stop := 2331 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2344, root := ~, start := 2206, stop := 2343 ), rec( attributes := rec( ), content := [ rec( content := "James D. Mitchell\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "jdm3@st-and.ac.uk", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2410, root := ~, start := 2378, stop := 2409 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "https://jdbm.me", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2451, root := ~, start := 2415, stop := 2450 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2463, root := ~, start := 2348, stop := 2462 ), rec( attributes := rec( ), content := [ rec( content := "Michael Torpey\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "mct25@st-andrews.ac.uk", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2531, root := ~, start := 2494, stop := 2530 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "https://mtorpey.github.io\ ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2582, root := ~, start := 2536, stop := 2581 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2594, root := ~, start := 2467, stop := 2593 ), rec( attributes := rec( ), content := [ rec( content := "Wilf A. Wilson\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "gap@wilf-wilson.net", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Email", next := 2659, root := ~, start := 2625, stop := 2658 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "http://wilf.me", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "Homepage", next := 2699, root := ~, start := 2664, stop := 2698 ), rec( content := "\n ", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2711, root := ~, start := 2598, stop := 2710 ), rec( attributes := rec( ), content := [ rec( content := "Stuart Burrell", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2746, root := ~, start := 2715, stop := 2745 ), rec( attributes := rec( ), content := [ rec( content := "Reinis Cirpons", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2781, root := ~, start := 2750, stop := 2780 ), rec( attributes := rec( ), content := [ rec( content := "Luke Elliott", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2814, root := ~, start := 2785, stop := 2813 ), rec( attributes := rec( ), content := [ rec( content := "Max Horn", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2843, root := ~, start := 2818, stop := 2842 ), rec( attributes := rec( ), content := [ rec( content := "Christopher Jefferson", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2885, root := ~, start := 2847, stop := 2884 ), rec( attributes := rec( ), content := [ rec( content := "Markus Pfeiffer", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2921, root := ~, start := 2889, stop := 2920 ), rec( attributes := rec( ), content := [ rec( content := "Chris Russell", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2955, root := ~, start := 2925, stop := 2954 ), rec( attributes := rec( ), content := [ rec( content := "Finn Smith", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 2986, root := ~, start := 2959, stop := 2985 ), rec( attributes := rec( ), content := [ rec( content := "Murray White", count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 0, 1 ], name := "Author", next := 3019, root := ~, start := 2990, stop := 3018 ), rec( attributes := rec( ), content := [ rec( content := "© 2014-20 by\n Jan De Beule, \ Julius Jonušas, James D. Mitchell, Michael\n Torpey, Wilf A. Wilson et al.\ ", count := [ 0, 0, 1, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "P", next := 3169, root := ~, start := 3165, stop := 3168 ), rec( content := "\n\n ", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is free software; you can redis\ tribute it and/or modify it\n under the terms of the ", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "GNU General Public License" ), content := [ rec( content := "\n http://www.fsf.org/\ licenses/gpl.html", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "URL", next := 3358, root := ~, start := 3272, stop := 3357 ), rec( content := " as published by the Free Softwa\ re\n Foundation; either version 3 of the License, or (at your option) any l\ ater\n version.\n ", count := [ 0, 0, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 1, 1 ], name := "Copyright", next := 3499, root := ~, start := 3023, stop := 3498 ), rec( attributes := rec( ), content := [ rec( content := "\n The ", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package is a ", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " package containing methods for \ graphs,\n digraphs, and multidigraphs.\n ", count := [ 0, 0, 2, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 2, 1 ], name := "Abstract", next := 3637, root := ~, start := 3503, stop := 3636 ), rec( attributes := rec( ), content := [ rec( content := "\n We would like to thank Chri\ stopher Jefferson for his help in including \n ", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ) , content := [ rec( content := "http://www.tcs.tk\ k.fi/Software/bliss/", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 0, 0, 3, 1 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 0, 0, 3, 1 ], name := "PCDATA", root := ~ ) ], count := [ 0, 0, 3, 1 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " in ", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ".\n\n This package's methods \ for computing digraph homomorphisms are based on work\n by Max Neunhöffer,\ and independently Artur Schäfer.\n ", count := [ 0, 0, 3, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 3, 1 ], name := "Acknowledgements", next := 3932, root := ~, start := 3641, stop := 3931 ) ], count := [ 0, 0, 0, 1 ], mathmode := "Text", name := "TitlePage", next := 3946, root := ~, start := 1985, stop := 3945 ), rec( attributes := rec( ), content := 0, count := [ 0, 0, 4, 1 ], mathmode := "Text", name := "TableOfContents", next := 3967, root := ~, start := 3949, stop := 3966 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 0, 0, 4, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "The Digraphs package" ), content := [ rec( content := "\n ", count := [ 1, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The ", count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\n ", count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 4068, root := ~, start := 4019, stop := 4067 ), rec( content := "\n ", count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "Digraphs" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package overview", count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "Index", next := 4128, root := ~, start := 4071, stop := 4127 ), rec( content := "\n\n", count := [ 1, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Introduction", count := [ 1, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 4170, root := ~, start := 4139, stop := 4169 ), rec( content := "\n\n This is the manual\ for version 1.4.0 of the ", count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package.\n This packag\ e was developed at the University of St Andrews by:\n ", count := [ 1, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Jan De Be\ ule,", count := [ 1, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 5 ], name := "Item", next := 4346, root := ~, start := 4320, stop := 4345 ), rec( attributes := rec( ), content := [ rec( content := "Julius Jo\ nušas,", count := [ 1, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 7 ], name := "Item", next := 4384, root := ~, start := 4351, stop := 4383 ), rec( attributes := rec( ), content := [ rec( content := "James D. \ Mitchell,", count := [ 1, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 9 ], name := "Item", next := 4420, root := ~, start := 4389, stop := 4419 ), rec( attributes := rec( ), content := [ rec( content := "Michael C\ . Torpey, and", count := [ 1, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 11 ], name := "Item", next := 4460, root := ~, start := 4425, stop := 4459 ), rec( attributes := rec( ), content := [ rec( content := "Wilf A. W\ ilson.", count := [ 1, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 13 ], name := "Item", next := 4493, root := ~, start := 4465, stop := 4492 ) ], count := [ 1, 1, 0, 4 ], mathmode := "Text", name := "List", next := 4503, root := ~, start := 4309, stop := 4502 ), rec( content := "\n Additional contribut\ ions were made by:\n ", count := [ 1, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Stuart Bu\ rrell,", count := [ 1, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 17 ], name := "Item", next := 4586, root := ~, start := 4558, stop := 4585 ), rec( attributes := rec( ), content := [ rec( content := "Reinis Ci\ rpons,", count := [ 1, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 19 ], name := "Item", next := 4619, root := ~, start := 4591, stop := 4618 ), rec( attributes := rec( ), content := [ rec( content := "Luke Elli\ ott,", count := [ 1, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 21 ], name := "Item", next := 4650, root := ~, start := 4624, stop := 4649 ), rec( attributes := rec( ), content := [ rec( content := "Max Horn,\ ", count := [ 1, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 23 ], name := "Item", next := 4677, root := ~, start := 4655, stop := 4676 ), rec( attributes := rec( ), content := [ rec( content := "Christoph\ er Jefferson,", count := [ 1, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 25 ], name := "Item", next := 4717, root := ~, start := 4682, stop := 4716 ), rec( attributes := rec( ), content := [ rec( content := "Markus Pf\ eiffer,", count := [ 1, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 27 ], name := "Item", next := 4751, root := ~, start := 4722, stop := 4750 ), rec( attributes := rec( ), content := [ rec( content := "Chris Rus\ sell,", count := [ 1, 1, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 29 ], name := "Item", next := 4783, root := ~, start := 4756, stop := 4782 ), rec( attributes := rec( ), content := [ rec( content := "Finn Smit\ h, and", count := [ 1, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 31 ], name := "Item", next := 4816, root := ~, start := 4788, stop := 4815 ), rec( attributes := rec( ), content := [ rec( content := "Maria Tsa\ lakou,", count := [ 1, 1, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 33 ], name := "Item", next := 4849, root := ~, start := 4821, stop := 4848 ), rec( attributes := rec( ), content := [ rec( content := "Murray Wh\ yte.", count := [ 1, 1, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 35 ], name := "Item", next := 4880, root := ~, start := 4854, stop := 4879 ) ], count := [ 1, 1, 0, 16 ], mathmode := "Text", name := "List", next := 4890, root := ~, start := 4547, stop := 4889 ), rec( content := "\n\n The ", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package contains a vari\ ety of methods for efficiently creating\n and storing mutable and immutable d\ igraphs and computing information about\n them. Full explanations of all t\ he functions contained in the package are\n provided below. ", count := [ 1, 1, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "P", next := 5152, root := ~, start := 5148, stop := 5151 ), rec( content := "\n\n If the ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://ww\ w.maths.qmul.ac.uk/~leonard/grape/", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 1, 1, 0, 38 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 38 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package is available, i\ t will be loaded automatically.\n Digraphs created with the ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package can be converte\ d to ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://ww\ w.maths.qmul.ac.uk/~leonard/grape/", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 1, 1, 0, 38 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 38 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := "\n graphs with ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Graph" ) , content := 0, count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Ref", next := 5334, root := ~, start := 5315, stop := 5333 ), rec( content := ", and conversely ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://ww\ w.maths.qmul.ac.uk/~leonard/grape/", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 1, 1, 0, 38 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 38 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " graphs can be\n conver\ ted to ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " objects with ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph" ), content := 0, count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Ref", next := 5433, root := ~, start := 5412, stop := 5432 ), rec( content := ". ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://ww\ w.maths.qmul.ac.uk/~leonard/grape/", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 1, 1, 0, 38 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ 1, 1, 0, 38 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " is not\n required for \ ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " to run. ", count := [ 1, 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "P", next := 5489, root := ~, start := 5485, stop := 5488 ), rec( content := "\n\n The ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), content := [ rec( content := "http://ww\ w.tcs.tkk.fi/Software/bliss/", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 1, 1, 0, 39 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 1, 1, 0, 39 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " tool ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "JK07" ), content := 0, count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Cite", next := 5528, root := ~, start := 5510, stop := 5527 ), rec( content := " is included in this pac\ kage. It\n is an open-source tool for computing automorphism groups and cano\ nical forms\n of graphs, written by Tommi Junttila and Petteri Kaski. Several\ of the methods\n in the ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package rely on ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), content := [ rec( content := "http://ww\ w.tcs.tkk.fi/Software/bliss/", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 1, 1, 0, 39 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 1, 1, 0, 39 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := ". If the ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "NautyTracesInte\ rface" ), content := [ rec( content := "https://github.com/gap-packages/NautyT\ racesInterface", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 1, 1, 0, 39 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ) , rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "NautyTrac\ esInterface", count := [ 1, 1, 0, 39 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], name := "Package", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := "\n package for GAP is a\ vailable then it is also possible to use ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "nauty" ), content := [ rec( content := "http://pa\ llini.di.uniroma1.it/", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 1, 1, 0, 39 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "nauty", count := [ 1, 1, 0, 39 ], name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], name := "Package", next := 124, root := ~, start := 100, stop := 123 ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "MP14" ), content := 0, count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Cite", next := 5890, root := ~, start := 5868, stop := 5889 ), rec( content := " for computing automorph\ ism groups and canonical forms\n in ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ". See Section \n ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Isomorphisms and canoni\ cal labellings" ), content := 0, count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "Ref", next := 6028, root := ~, start := 5977, stop := 6027 ), rec( content := " for more details. \n \ ", count := [ 1, 1, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "P", next := 6055, root := ~, start := 6051, stop := 6054 ), rec( content := "\n\n From version 1.0.0\ of this package, digraphs can be either mutable or immutable.\n Mutable digr\ aphs can be changed in-place by many of the methods in the\n package, which a\ voids unnecessary copying. Immutable digraphs cannot be\n changed in-place, b\ ut their advantage is that the value of an attribute of\n an immutable digrap\ h is only ever computed once. Mutable digraphs can be\n converted into immuta\ ble digraphs in-place using ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref", Func := "MakeImmutable" ), content := 0, count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "Ref", next := 6536, root := ~, start := 6490, stop := 6535 ), rec( content := ". One of the motivation\ s for introducing mutable\n digraphs in version 1.0.0 was that in practice th\ e authors often wanted\n to create a digraph and immediately modify it (remov\ ing certain edges,\n loops, and so on). Before version 1.0.0, this involved c\ opying the digraph\n several times, with each copy being discarded almost imm\ ediately.\n From version 1.0.0, this unnecessary copying can be eliminated by\ \n first creating a mutable digraph, then changing it in-place, and finally\ \n converting the mutable digraph to an immutable one in-place\n (if desirab\ le).\n\n ", count := [ 1, 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Definitions" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Definitio\ ns", count := [ 1, 1, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 2 ], mathmode := "Text", name := "Heading", next := 7167, root := ~, start := 7137, stop := 7166 ), rec( content := "\n\n For the \ purposes of this package and its documentation, the following\n definitions\ apply: ", count := [ 1, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "P", next := 7270, root := ~, start := 7266, stop := 7269 ), rec( content := "\n\n A ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7292, root := ~, start := 7278, stop := 7291 ), rec( content := " ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E=(E^0,E^\ 1,r,s)", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7315, root := ~, start := 7293, stop := 7314 ), rec( content := ", also known as \ a ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed\\ n graph", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7360, root := ~, start := 7333, stop := 7359 ), rec( content := ", consists of a \ set of vertices ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E^0", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7402, root := ~, start := 7392, stop := 7401 ), rec( content := " and a set of ed\ ges\n ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E^1", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7436, root := ~, start := 7426, stop := 7435 ), rec( content := " together with f\ unctions ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s, r: E^1\ \\to E^0", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7485, root := ~, start := 7461, stop := 7484 ), rec( content := ", called the\n \ ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "source", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7515, root := ~, start := 7502, stop := 7514 ), rec( content := " and ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "range", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7532, root := ~, start := 7520, stop := 7531 ), rec( content := ", respectively.\\ n\n The source and range of an edge is respectively the values of ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s, r", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "M", next := 7626, root := ~, start := 7615, stop := 7625 ), rec( content := " at\n that ed\ ge.\n\n An edge is called a ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "loop", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7681, root := ~, start := 7670, stop := 7680 ), rec( content := " if its source a\ nd range are the same.\n\n A digraph is called a ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "multidigr\ aph", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "E", next := 7766, root := ~, start := 7747, stop := 7765 ), rec( content := " if there exist \ two or more edges\n with the same source and the same range. ", count := [ 1, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "P", next := 7849, root := ~, start := 7845, stop := 7848 ), rec( content := "\n\n A ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed \ walk", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "E", next := 7877, root := ~, start := 7857, stop := 7876 ), rec( content := " on a digraph is\ a sequence of alternating vertices\n and edges ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(v_1, e_1\ , v_2, e_2, ..., e_{n-1}, v_n)", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 7989, root := ~, start := 7943, stop := 7988 ), rec( content := " such that each \ edge\n ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e_i", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8024, root := ~, start := 8014, stop := 8023 ), rec( content := " has source ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v_i", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8046, root := ~, start := 8036, stop := 8045 ), rec( content := " and range ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v_{i+1}", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8071, root := ~, start := 8057, stop := 8070 ), rec( content := ". A ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed\\ n path", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "E", next := 8101, root := ~, start := 8075, stop := 8100 ), rec( content := " is a directed w\ alk where no vertex (and hence no edge) is\n repeated. A ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed \ circuit", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "E", next := 8200, root := ~, start := 8177, stop := 8199 ), rec( content := " is a directed w\ alk where ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v_1 =\n \ v_n", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8248, root := ~, start := 8226, stop := 8247 ), rec( content := ", and a ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "directed \ cycle", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "E", next := 8277, root := ~, start := 8256, stop := 8276 ), rec( content := " is a directed c\ ircuit where where no\n vertex is repeated, except for ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v_1 = v_n\ ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "M", next := 8366, root := ~, start := 8350, stop := 8365 ), rec( content := ". ", count := [ 1, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "P", next := 8373, root := ~, start := 8369, stop := 8372 ), rec( content := "\n\n The ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "length", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 8396, root := ~, start := 8383, stop := 8395 ), rec( content := " of a directed w\ alk ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(v_1, e_1\ , v_2, e_2, ..., e_{n-1},\n v_n)", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8468, root := ~, start := 8416, stop := 8467 ), rec( content := " is equal to ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n-1", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8491, root := ~, start := 8481, stop := 8490 ), rec( content := ", the number of \ edges it contains. A\n directed walk (or path) ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(v_1, e_1\ , v_2, e_2, ..., e_{n-1}, v_n)", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8603, root := ~, start := 8557, stop := 8602 ), rec( content := " is\n sometim\ es called a directed walk (or path) ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "from vert\ ex ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ _1", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8679, root := ~, start := 8669, stop := 8678 ), rec( content := " to\n \ vertex ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ _n", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8706, root := ~, start := 8696, stop := 8705 ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 8710, root := ~, start := 8654, stop := 8709 ), rec( content := ". A directed wa\ lk of zero length, i.e. a sequence\n ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(v)", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8775, root := ~, start := 8765, stop := 8774 ), rec( content := " for some vertex\ ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8800, root := ~, start := 8792, stop := 8799 ), rec( content := ", is called ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "trivial", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 8826, root := ~, start := 8812, stop := 8825 ), rec( content := ". A trivial\n \ directed walk is considered to be both a circuit and a cycle, as is the\n \ empty directed walk ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "()", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "M", next := 8947, root := ~, start := 8938, stop := 8946 ), rec( content := ". A ", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "simple ci\ rcuit", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "E", next := 8972, root := ~, start := 8951, stop := 8971 ), rec( content := " is another name\ for a\n non-trivial and non-empty directed cycle.", count := [ 1, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "P", next := 9044, root := ~, start := 9040, stop := 9043 ), rec( content := "\n\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 1, 1 ], mathmode := "Text", name := "Subsection", next := 9061, root := ~, start := 7105, stop := 9060 ), rec( content := "\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 9074, root := ~, start := 4130, stop := 9073 ), rec( content := "\n", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 1, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 9085, root := ~, start := 3978, stop := 9084 ), rec( content := "\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " intro ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 1, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Installing Digraphs" ) , content := [ rec( content := "\n ", count := [ 2, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Installing ", count := [ 2, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 0, 0, 2 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ) ], count := [ 2, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 9184, root := ~, start := 9144, stop := 9183 ), rec( content := "\n\n ", count := [ 2, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "For those in a hurry" ), content := [ rec( content := "\n ", count := [ 2, 1, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "For those in a hu\ rry", count := [ 2, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 9270, root := ~, start := 9231, stop := 9269 ), rec( content := "\n In this section we\ give a brief description of how to start using\n ", count := [ 2, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ".", count := [ 2, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "P", next := 9360, root := ~, start := 9356, stop := 9359 ), rec( content := "\n\n It is assumed th\ at you have a working copy of ", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " with version number\n \ 4.10.0 or higher. The most up-to-date version of ", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " and instructions\n o\ n how to install it can be obtained from the main ", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " webpage\n ", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "http://www.gap-sy\ stem.org", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "URL", next := 9626, root := ~, start := 9590, stop := 9625 ), rec( content := ".", count := [ 2, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "P", next := 9631, root := ~, start := 9627, stop := 9630 ), rec( content := "\n\n The following is\ a summary of the steps that should lead to a\n successful installation of \ ", count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ":\n\n ", count := [ 2, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ensure that the ", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 2, 1, 0, 7 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 2, 1, 0, 7 ], name := \ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 7 ], name := "Package", next := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " package\ version 4.5.1 or higher is available.\n ", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 2, 1, 0, 7 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 2, 1, 0, 7 ], name := \ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 7 ], name := "Package", next := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " must be\ compiled before ", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " can be \ loaded.\n ", count := [ 2, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 7 ], name := "Item", next := 9922, root := ~, start := 9761, stop := 9921 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ensure that the ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Orb" ), content := [ rec( content := "\n http://http://gap-\ packages.github.io/orb/", count := [ 2, 1, 0, 9 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "URL\ ", next := 86, root := ~, start := 17, stop := 85 ), rec( content := "\n " , count := [ 2, 1, 0, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Alt", next := 98, root := ~, start := 1, stop := 97 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Orb", count := [ 2, 1, 0, 9 ], name :=\ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Package", next := 137, root := ~, start := 115 , stop := 136 ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 98, stop := 142 ), rec( content := " package\ version 4.8.2 or higher is available.\n ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Orb" ), content := [ rec( content := "\n http://http://gap-\ packages.github.io/orb/", count := [ 2, 1, 0, 9 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "URL\ ", next := 86, root := ~, start := 17, stop := 85 ), rec( content := "\n " , count := [ 2, 1, 0, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Alt", next := 98, root := ~, start := 1, stop := 97 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Orb", count := [ 2, 1, 0, 9 ], name :=\ "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Package", next := 137, root := ~, start := 115 , stop := 136 ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 98, stop := 142 ), rec( content := " has bet\ ter performance when compiled, but although compilation is\n recommende\ d, it is not required to be compiled for ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " to be\n\ loaded.\n ", count := [ 2, 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 9 ], name := "Item", next := 10202, root := ~, start := 9930, stop := 10201 ), rec( attributes := rec( ), content := [ rec( content := " ensure t\ hat the ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "datastructures" ), content := [ rec( content := "https://github\ .com/gap-packages/datastructures", count := [ 2, 1, 0, 11 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], mathmode := "Tex\ t", name := "URL", next := 103, root := ~, start := 17, stop := 102 ) ], count := [ 2, 1, 0, 11 ], name := "Alt", next := 109, root := ~, start := 1, stop := 108 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "datastructures", count := [ 2, 1, 0, 1\ 1 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], name := "Packa\ ge", next := 162, root := ~, start := 129, stop := 161 ) ], count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "Alt", next := 168, root := ~, start := 109, stop := 167 ), rec( content := " package\ version\n 0.2.5 or higher is available. \n ", count := [ 2, 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 11 ], name := "Item", next := 10331, root := ~, start := 10210, stop := 10330 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ his step is optional:", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "B", next := 10383, root := ~, start := 10354, stop := 10382 ), rec( content := " certain\ functions in ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " require\ \n the ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 1, 0, 13 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 2, 1, 0, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ to be available; see Section\n ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Subsect := "The \ Grape package" ), content := 0, count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Ref", next := 10523, root := ~, start := 10489, stop := 10522 ), rec( content := " for ful\ l details. To use these\n functions make sure\n that the ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 1, 0, 13 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 2, 1, 0, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ version 4.8.1 or higher is available.\n If ", count := [ 2, 1, 0, 13 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 1, 0, 13 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 2, 1, 0, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " is not \ available, then ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " can be \ used as normal with\n the exception that the functions listed in Subsec\ tion\n ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Subsect := "The \ Grape package" ), content := 0, count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "Ref", next := 10845, root := ~, start := 10811, stop := 10844 ), rec( content := " will no\ t work.\n ", count := [ 2, 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 13 ], name := "Item", next := 10874, root := ~, start := 10339, stop := 10873 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ his step is optional:", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "B", next := 10932, root := ~, start := 10903, stop := 10931 ), rec( content := " certain\ functions in ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " require\ \n the ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 2, 1, 0, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 2, 1, 0, 15 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 2, 1\ , 0, 15 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " package\ to be available.\n ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " TODO \n\ see Section\n for full d\ etails. ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "XMLCOMMENT" , root := ~ ), rec( content := "\n \ If you want to make use of these functions, please ensure that the\n \ ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 2, 1, 0, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 2, 1, 0, 15 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 2, 1\ , 0, 15 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " package\ version 0.2 or\n higher is available. If ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 2, 1, 0, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 2, 1, 0, 15 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 2, 1\ , 0, 15 ], name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is not \ available,\n then ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " can be \ used as normal with the exception that \n functions whose names conta\ in ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ auty", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "Q", next := 11491, root := ~, start := 11479, stop := 11490 ), rec( content := " will no\ t work.\n ", count := [ 2, 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 15 ], name := "Item", next := 11520, root := ~, start := 10888, stop := 11519 ), rec( attributes := rec( ), content := [ rec( content := "\n \ download the package archive ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-1.4.0.tar.gz", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "F", next := 11599, root := ~, start := 11572, stop := 11598 ), rec( content := " from\n \ ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "the Dig\ raph package webpage" ), content := [ rec( content := "http://digraphs.github.\ io/Digraphs/", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "URL", next := 11694, root := ~, start := 11613, stop := 11693 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 17 ], name := "Item", next := 11709, root := ~, start := 11528, stop := 11708 ), rec( attributes := rec( ), content := [ rec( content := "\n \ unzip and untar the file, this should create a directory called\n ", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-1.4.0", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "F", next := 11824, root := ~, start := 11804, stop := 11823 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 19 ], name := "Item", next := 11839, root := ~, start := 11717, stop := 11838 ), rec( attributes := rec( ), content := [ rec( content := "\n \ locate the ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ kg", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 11883, root := ~, start := 11873, stop := 11882 ), rec( content := " directo\ ry of your ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " directo\ ry, which contains\n the directories ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ib", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 11968, root := ~, start := 11958, stop := 11967 ), rec( content := ", ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ oc", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 11980, root := ~, start := 11970, stop := 11979 ), rec( content := " and so \ on. Move the directory\n ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-1.4.0", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 12039, root := ~, start := 12019, stop := 12038 ), rec( content := " into th\ e ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ kg", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "F", next := 12059, root := ~, start := 12049, stop := 12058 ), rec( content := " directo\ ry.\n ", count := [ 2, 1, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 21 ], name := "Item", next := 12084, root := ~, start := 11847, stop := 12083 ), rec( attributes := rec( ), content := [ rec( content := "\n \ it is necessary to compile the ", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\ .\n Inside the ", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ kg/digraphs-1.4.0", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "F", next := 12201, root := ~, start := 12177, stop := 12200 ), rec( content := " directo\ ry, type\n ", count := [ 2, 1, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ n./configure\nmake", count := [ 2, 1, 0, 24 ], name := "PCDATA", root := ~ ) ] , count := [ 2, 1, 0, 24 ], mathmode := "Text", name := "Listing", next := 12274, root := ~, start := 12226, stop := 12273 ), rec( content := "\n \ Further information about this step can be found in Section\n ", count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Compili\ ng the kernel module" ), content := 0, count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "Ref", next := 12392, root := ~, start := 12351, stop := 12391 ), rec( content := ".\n \ ", count := [ 2, 1, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 23 ], name := "Item", next := 12407, root := ~, start := 12092, stop := 12406 ), rec( attributes := rec( ), content := [ rec( content := "\n \ start ", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " in the \ usual way (i.e. type ", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "C", next := 12480, root := ~, start := 12470, stop := 12479 ), rec( content := " at the \ command line).\n ", count := [ 2, 1, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 27 ], name := "Item", next := 12516, root := ~, start := 12415, stop := 12515 ), rec( attributes := rec( ), content := [ rec( content := "\n \ type ", count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ oadPackage(\"digraphs\");", count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "C", next := 12575, root := ~, start := 12544, stop := 12574 ), rec( content := "\n \ ", count := [ 2, 1, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 29 ], name := "Item", next := 12589, root := ~, start := 12524, stop := 12588 ) ], count := [ 2, 1, 0, 6 ], mathmode := "Text", name := "List", next := 12601, root := ~, start := 9747, stop := 12600 ), rec( content := "\n\n If you want to c\ heck that the package is working correctly, you\n should run some of the te\ sts described in Section\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "Testing your installati\ on" ), content := 0, count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "Ref", next := 12768, root := ~, start := 12729, stop := 12767 ), rec( content := ".\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 12782, root := ~, start := 9188, stop := 12781 ), rec( content := "\n\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 1, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Optional package dependencies"\ ), content := [ rec( content := "\n ", count := [ 2, 2, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Optional package \ dependencies", count := [ 2, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 13047, root := ~, start := 12999, stop := 13046 ), rec( content := "\n\n ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "************************\ ********************************************", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n The ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package is written in " , count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " and C code and requires\ the\n ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ rec( content := "\n ht\ tp://gap-packages.github.io/io/\n", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 2, 2, 0, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "IO", count := [ 2, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], name := "Package", next := 122, root := ~, start := 101, stop := 121 ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " package. The ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ rec( content := "\n ht\ tp://gap-packages.github.io/io/\n", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 2, 2, 0, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "IO", count := [ 2, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 3 ], name := "Package", next := 122, root := ~, start := 101, stop := 121 ) ], count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " package is used to read\ and write transformations,\n partial permutations, and bipartitions to a f\ ile. ", count := [ 2, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 2, 0, 4 ], mathmode := "Text", name := "P", next := 13342, root := ~, start := 13338, stop := 13341 ), rec( content := "\n\n ", count := [ 2, 2, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "************************\ ********************************************", count := [ 2, 2, 0, 4 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 2, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "The Grape package" ), content := [ rec( content := "\n ", count := [ 2, 2, 1, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "The Grape\ package", count := [ 2, 2, 1, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 2 ], mathmode := "Text", name := "Heading", next := 13510, root := ~, start := 13474, stop := 13509 ), rec( content := "\n\n The ", count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ 2, 2, 1, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 2, 1, 3 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ) , content := [ rec( content := "G\ rape", count := [ 2, 2, 1, 3 ], name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 3 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package must be\ available for the following operations to be\n available:\n\n ", count := [ 2, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( attributes :=\ rec( Oper := "Graph" ), content := 0, count := [ 2, 2, 1, 5 ], mathmode := "T\ ext", name := "Ref", next := 13655, root := ~, start := 13636, stop := 13654 ) , rec( content := "\ with a digraph argument", count := [ 2, 2, 1, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 2, 2, 1, 5 ], name := "Item", next := 13686, root := ~, start := 13630, stop := 13685 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( Attr := "AsGraph" ), content := 0, count := [ 2, 2, 1, 7 ], mathmode := \ "Text", name := "Ref", next := 13722, root := ~, start := 13701, stop := 13721\ ), rec( content := " with a digraph argument", count := [ 2, 2, 1, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 7 ], name := "Item", next := 13753, root := ~, start := 13695, stop := 13752 ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( Oper := "Digraph" ), content := 0, count := [ 2, 2, 1, 9 ], mathmode := \ "Text", name := "Ref", next := 13789, root := ~, start := 13768, stop := 13788\ ), rec( content := " with a ", count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( content := "http://www.math\ s.qmul.ac.uk/~leonard/grape/", count := [ 2, 2, 1, 9 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 9 ], mathmode := "Text", name\ := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 2, 1, 9 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "Grape", count := [ 2, 2, 1, 9 ], name := "PCDATA", root := ~ ) ], count := [ \ 2, 2, 1, 9 ], name := "Package", next := 137, root := ~, start := 113, stop :=\ 136 ) ], count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := "\ graph argument", count := [ 2, 2, 1, 9 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 2, 2, 1, 9 ], name := "Item", next := 13826, root := ~, start := 13762, stop := 13825 ) ], count := [ 2, 2, 1, 4 ], mathmode := "Text", name := "List", next := 13840, root := ~, start := 13615, stop := 13839 ), rec( content := "\n\n If ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ 2, 2, 1, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 2, 2, 1, 11 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ) , content := [ rec( content := "G\ rape", count := [ 2, 2, 1, 11 ], name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 11 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " is not availabl\ e, then ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs" , count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " can be used as \ normal with\n the exception that the functions above will not work.\n \ ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 1, 1 ], mathmode := "Text", name := "Subsection", next := 13997, root := ~, start := 13429, stop := 13996 ), rec( content := "\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 14010, root := ~, start := 12947, stop := 14009 ), rec( content := "\n\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 2, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Compiling the kernel module" ) , content := [ rec( content := "\n ", count := [ 2, 3, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Compiling the ker\ nel module", count := [ 2, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 14271, root := ~, start := 14225, stop := 14270 ), rec( content := "\n The ", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package has a ", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " kernel component in\n \ C which should be compiled. This component contains certain low-level\n f\ unctions required by ", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ".\n ", count := [ 2, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "P", next := 14451, root := ~, start := 14447, stop := 14450 ), rec( content := "\n\n It is not possib\ le to use the ", count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package without compili\ ng it.", count := [ 2, 3, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "P", next := 14531, root := ~, start := 14527, stop := 14530 ), rec( content := "\n\n To compile the k\ ernel component inside the ", count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "pkg/digraphs-1.4.\ 0", count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "F", next := 14604, root := ~, start := 14580, stop := 14603 ), rec( content := "\n directory, type\n" , count := [ 2, 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n\n./configure\n\ make", count := [ 2, 3, 0, 6 ], name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 6 ], mathmode := "Text", name := "Listing", next := 14674, root := ~, start := 14625, stop := 14673 ), rec( content := "\n ", count := [ 2, 3, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "P", next := 14683, root := ~, start := 14679, stop := 14682 ), rec( content := "\n\n If you installed\ the package in another 'pkg' directory than the standard 'pkg'\n directory\ in your ", count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " installation, then you \ have to do two things. Firstly\n during compilation you have to use the opt\ ion '--with-gaproot=PATH' of the\n 'configure' script where 'PATH' is a pat\ h to the main GAP root directory (if\n not given the default '../..' is ass\ umed).", count := [ 2, 3, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "P", next := 15061, root := ~, start := 15057, stop := 15060 ), rec( content := "\n\n If you installed\ ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "GAP", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " on several architecture\ s, you must execute the\n configure/make step for each of the architectures\ . You can either do this\n immediately after configuring and compiling GAP \ itself on this architecture, or\n alternatively set the environment\n va\ riable 'CONFIGNAME' to the name of the configuration you used when compiling\n\ GAP before running './configure'. Note however that your compiler choice \ and\n flags (environment variables 'CC' and 'CFLAGS') need to be chosen to \ match the\n setup of the original GAP compilation. For example you have to \ specify 32-bit\n or 64-bit mode correctly!\n ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 15709, root := ~, start := 14175, stop := 15708 ), rec( content := "\n\n ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Rebuilding the documentation" ), content := [ rec( content := "\n ", count := [ 2, 4, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Rebuilding the do\ cumentation", count := [ 2, 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 15972, root := ~, start := 15925, stop := 15971 ), rec( content := "\n The ", count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package comes complete \ with pdf, html, and text versions of\n the documentation. However, you migh\ t find it necessary, at some point, to\n rebuild the documentation. To rebu\ ild the documentation, please use the\n function ", count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "DigraphsMakeDoc" ), content := 0, count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "Ref", next := 16249, root := ~, start := 16220, stop := 16248 ), rec( content := ".\n\n ", count := [ 2, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsMakeDoc\ " ), content := 0, count := [ 2, 4, 1, 2 ], mathmode := "Text", name := "Func", next := 16308, root := ~, start := 16271, stop := 16307 ), rec( attributes := rec( ), content := [ rec( content := "Nothing", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "Returns", next := 16337, root := ~, start := 16311, stop := 16336 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function should be called with no argument to compile the ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := "\n do\ cumentation.\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 16467, root := ~, start := 16340, stop := 16466 ) ], count := [ 2, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 16481, root := ~, start := 16256, stop := 16480 ), rec( content := "\n\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 16495, root := ~, start := 15874, stop := 16494 ), rec( content := "\n\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 2, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Testing your installation" ), content := [ rec( content := "\n ", count := [ 2, 5, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Testing your inst\ allation", count := [ 2, 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 16752, root := ~, start := 16708, stop := 16751 ), rec( content := "\n\n In this section \ we describe how to test that ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is working as\n inte\ nded. To test that ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is installed correctly \ use\n ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "DigraphsTestInstall" ), content := 0, count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "Ref", next := 16931, root := ~, start := 16898, stop := 16930 ), rec( content := " or for more extensive t\ ests use\n ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "DigraphsTestStandard" ) , content := 0, count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "Ref", next := 17002, root := ~, start := 16968, stop := 17001 ), rec( content := ". ", count := [ 2, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "P", next := 17008, root := ~, start := 17004, stop := 17007 ), rec( content := "\n\n If something goe\ s wrong, then please review the instructions in Section ", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Sect := "For those in a hurry" ) , content := 0, count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "Ref", next := 17126, root := ~, start := 17086, stop := 17125 ), rec( content := " and ensure that ", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " has been\n properly \ installed. If you continue having problems, please use the ", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "issue tracker" ), content := [ rec( content := "https://github.co\ m/digraphs/Digraphs/issues", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "URL", next := 17317, root := ~, start := 17236, stop := 17316 ), rec( content := "\n to report the issu\ es you are having.\n\n ", count := [ 2, 5, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsTestIns\ tall" ), content := 0, count := [ 2, 5, 1, 2 ], mathmode := "Text", name := "Func", next := 17420, root := ~, start := 17379, stop := 17419 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "K", next := 17443, root := ~, start := 17432, stop := 17442 ), rec( content := " or ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "K", next := 17459, root := ~, start := 17447, stop := 17458 ), rec( content := ".", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 17470, root := ~, start := 17423, stop := 17469 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function can be called without arguments to test your installation of\n " , count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is work\ ing correctly. These tests should take no more than a\n few seconds to comp\ lete. To test more comprehensively that\n ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is work\ ing correctly, use ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sTestStandard" ), content := 0, count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Ref", next := 17781, root := ~, start := 17747, stop := 17780 ), rec( content := ".\n ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 17799, root := ~, start := 17473, stop := 17798 ) ], count := [ 2, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 17813, root := ~, start := 17364, stop := 17812 ), rec( content := "\n\n ", count := [ 2, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsTestSta\ ndard" ), content := 0, count := [ 2, 5, 2, 2 ], mathmode := "Text", name := "Func", next := 17876, root := ~, start := 17834, stop := 17875 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "K", next := 17899, root := ~, start := 17888, stop := 17898 ), rec( content := " or ", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "K", next := 17915, root := ~, start := 17903, stop := 17914 ), rec( content := ".", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 17926, root := ~, start := 17879, stop := 17925 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function can be called without arguments to test all of the methods\n inc\ luded in ", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ".\n T\ hese tests should take less than a minute to complete. ", count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "P", next := 18112, root := ~, start := 18108, stop := 18111 ), rec( content := "\n\n \ To quickly test that ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " is inst\ alled correctly use ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sTestInstall" ), content := 0, count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 18214, root := ~, start := 18177, stop := 18213 ), rec( content := ". For a\ more thorough test, use ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sTestExtreme" ), content := 0, count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 18284, root := ~, start := 18247, stop := 18283 ), rec( content := ".\n ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 18302, root := ~, start := 17929, stop := 18301 ) ], count := [ 2, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 18316, root := ~, start := 17819, stop := 18315 ), rec( content := "\n\n ", count := [ 2, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsTestExt\ reme" ), content := 0, count := [ 2, 5, 3, 2 ], mathmode := "Text", name := "Func", next := 18378, root := ~, start := 18337, stop := 18377 ), rec( attributes := rec( ), content := [ rec( content := "Nothing." , count := [ 2, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 3 ], mathmode := "Text", name := "Returns", next := 18408, root := ~, start := 18381, stop := 18407 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function should be called with no argument. It executes a series of\n ver\ y demanding tests, which measure the performance of a variety of\n function\ s on large examples. These tests take a long time to\n complete, at least \ several minutes. ", count := [ 2, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "P", next := 18684, root := ~, start := 18680, stop := 18683 ), rec( content := "\n\n \ For these tests to complete, the digraphs library ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-lib", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "C", next := 18759, root := ~, start := 18740, stop := 18758 ), rec( content := " must\n \ be downloaded and placed in the ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "C", next := 18816, root := ~, start := 18801, stop := 18815 ), rec( content := " directo\ ry in a subfolder\n named ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs-lib", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "F", next := 18871, root := ~, start := 18852, stop := 18870 ), rec( content := ".\n T\ his library can be found on the ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " website\ .\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 18946, root := ~, start := 18411, stop := 18945 ) ], count := [ 2, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 18960, root := ~, start := 18322, stop := 18959 ), rec( content := "\n\n\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 18975, root := ~, start := 16660, stop := 18974 ), rec( content := "\n\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "********************************\ **************************************", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n\n", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 2, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 19149, root := ~, start := 9104, stop := 19148 ), rec( content := "\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " installing ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 2, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Creating digraphs" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Creating digraphs", count := [ 3, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 19244, root := ~, start := 19208, stop := 19243 ), rec( content := "\n\n In this chapter we describ\ e how to create digraphs.", count := [ 3, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 0, 0, 4 ], mathmode := "Text", name := "P", next := 19303, root := ~, start := 19299, stop := 19302 ), rec( content := "\n\n ", count := [ 3, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Creating digraphs\ ", count := [ 3, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 19352, root := ~, start := 19316, stop := 19351 ), rec( content := "\n ", count := [ 3, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsDigraph", Type := "Category" ), content := 0, count := [ 3, 1, 1, 2 ], mathmode := "Text", name := "Filt", next := 19412, root := ~, start := 19372, stop := 19411 ), rec( attributes := rec( ), content := [ rec( content := "\n Eve\ ry digraph in ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " belong\ s to the category\n ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraph", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "C", next := 19511, root := ~, start := 19495, stop := 19510 ), rec( content := ". Some \ basic attributes and operations for digraphs\n are ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Vertices" ), content := 0, count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Ref", next := 19601, root := ~, start := 19572, stop := 19600 ), rec( content := ", ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Edges" ), content := 0, count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Ref", next := 19629, root := ~, start := 19603, stop := 19628 ), rec( content := ", and\n \ ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutNeig\ hbours" ), content := 0, count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Ref", next := 19666, root := ~, start := 19639, stop := 19665 ), rec( content := ".\n " , count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 19686, root := ~, start := 19415, stop := 19685 ) ], count := [ 3, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 19700, root := ~, start := 19357, stop := 19699 ), rec( content := "\n\n ", count := [ 3, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsMutableDigrap\ h", Type := "Category" ), content := 0, count := [ 3, 1, 2, 2 ], mathmode := "Text", name := "Filt", next := 19768, root := ~, start := 19721, stop := 19767 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMutableDigraph", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "C", next := 19812, root := ~, start := 19789, stop := 19811 ), rec( content := " is a sy\ nonym for ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsDigra\ ph" ), content := 0, count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Ref", next := 19854, root := ~, start := 19830, stop := 19853 ), rec( content := " and ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsMutable" ), content := 0, count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Ref", next := 19902, root := ~, start := 19859, stop := 19901 ), rec( content := ". A mut\ able digraph may be changed\n in-place by methods in the ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\ , and is not attribute-storing\n – see ", count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsAttributeStoringRep" ), content := 0, count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Ref", next := 20081, root := ~, start := 20030, stop := 20080 ), rec( content := ".\n " , count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "P", next := 20091, root := ~, start := 20087, stop := 20090 ), rec( content := "\n\n \ A mutable digraph may be converted into an immutable attribute-storing\n di\ graph by calling ", count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "MakeImmutable" ), content := 0, count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "Ref", next := 20233, root := ~, start := 20191, stop := 20232 ), rec( content := " on the\\ n digraph.\n ", count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 20272, root := ~, start := 19771, stop := 20271 ) ], count := [ 3, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 20286, root := ~, start := 19706, stop := 20285 ), rec( content := "\n\n ", count := [ 3, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsImmutableDigr\ aph", Type := "Category" ), content := 0, count := [ 3, 1, 3, 2 ], mathmode := "Text", name := "Filt", next := 20356, root := ~, start := 20307, stop := 20355 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sImmutableDigraph", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "C", next := 20402, root := ~, start := 20377, stop := 20401 ), rec( content := " is a su\ bcategory of ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsDigra\ ph" ), content := 0, count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 20447, root := ~, start := 20423, stop := 20446 ), rec( content := ".\n D\ igraphs that lie in ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sImmutableDigraph", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "C", next := 20499, root := ~, start := 20474, stop := 20498 ), rec( content := " are imm\ utable and\n attribute-storing. In particular, they lie in\n ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsAttributeStoringRep" ), content := 0, count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 20623, root := ~, start := 20572, stop := 20622 ), rec( content := ". ", count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "P", next := 20629, root := ~, start := 20625, stop := 20628 ), rec( content := "\n\n \ A mutable digraph may be converted to an immutable\n digraph that lies in t\ he category ", count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sImmutableDigraph", count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "C", next := 20749, root := ~, start := 20724, stop := 20748 ), rec( content := " by call\ ing\n ", count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "MakeImmutable" ), content := 0, count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "Ref", next := 20807, root := ~, start := 20765, stop := 20806 ), rec( content := " on the \ digraph.", count := [ 3, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "P", next := 20827, root := ~, start := 20823, stop := 20826 ), rec( content := "\n\n \ The operation ", count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ MutableCopy" ), content := 0, count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "Ref", next := 20879, root := ~, start := 20847, stop := 20878 ), rec( content := " can be \ used\n to construct a mutable copy of an immutable digraph. It is however n\ ot\n possible to convert an immutable digraph into a mutable digraph in-pla\ ce.\n ", count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 21061, root := ~, start := 20359, stop := 21060 ) ], count := [ 3, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 21075, root := ~, start := 20292, stop := 21074 ), rec( content := "\n\n ", count := [ 3, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsCayleyDigraph\ ", Type := "Category" ), content := 0, count := [ 3, 1, 4, 2 ], mathmode := "Text", name := "Filt", next := 21142, root := ~, start := 21096, stop := 21141 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sCayleyDigraph", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "C", next := 21185, root := ~, start := 21163, stop := 21184 ), rec( content := " is a su\ bcategory of ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraph", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "C", next := 21222, root := ~, start := 21206, stop := 21221 ), rec( content := ". Digrap\ hs that\n are Cayley digraphs of a group and that are constructed by the op\ eration\n ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CayleyD\ igraph" ), content := 0, count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "Ref", next := 21346, root := ~, start := 21319, stop := 21345 ), rec( content := " are con\ structed in this category, and are\n always immutable.\n ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 21429, root := ~, start := 21145, stop := 21428 ) ], count := [ 3, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 21443, root := ~, start := 21081, stop := 21442 ), rec( content := "\n\n ", count := [ 3, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "IsDigraphWithAd\ jacencyFunction", Type := "Category" ), content := 0, count := [ 3, 1, 5, 2 ], mathmode := "Text", name := "Filt", next := 21525, root := ~, start := 21464, stop := 21524 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphWithAdjacencyFunction", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "C", next := 21583, root := ~, start := 21546, stop := 21582 ), rec( content := " is a su\ bcategory of ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraph", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "C", next := 21620, root := ~, start := 21604, stop := 21619 ), rec( content := ".\n D\ igraphs that are ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ reated", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "E", next := 21658, root := ~, start := 21644, stop := 21657 ), rec( content := " using a\ n adjacency function are constructed\n in this category.\n ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 21741, root := ~, start := 21528, stop := 21740 ) ], count := [ 3, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 21755, root := ~, start := 21449, stop := 21754 ), rec( content := "\n\n ", count := [ 3, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Name := "DigraphByOutNei\ ghboursType" ), content := 0, count := [ 3, 1, 6, 2 ], mathmode := "Text", name := "Var", next := 21816, root := ~, start := 21776, stop := 21815 ), rec( attributes := rec( Name := "DigraphFamily" ), content := 0, count := [ 3, 1, 6, 4 ], mathmode := "Text", name := "Fam", next := 21846, root := ~, start := 21819, stop := 21845 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ type of all digraphs is ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphByOutNeighboursType", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "C", next := 21928, root := ~, start := 21895, stop := 21927 ), rec( content := ".\n T\ he family of all digraphs is ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFamily", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "C", next := 21984, root := ~, start := 21964, stop := 21983 ), rec( content := ".\n ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "Description", next := 22002, root := ~, start := 21849, stop := 22001 ) ], count := [ 3, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 22016, root := ~, start := 21761, stop := 22015 ), rec( content := "\n\n ", count := [ 3, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt,] obj[, so\ urce, range]", Name := "Digraph" ), content := 0, count := [ 3, 1, 7, 2 ], mathmode := "Text", name := "Oper", next := 22098, root := ~, start := 22037, stop := 22097 ), rec( attributes := rec( Arg := "[filt,] list, fu\ nc", Label := "for a list and function", Name := "Digraph" ), content := 0, count := [ 3, 1, 7, 4 ], mathmode := "Text", name := "Oper", next := 22191, root := ~, start := 22101, stop := 22190 ), rec( attributes := rec( Arg := "[filt,] G, list,\ act, adj", Label := "for a group, list, function, and function", Name := "Digraph" ), content := 0, count := [ 3, 1, 7, 6 ], mathmode := "Text", name := "Oper", next := 22308, root := ~, start := 22194, stop := 22307 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Returns", next := 22340, root := ~, start := 22311, stop := 22339 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "A", next := 22403, root := ~, start := 22392, stop := 22402 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "A", next := 22549, root := ~, start := 22538, stop := 22548 ), rec( content := " is ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Ref", next := 22583, root := ~, start := 22553, stop := 22582 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "A", next := 22651, root := ~, start := 22640, stop := 22650 ), rec( content := " is ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Ref", next := 22691, root := ~, start := 22655, stop := 22690 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "A", next := 22776, root := ~, start := 22765, stop := 22775 ), rec( content := " is not \ present, then ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Ref", next := 22834, root := ~, start := 22798, stop := 22833 ), rec( content := " is used\ by default.\n ", count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "for a list (i.e. an adjacency list)", coun\ t := [ 3, 1, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 8 ], name := "Mar\ k", next := 22920, root := ~, start := 22872, stop := 22919 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 9\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "obj", count := [ 3, 1, 7, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode := \ "Text", name := "A", next := 22955, root := ~, start := 22945, stop := 22954 ) , rec( cont\ ent := " is a list of lists of positive integers in\n the range from ", coun\ t := [ 3, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "1", count := [ 3, 1, 7, 9 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], math\ mode := "Text", name := "C", next := 23030, root := ~, start := 23022, stop :=\ 23029 ), rec( content := " to ", count := [ 3, 1, 7, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "Length(", count := [ 3, 1, 7, 9 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ), rec( attributes := rec( ), content := [ rec( content := "obj",\ count := [ 3, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ],\ count := [ 3, 1, 7, 9 ], mathmode := "Text", name := "A", next := 23054, root\ := ~, start := 23044, stop := 23053 ), rec( content := ")", count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "C", next := 23059, root := ~, start := 23\ 034, stop := 23058 ), rec( content := ", then this function\n returns t\ he digraph with vertices\n ", count := [ 3, 1, 7, 9 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "E ^ 0 = ", count := [ 3, 1, 7, 9 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode := "Text", name :=\ "M", next := 23145, root := ~, start := 23130, stop := 23144 ), rec( attribut\ es := rec( ), content := [ rec( content := "[1 .. Length(", count := [ 3, 1, \ 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := \ rec( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 9 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode\ := "Text", name := "A", next := 23171, root := ~, start := 23161, stop := 231\ 70 ), rec( content := ")]", count := [ 3, 1, 7, 9 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmode := "Text", name \ := "C", next := 23177, root := ~, start := 23145, stop := 23176 ), rec( conten\ t := ",\n and edges corresponding to the entries of ", count := [ 3, 1,\ 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 9 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], mathmod\ e := "Text", name := "A", next := 23239, root := ~, start := 23229, stop := 23\ 238 ), rec( content := ". ", count := [ 3, 1, 7, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ \ 3, 1, 7, 10 ], mathmode := "Text", name := "P", next := 23245, root := ~, star\ t := 23241, stop := 23244 ), rec( content := "\n\n More precisely, ther\ e is an edge from vertex ", count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "i", count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "C", next :=\ 23308, root := ~, start := 23300, stop := 23307 ), rec( content := " to ", coun\ t := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "j", count := [ 3, 1, 7, 10 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ] , math\ mode := "Text", name := "C", next := 23320, root := ~, start := 23312, stop :=\ 23319 ), rec( content := " if and\n only if ", count := [ 3, 1, 7, 10 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "j", count := [ 3, 1, 7, 10 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 23352, root := ~, start := 23344, stop := 23351 ), rec( conte\ nt := " is in ", count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := re\ c( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode\ := "Text", name := "A", next := 23372, root := ~, start := 23362, stop := 233\ 71 ), rec( content := "[i]", count := [ 3, 1, 7, 10 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 23379, root := ~, start := 23359, stop := 23378 ), rec( conte\ nt := "; the source of this edge is\n ", count := [ 3, 1, 7, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 3, 1, 7, 10 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 23424, root := ~, start := 23416, stop := 23423 ), rec( conte\ nt := " and the range is ", count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "j", count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "C", next :=\ 23450, root := ~, start := 23442, stop := 23449 ), rec( content := ". If ", coun\ t := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "j", count := [ 3, 1, 7, 10 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ] , math\ mode := "Text", name := "C", next := 23463, root := ~, start := 23455, stop :=\ 23462 ), rec( content := " occurs in\n ", count := [ 3, 1, 7, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( attributes := rec( ), content := [ rec( content := "obj", count\ := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 10 ], mathmode := "Text", name := "A", next := 23495, root := \ ~, start := 23485, stop := 23494 ), rec( content := "[i]", count := [ 3, 1, 7,\ 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7\ , 10 ], mathmode := "Text", name := "C", next := 23502, root := ~, start := 23\ 482, stop := 23501 ), rec( content := " with multiplicity ", count := [ 3, 1, \ 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "k", count := [ 3, 1, 7, 10 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmod\ e := "Text", name := "C", next := 23529, root := ~, start := 23521, stop := 23\ 528 ), rec( content := ", then there are ", count := [ 3, 1, 7, 10 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "k", count := [ 3, 1, 7, 10 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "C", next := 23554, root := ~, start := 23546, stop := 23553 ), rec( conte\ nt := "\n edges from ", count := [ 3, 1, 7, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "i", count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 10 ], mathmode := "Text", name := "C", next :=\ 23582, root := ~, start := 23574, stop := 23581 ), rec( content := " to ", coun\ t := [ 3, 1, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "j", count := [ 3, 1, 7, 10 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 10 ] , math\ mode := "Text", name := "C", next := 23594, root := ~, start := 23586, stop :=\ 23593 ), rec( content := ".\n ", count := [ 3, 1, 7, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 9 ], name := "Item\ ", next := 23609, root := ~, start := 22927, stop := 23608 ), rec( attributes :\ = rec( ), content := [ rec( content := "for three lists", count := [ 3, 1, 7,\ 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 11 ], name := "Mar\ k", next := 23645, root := ~, start := 23617, stop := 23644 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 1\ 2 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 12 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode \ := "Text", name := "A", next := 23680, root := ~, start := 23670, stop := 2367\ 9 ), rec( content := " is a duplicate-free list, and ", count := [ 3, 1, 7, 12\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "source", count := [ 3, 1, 7, 12 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmod\ e := "Text", name := "A", next := 23724, root := ~, start := 23711, stop := 23\ 723 ), rec( content := " and\n ", count := [ 3, 1, 7, 12 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "range", count := [ 3, 1, 7, 12 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name\ := "A", next := 23749, root := ~, start := 23737, stop := 23748 ), rec( conte\ nt := " are lists of equal length consisting of positive integers\n in \ the list ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "[1 .. Lengt\ h(", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "obj", count := [\ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := \ [ 3, 1, 7, 12 ], mathmode := "Text", name := "A", next := 23854, root := ~, st\ art := 23844, stop := 23853 ), rec( content := ")]", count := [ 3, 1, 7, 12 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ] , math\ mode := "Text", name := "C", next := 23860, root := ~, start := 23828, stop :=\ 23859 ), rec( content := ", then this function\n returns a digraph wit\ h vertices\n ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "E ^ 0 = ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "M", next \ := 23944, root := ~, start := 23929, stop := 23943 ), rec( attributes := rec( \ ), content := [ rec( content := "[1 .. Length(", count := [ 3, 1, 7, 12 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), c\ ontent := [ rec( content := "obj", count := [ 3, 1, 7, 12 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Tex\ t", name := "A", next := 23970, root := ~, start := 23960, stop := 23969 ), re\ c( content := ")]", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "C" , next\ := 23976, root := ~, start := 23944, stop := 23975 ), rec( content := ", and\ \n ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "Length(", c\ ount := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), re\ c( attributes := rec( ), content := [ rec( content := "source", count := [ 3,\ 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3\ , 1, 7, 12 ], mathmode := "Text", name := "A", next := 24013, root := ~, start\ := 24000, stop := 24012 ), rec( content := ")", count := [ 3, 1, 7, 12 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], math\ mode := "Text", name := "C", next := 24018, root := ~, start := 23990, stop :=\ 24017 ), rec( content := " edges. For each ", count := [ 3, 1, 7, 12 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 3, 1, 7, 12 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name\ := "C", next := 24044, root := ~, start := 24036, stop := 24043 ), rec( conte\ nt := " in\n ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "[1 .. Length(", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "sou\ rce", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "A", next := 2408\ 5, root := ~, start := 24072, stop := 24084 ), rec( content := ")]", count := \ [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 12 ], mathmode := "Text", name := "C", next := 24091, root := ~, star\ t := 24056, stop := 24090 ), rec( content := " there exists an edge with sourc\ e\n vertex ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PC\ DATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\ source[i]", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "C", next :\ = 24156, root := ~, start := 24140, stop := 24155 ), rec( content := " and ran\ ge vertex ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "range[i]", \ count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 12 ], mathmode := "Text", name := "C", next := 24189, root := \ ~, start := 24174, stop := 24188 ), rec( content := ".\n See ", count :\ = [ 3, 1, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( Attr := "DigraphSource" ), content := 0, count := [ 3, 1, 7, 12\ ], mathmode := "Text", name := "Ref", next := 24230, root := ~, start := 2420\ 3, stop := 24229 ), rec( content := " and ", count := [ 3, 1, 7, 12 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "D\ igraphRange" ), content := 0, count := [ 3, 1, 7, 12 ], mathmode := "Text", name\ := "Ref", next := 24261, root := ~, start := 24235, stop := 24260 ), rec( cont\ ent := ".\n ", count := [ 3, 1, 7, 12 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1\ , 7, 13 ], mathmode := "Text", name := "P", next := 24275, root := ~, start :=\ 24271, stop := 24274 ), rec( content := "\n\n The vertices of the digr\ aph will be labelled by the elements of\n ", count := [ 3, 1, 7, 13 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "obj", count := [ 3, 1, 7, 13 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 13 ], mathmode := "Text", name\ := "A", next := 24367, root := ~, start := 24357, stop := 24366 ), rec( conte\ nt := ".\n ", count := [ 3, 1, 7, 13 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 1, 7, 12 ], name := "Item", next := 24382, root := ~, start := 236\ 52, stop := 24381 ), rec( attributes := rec( ), content := [ rec( content := \ "for an integer, and two lists", count := [ 3, 1, 7, 14 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 14 ], name := "Mark", next := 2443\ 2, root := ~, start := 24390, stop := 24431 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n if ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "obj", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "A", next := 2\ 4467, root := ~, start := 24457, stop := 24466 ), rec( content := " is an inte\ ger, and ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "source", co\ unt := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 15 ], mathmode := "Text", name := "A", next := 24500, root := \ ~, start := 24487, stop := 24499 ), rec( content := " and\n ", count :=\ [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "range", count := [ 3, 1, 7, 1\ 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, \ 15 ], mathmode := "Text", name := "A", next := 24525, root := ~, start := 2451\ 3, stop := 24524 ), rec( content := " are lists of equal length consisting of \ positive integers\n in the list ", count := [ 3, 1, 7, 15 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "[1 .. ", count := [ 3, 1, 7, 15 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( con\ tent := "obj", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "A", nex\ t := 24623, root := ~, start := 24613, stop := 24622 ), rec( content := "]", c\ ount := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next := 24628, root := \ ~, start := 24604, stop := 24627 ), rec( content := ", then this function\n \ returns a digraph with vertices\n ", count := [ 3, 1, 7, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "E ^ 0 = ", count := [ 3, 1, 7, 15 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "T\ ext", name := "M", next := 24712, root := ~, start := 24697, stop := 24711 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[1 .. ", count := [ 3, 1, 7, \ 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 15 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode\ := "Text", name := "A", next := 24731, root := ~, start := 24721, stop := 247\ 30 ), rec( content := "]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name\ := "C", next := 24736, root := ~, start := 24712, stop := 24735 ), rec( conte\ nt := ", and\n ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "Length(", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", \ root := ~ ), rec( attributes := rec( ), content := [ rec( content := "source"\ , count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) \ ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "A", next := 24773, r\ oot := ~, start := 24760, stop := 24772 ), rec( content := ")", count := [ 3, \ 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3,\ 1, 7, 15 ], mathmode := "Text", name := "C", next := 24778, root := ~, start \ := 24750, stop := 24777 ), rec( content := " edges. For each ", count := [ 3,\ 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "i", count := [ 3, 1, 7, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], math\ mode := "Text", name := "C", next := 24804, root := ~, start := 24796, stop :=\ 24803 ), rec( content := " in\n ", count := [ 3, 1, 7, 15 ], mathmode \ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content \ := [ rec( content := "[1 .. Length(", count := [ 3, 1, 7, 15 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "source", count := [ 3, 1, 7, 15 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name \ := "A", next := 24845, root := ~, start := 24832, stop := 24844 ), rec( conten\ t := ")]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next :=\ 24851, root := ~, start := 24816, stop := 24850 ), rec( content := " there ex\ ists an edge with source\n vertex ", count := [ 3, 1, 7, 15 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "source[i]", count := [ 3, 1, 7, 15 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Tex\ t", name := "C", next := 24916, root := ~, start := 24900, stop := 24915 ), rec( cont\ ent := " and range vertex ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "range[i]", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "C", next\ := 24949, root := ~, start := 24934, stop := 24948 ), rec( content := ".\n \ See ", count := [ 3, 1, 7, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( Attr := "DigraphSource" ), content := 0, coun\ t := [ 3, 1, 7, 15 ], mathmode := "Text", name := "Ref", next := 24990, root :\ = ~, start := 24963, stop := 24989 ), rec( content := " and ", count := [ 3, 1\ , 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( Attr := "DigraphRange" ), content := 0, count := [ 3, 1, 7, 15 ], math\ mode := "Text", name := "Ref", next := 25021, root := ~, start := 24995, stop \ := 25020 ), rec( content := ".\n ", count := [ 3, 1, 7, 15 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 15 ], name := "Ite\ m", next := 25036, root := ~, start := 24439, stop := 25035 ), rec( attributes :\ = rec( ), content := [ rec( content := "for a list and a function", count := [ \ 3, 1, 7, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 16 ], name := "Mar\ k", next := 25082, root := ~, start := 25044, stop := 25081 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 1\ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "list", count := [ 3, 1, 7, 17 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode\ := "Text", name := "A", next := 25118, root := ~, start := 25107, stop := 251\ 17 ), rec( content := " is a list and ", count := [ 3, 1, 7, 17 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "func", count := [ 3, 1, 7, 17 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name\ := "A", next := 25144, root := ~, start := 25133, stop := 25143 ), rec( conte\ nt := " is a function taking 2\n arguments that are elements of ", coun\ t := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "list", count := [ 3, 1, 7, 17\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 1\ 7 ], mathmode := "Text", name := "A", next := 25218, root := ~, start := 25207 , stop\ := 25217 ), rec( content := ", and ", count := [ 3, 1, 7, 17 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "func", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name :=\ "A", next := 25235, root := ~, start := 25224, stop := 25234 ), rec( content \ := " returns\n ", count := [ 3, 1, 7, 17 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "true", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "K", next :=\ 25263, root := ~, start := 25252, stop := 25262 ), rec( content := " or ", coun\ t := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "false", count := [ 3, 1, 7, 1\ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, \ 17 ], mathmode := "Text", name := "K", next := 25279, root := ~, start := 2526\ 7, stop := 25278 ), rec( content := ", then this operation creates a digraph w\ ith\n vertices ", count := [ 3, 1, 7, 17 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "[1 .. Length(", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "A", next := 2536\ 8, root := ~, start := 25357, stop := 25367 ), rec( content := ")]", count := \ [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 25374, root := ~, star\ t := 25341, stop := 25373 ), rec( content := " and an edge from vertex\n \ ", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "i", count := [ 3\ , 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 25415, root := ~, star\ t := 25407, stop := 25414 ), rec( content := " to vertex ", count := [ 3, 1, 7\ , 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "j", count := [ 3, 1, 7, 17 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode :\ = "Text", name := "C", next := 25434, root := ~, start := 25426, stop := 25433\ ), rec( content := " if and only if\n ", count := [ 3, 1, 7, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( attributes := rec( ), content := [ rec( content := "func", coun\ t := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], cou\ nt := [ 3, 1, 7, 17 ], mathmode := "Text", name := "A", next := 25472, root :=\ ~, start := 25461, stop := 25471 ), rec( content := "(", count := [ 3, 1, 7, \ 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "list", count := [ 3, 1, 7, 17 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmod\ e := "Text", name := "A", next := 25484, root := ~, start := 25473, stop := 25\ 483 ), rec( content := "[i], ", count := [ 3, 1, 7, 17 ], mathmode := "Text", \ name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( c\ ontent := "list", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "A", \ next := 25500, root := ~, start := 25489, stop := 25499 ), rec( content := "[j\ ])", count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ 3, 1, 7, 17 ], mathmode := "Text", name := "C", next := 25508 , root\ := ~, start := 25458, stop := 25507 ), rec( content := " returns ", count := [ \ 3, 1, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "true", count := [ 3, 1, 7, 17 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ] , math\ mode := "Text", name := "K", next := 25528, root := ~, start := 25517, stop :=\ 25527 ), rec( content := ".\n ", count := [ 3, 1, 7, 17 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 17 ], name := "Ite\ m", next := 25543, root := ~, start := 25089, stop := 25542 ), rec( attributes :\ = rec( ), content := [ rec( content := "for a group, a list, and two function\ s", count := [ 3, 1, 7, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 18 ], name := "Mar\ k", next := 25602, root := ~, start := 25551, stop := 25601 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n The arguments will be ", coun\ t := [ 3, 1, 7, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G, list, act, adj", count := \ [ 3, 1, 7, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 19 ], mathmode := "Text", name := "A", next := 25670, root := ~, star\ t := 25646, stop := 25669 ), rec( content := ". ", count := [ 3, 1, 7, 19 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := 0, count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "P", next := 2\ 5676, root := ~, start := 25672, stop := 25675 ), rec( content := "\n\n \ Let ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count := \ [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next := 25698, root := ~, star\ t := 25690, stop := 25697 ), rec( content := " be a group acting on the object\ s in ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ), rec( attributes := rec( ), content := [ rec( content := "list", count \ := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next := 25746, root := ~ , star\ t := 25735, stop := 25745 ), rec( content := " via\n the action ", coun\ t := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "act", count := [ 3, 1, 7, 20 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20\ ], mathmode := "Text", name := "A", next := 25780, root := ~, start := 25770, stop\ := 25779 ), rec( content := ", and let ", count := [ 3, 1, 7, 20 ], mathmode \ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content \ := [ rec( content := "adj", count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "A", next := 25800, root := ~, start := 25790, stop := 25799 ), rec( conte\ nt := " be a function taking\n two objects from ", count := [ 3, 1, 7, \ 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "list", count := [ 3, 1, 7, 20 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmod\ e := "Text", name := "A", next := 25858, root := ~, start := 25847, stop := 25\ 857 ), rec( content := " as arguments and returning ", count := [ 3, 1, 7, 20 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "true", count := [ 3, 1, 7, 20 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text" , name\ := "C", next := 25897, root := ~, start := 25886, stop := 25896 ), rec( conte\ nt := " or\n ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "false", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "C", next := \ 25921, root := ~, start := 25909, stop := 25920 ), rec( content := ". The func\ tion ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ), rec( attributes := rec( ), content := [ rec( content := "adj", count :\ = [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next := 25946, root := ~, star\ t := 25936, stop := 25945 ), rec( content := " will describe the adjacency\n \ between objects from ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "list", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next\ := 26015, root := ~, start := 26004, stop := 26014 ), rec( content := ", whic\ h is invariant under the action of\n ", count := [ 3, 1, 7, 20 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "G", count := [ 3, 1, 7, 20 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "A", next := 26072, root := ~, start := 26064, stop := 26071 ), rec( conte\ nt := ". This variant of the constructor returns a digraph with\n verti\ ces the objects of ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "list", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "A", next := 26\ 172, root := ~, start := 26161, stop := 26171 ), rec( content := " and directe\ d edges ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "[x, y]", co\ unt := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 20 ], mathmode := "Text", name := "C", next := 26205, root := \ ~, start := 26192, stop := 26204 ), rec( content := "\n when ", count :\ = [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "f(x, y)", count := [ 3, 1, 7,\ 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7\ , 20 ], mathmode := "Text", name := "C", next := 26233, root := ~, start := 26\ 219, stop := 26232 ), rec( content := " is ", count := [ 3, 1, 7, 20 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "true", count := [ 3, 1, 7, 20 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 20 ], mathmode := "Text", name\ := "C", next := 26248, root := ~, start := 26237, stop := 26247 ), rec( conte\ nt := ". ", count := [ 3, 1, 7, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 7, 21 ], math\ mode := "Text", name := "P", next := 26254, root := ~, start := 26250, stop :=\ 26253 ), rec( content := "\n\n The action of the group ", count := [ 3\ , 1, 7, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "G", count := [ 3, 1, 7, 21 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 21 ], math\ mode := "Text", name := "A", next := 26296, root := ~, start := 26288, stop :=\ 26295 ), rec( content := " on the objects in ", count := [ 3, 1, 7, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "list", count := [ 3, 1, 7, 21 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 21 ], mathmode := "Text" , name\ := "A", next := 26326, root := ~, start := 26315, stop := 26325 ), rec( conte\ nt := " is stored\n in the attribute ", count := [ 3, 1, 7, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := \ "DigraphGroup" ), content := 0, count := [ 3, 1, 7, 21 ], mathmode := "Text", name\ := "Ref", next := 26388, root := ~, start := 26362, stop := 26387 ), rec( cont\ ent := ", and is used to speed up\n operations like ", count := [ 3, 1,\ 7, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Attr := "DigraphDiameter" ), content := 0, count := [ 3, 1, 7, 21 ], math\ mode := "Text", name := "Ref", next := 26467, root := ~, start := 26438, stop \ := 26466 ), rec( content := ".\n ", count := [ 3, 1, 7, 21 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 19 ], name := "Ite\ m", next := 26482, root := ~, start := 25609, stop := 26481 ), rec( attributes :\ = rec( ), content := [ rec( content := "for a Grape package graph", count := [ \ 3, 1, 7, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 22 ], name := "Mar\ k", next := 26528, root := ~, start := 26490, stop := 26527 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 2\ 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 23 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], mathmode \ := "Text", name := "A", next := 26563, root := ~, start := 26553, stop := 2656\ 2 ), rec( content := " is a ", count := [ 3, 1, 7, 23 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content :=\ [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "ht\ tp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ 3, 1, 7, 23 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], mathmod\ e := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ] , coun\ t := [ 3, 1, 7, 23 ], name := "Alt", next := 93, root := ~, start := 1, stop :\ = 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes\ := rec( ), content := [ rec( content := "Grape", count := [ 3, 1, 7, 23 ], n\ ame := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], name := "Package", n\ ext := 137, root := ~, start := 113, stop := 136 ) ], count := [ 3, 1, 7, 23 ] , math\ mode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 14\ 2 ), rec( content := " package graph (i.e. a record for which the\n fun\ ction ", count := [ 3, 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root \ := ~ ), rec( attributes := rec( ), content := [ rec( content := "IsGraph", co\ unt := [ 3, 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 23 ], mathmode := "Text", name := "C", next := 26651, root := \ ~, start := 26637, stop := 26650 ), rec( content := " returns ", count := [ 3,\ 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "true", count := [ 3, 1, 7, 23 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], math\ mode := "Text", name := "K", next := 26671, root := ~, start := 26660, stop :=\ 26670 ), rec( content := "), then this function returns\n a digraph is\ omorphic to ", count := [ 3, 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "obj", count\ := [ 3, 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 23 ], mathmode := "Text", name := "A", next := 26743, root := \ ~, start := 26733, stop := 26742 ), rec( content := ".\n ", count := [ 3,\ 1, 7, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 23 ], name := "Ite\ m", next := 26758, root := ~, start := 26535, stop := 26757 ), rec( attributes :\ = rec( ), content := [ rec( content := "for a binary relation", count := [ 3,\ 1, 7, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 24 ], name := "Mar\ k", next := 26800, root := ~, start := 26766, stop := 26799 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n if ", count := [ 3, 1, 7, 2\ 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "obj", count := [ 3, 1, 7, 25 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], mathmode \ := "Text", name := "A", next := 26835, root := ~, start := 26825, stop := 2683\ 4 ), rec( content := " is a binary relation on the points ", count := [ 3, 1, \ 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "[1 .. n]", count := [ 3, 1, 7, 25 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], math\ mode := "Text", name := "C", next := 26886, root := ~, start := 26871, stop :=\ 26885 ), rec( content := " for\n some posititve integer ", count := [ \ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "n", count := [ 3, 1, 7, 25 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], math\ mode := "Text", name := "M", next := 26930, root := ~, start := 26922, stop :=\ 26929 ), rec( content := ", then this function returns the digraph\n d\ efined by ", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "obj", count\ := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 25 ], mathmode := "Text", name := "A", next := 27000, root := \ ~, start := 26990, stop := 26999 ), rec( content := ". Specifically, this func\ tion returns a digraph\n which has ", count := [ 3, 1, 7, 25 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "n", count := [ 3, 1, 7, 25 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], mathmode := "Text", name\ := "M", next := 27074, root := ~, start := 27066, stop := 27073 ), rec( conte\ nt := " vertices, and which has an edge with source ", count := [ 3, 1, 7, 25 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 3, 1, 7, 25 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], mathmode := "Text", name\ := "C", next := 27127, root := ~, start := 27119, stop := 27126 ), rec( conte\ nt := "\n and range ", count := [ 3, 1, 7, 25 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "j", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "C", next :=\ 27154, root := ~, start := 27146, stop := 27153 ), rec( content := " if and o\ nly if ", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "[i,j]", cou\ nt := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 3, 1, 7, 25 ], mathmode := "Text", name := "C", next := 27182, root := \ ~, start := 27170, stop := 27181 ), rec( content := " is a pair in\n th\ e binary relation ", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PC\ DATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\ obj", count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 3, 1, 7, 25 ], mathmode := "Text", name := "A", next := 2723\ 4, root := ~, start := 27224, stop := 27233 ), rec( content := ".\n ", coun\ t := [ 3, 1, 7, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 25 ], name := "Ite\ m", next := 27249, root := ~, start := 26807, stop := 27248 ) ], count := [ 3, 1, 7, 8 ], mathmode := "Text", name := "List", next := 27261, root := ~, start := 22859, stop := 27260 ), rec( content := "\n\n \ ", count := [ 3, 1, 7, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\n> [2, 5, 8, 10], [2, 3, 4, 2, 5, 6, 8, 9, 10], [1],\n> \ [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],\n> [1, 5, 9], [1, 2, 7,\ 8], [3, 5]]);\n\ngap> gr :\ = Digraph([\"a\", \"b\", \"c\"], [\"a\"], [\"b\"]);\n\ngap> gr := Digraph(5, [1, 2, 2, 4, 1, 1], [2, 3, 5, 5, 1, \ 1]);\n\ngap> Petersen := Grap\ h(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x, y) return Intersection(x\ , y) = []; end);;\ngap> Digraph(Petersen);\n\ngap> gr := Digraph([1 .. 10], ReturnTrue);\n", count := [ 3, 1, 7, 28 ], name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 28 ], mathmode := "Text", name := "Example", next := 28003, root := ~, start := 27267, stop := 28002 ), rec( content := "\n\n \ The next example illustrates the uses of the fourth and fifth variants of\n \ this constructor. The resulting digraph is a strongly regular graph, and\ \n it is actually the point graph of the van Lint-Schrijver partial\n \ geometry, ", count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "vLS81" ) , content := 0, count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "Cite", next := 28270, root := ~, start := 28251, stop := 28269 ), rec( content := ". The a\ lgebraic description\n is taken from the seminal paper of Calderbank and \ Kantor ", count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "CK86" ), content := 0, count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "Cite", next := 28388, root := ~, start := 28362, stop := 28387 ), rec( content := ".\n", count := [ 3, 1, 7, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> f := GF(3 ^ 4);\nGF(3^4)\ngap> gamma := First(f, x -> Order(x) = 5);\nZ(\ 3^4)^64\ngap> L := Union([Zero(f)], List(Group(gamma)));\n[ 0*Z(3), Z(3)^0, Z(\ 3^4)^16, Z(3^4)^32, Z(3^4)^48, Z(3^4)^64 ]\ngap> omega := Union(List(L, x -> L\ ist(Difference(L, [x]), y -> x - y)));\n[ Z(3)^0, Z(3), Z(3^4)^5, Z(3^4)^7, Z(\ 3^4)^8, Z(3^4)^13, Z(3^4)^15, \n Z(3^4)^16, Z(3^4)^21, Z(3^4)^23, Z(3^4)^24, \ Z(3^4)^29, Z(3^4)^31, \n Z(3^4)^32, Z(3^4)^37, Z(3^4)^39, Z(3^4)^45, Z(3^4)^4\ 7, Z(3^4)^48, \n Z(3^4)^53, Z(3^4)^55, Z(3^4)^56, Z(3^4)^61, Z(3^4)^63, Z(3^4\ )^64, \n Z(3^4)^69, Z(3^4)^71, Z(3^4)^72, Z(3^4)^77, Z(3^4)^79 ]\ngap> adj :=\ function(x, y)\n> return x - y in omega;\n> end;\nfunction( x, y ) ... end\ \ngap> digraph := Digraph(AsList(f), adj);\n\ngap> group := Group(Z(3));;\ngap> act := \\*;\n\ngap> digraph := Digraph(group, List(f), act, adj);\n\n", count := [ 3, 1, 7, 30 ], name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 30 ], mathmode := "Text", name := "Example", next := 29358, root := ~, start := 28390, stop := 29357 ), rec( content := "\n ", count := [ 3, 1, 7, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 7, 7 ], mathmode := "Text", name := "Description", next := 29375, root := ~, start := 22343, stop := 29374 ) ], count := [ 3, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 29389, root := ~, start := 22022, stop := 29388 ), rec( content := "\n\n ", count := [ 3, 1, 7, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]list", Name := "DigraphByAdjace\ ncyMatrix" ), content := 0, count := [ 3, 1, 8, 2 ], mathmode := "Text", name := "Oper", next := 29468, root := ~, start := 29410, stop := 29467 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Returns", next := 29500, root := ~, start := 29471, stop := 29499 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "A", next := 29563, root := ~, start := 29552, stop := 29562 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "A", next := 29709, root := ~, start := 29698, stop := 29708 ), rec( content := " is ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 29743, root := ~, start := 29713, stop := 29742 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "A", next := 29811, root := ~, start := 29800, stop := 29810 ), rec( content := " is ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 29851, root := ~, start := 29815, stop := 29850 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "A", next := 29936, root := ~, start := 29925, stop := 29935 ), rec( content := " is not \ present, then ", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 29994, root := ~, start := 29958, stop := 29993 ), rec( content := " is used\ by default.", count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "P", next := 30018, root := ~, start := 30014, stop := 30017 ), rec( content := "\n\n \ If ", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "A", next := 30038, root := ~, start := 30027, stop := 30037 ), rec( content := " is the \ adjacency matrix of a digraph in the sense of\n ", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Adjacen\ cyMatrix" ), content := 0, count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "Ref", next := 30125, root := ~, start := 30096, stop := 30124 ), rec( content := ", then t\ his operation returns the digraph\n which is defined by ", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "A", next := 30202, root := ~, start := 30191, stop := 30201 ), rec( content := ". ", count := [ 3, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "P", next := 30208, root := ~, start := 30204, stop := 30207 ), rec( content := "\n\n \ Alternatively, if ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "A", next := 30243, root := ~, start := 30232, stop := 30242 ), rec( content := " is a sq\ uare boolean matrix, then this\n operation returns the digraph with ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ength(", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "C", next := 30335, root := ~, start := 30321, stop := 30334 ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "A", next := 30346, root := ~, start := 30335, stop := 30345 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "C", next := 30354, root := ~, start := 30346, stop := 30353 ), rec( content := "\n ve\ rtices which has the edge ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "C", next := 30399, root := ~, start := 30387, stop := 30398 ), rec( content := " if and \ only if\n ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "A", next := 30430, root := ~, start := 30419, stop := 30429 ), rec( attributes := rec( ), content := [ rec( content := "[\ i][j]", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "C", next := 30443, root := ~, start := 30430, stop := 30442 ), rec( content := " is ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "K", next := 30458, root := ~, start := 30447, stop := 30457 ), rec( content := ". ", count := [ 3, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 8, 6 ], mathmode := "Text", name := "P", next := 30464, root := ~, start := 30460, stop := 30463 ), rec( content := "\n\n \ ", count := [ 3, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphByAdjacencyMatrix([\n> [0, 1, 0, 2, 0],\n> [1, 1, 1, 0, 1],\n> [0\ , 3, 2, 1, 1],\n> [0, 0, 1, 0, 1],\n> [2, 0, 0, 0, 0]]);\n\ngap> D := DigraphByAdjacencyMatrix([\n> [true,\ false, true],\n> [false, false, true],\n> [false, true, false]]);\n\ngap> OutNeighbours(D);\n[ [ 1, 3 ], [ 3 ],\ [ 2 ] ]\ngap> D := DigraphByAdjacencyMatrix(IsMutableDigraph, \n> [[true, fal\ se, true],\n> [false, false, true],\n> [false, true, false]]);\n\n", count := [ 3, 1, 8, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 7 ], mathmode := "Text", name := "Example", next := 31060, root := ~, start := 30470, stop := 31059 ), rec( content := "\n ", count := [ 3, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 8, 3 ], mathmode := "Text", name := "Description", next := 31077, root := ~, start := 29503, stop := 31076 ) ], count := [ 3, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 31091, root := ~, start := 29395, stop := 31090 ), rec( content := "\n\n ", count := [ 3, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]list[, n\ ]", Name := "DigraphByEdges" ), content := 0, count := [ 3, 1, 9, 2 ], mathmode := "Text", name := "Oper", next := 31165, root := ~, start := 31112, stop := 31164 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Returns", next := 31197, root := ~, start := 31168, stop := 31196 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "A", next := 31260, root := ~, start := 31249, stop := 31259 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "A", next := 31406, root := ~, start := 31395, stop := 31405 ), rec( content := " is ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Ref", next := 31440, root := ~, start := 31410, stop := 31439 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "A", next := 31508, root := ~, start := 31497, stop := 31507 ), rec( content := " is ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Ref", next := 31548, root := ~, start := 31512, stop := 31547 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "A", next := 31633, root := ~, start := 31622, stop := 31632 ), rec( content := " is not \ present, then ", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Ref", next := 31691, root := ~, start := 31655, stop := 31690 ), rec( content := " is used\ by default.", count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "P", next := 31715, root := ~, start := 31711, stop := 31714 ), rec( content := "\n\n \ If ", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "A", next := 31735, root := ~, start := 31724, stop := 31734 ), rec( content := " is list\ of pairs of positive integers, then this function\n returns the digraph wi\ th the minimum number of vertices ", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "C", next := 31862, root := ~, start := 31854, stop := 31861 ), rec( content := " such th\ at\n its list equal ", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "A", next := 31903, root := ~, start := 31892, stop := 31902 ), rec( content := ".", count := [ 3, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "P", next := 31908, root := ~, start := 31904, stop := 31907 ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "A", next := 31954, root := ~, start := 31946, stop := 31953 ), rec( content := " is a po\ sitive integer with\n ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "n", count := [ 3, 1, 9, 5 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := \ "Text", name := "A", next := 31997, root := ~, start := 31989, stop := 31996 ) , rec( content := "\ >= m", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "C", next := 32006, root := ~, start := 31986, stop := 32005 ), rec( content := " (with " , count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "C", next := 32021, root := ~, start := 32013, stop := 32020 ), rec( content := " defined\ as above),\n then this function returns the digraph with ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "A", next := 32097, root := ~, start := 32089, stop := 32096 ), rec( content := " vertice\ s and list\n ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "A", next := 32131, root := ~, start := 32120, stop := 32130 ), rec( content := ". ", count := [ 3, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 9, 6 ], mathmode := "Text", name := "P", next := 32137, root := ~, start := 32133, stop := 32136 ), rec( content := "\n\n \ See ", count := [ 3, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Edges" ), content := 0, count := [ 3, 1, 9, 6 ], mathmode := "Text", name := "Ref", next := 32173, root := ~, start := 32147, stop := 32172 ), rec( content := ".\n\n \ ", count := [ 3, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphByEdges(\n> [[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],\n> [4, 6], \ [5, 2], [5, 4], [5, 6], [6, 6]]);\n\ngap> DigraphByEdges(\n> [[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],\n> [4, 6\ ], [5, 2], [5, 4], [5, 6], [6, 6]], 12);\n\ngap> DigraphByEdges(IsMutableDigraph, \n> [[1, 3], [2, 1], [2, 3],\ [2, 5], [3, 6],\n> [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]], 12);\n\n", count := [ 3, 1, 9, 7 ], name := "PCDAT\ A", root := ~ ) ], count := [ 3, 1, 9, 7 ], mathmode := "Text", name := "Example", next := 32703, root := ~, start := 32180, stop := 32702 ), rec( content := "\n ", count := [ 3, 1, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 9, 3 ], mathmode := "Text", name := "Description", next := 32720, root := ~, start := 31200, stop := 32719 ) ], count := [ 3, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 32734, root := ~, start := 31097, stop := 32733 ), rec( content := "\n\n ", count := [ 3, 1, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "G, edges[, n]", Name := "EdgeOrbitsDigra\ ph" ), content := 0, count := [ 3, 1, 10, 2 ], mathmode := "Text", name := "Oper", next := 32807, root := ~, start := 32755, stop := 32806 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ immutable digraph.\n ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "Returns", next := 32858, root := ~, start := 32810, stop := 32857 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 32890, root := ~, start := 32882, stop := 32889 ), rec( content := " is a pe\ rmutation group, ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 32927, root := ~, start := 32915, stop := 32926 ), rec( content := " is an e\ dge or list of\n edges, and ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 32973, root := ~, start := 32965, stop := 32972 ), rec( content := " is a no\ n-negative integer such that ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 33018, root := ~, start := 33010, stop := 33017 ), rec( content := " fixes\n\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 1, 10, 3 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := \ "Text", name := "A", next := 33046, root := ~, start := 33038, stop := 33045 ) , rec( content := "\ ]", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "C", next := 33051, root := ~, start := 33029, stop := 33050 ), rec( content := " setwise\ , then\n this operation returns an immutable digraph with ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 33127, root := ~, start := 33119, stop := 33126 ), rec( content := " vertice\ s and the union\n of the orbits of the edges in ", count := [ 3, 1, 10, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ edges ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 33199, root := ~, start := 33185, stop := 33198 ), rec( content := " under t\ he action of the\n permutation group ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "A", next := 33254, root := ~, start := 33246, stop := 33253 ), rec( content := ". An ed\ ge in this context is simply a pair of\n positive integers.\n ", count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "P", next := 33332, root := ~, start := 33328, stop := 33331 ), rec( content := "\n\n \ If the optional third argument ", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "A", next := 33377, root := ~, start := 33369, stop := 33376 ), rec( content := " is not \ present, then the largest\n moved point of the permutation group ", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "A", next := 33460, root := ~, start := 33452, stop := 33459 ), rec( content := " is used\ by default.", count := [ 3, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 10, 5 ], mathmode := "Text", name := "P", next := 33484, root := ~, start := 33480, stop := 33483 ), rec( content := "\n\n \ ", count := [ 3, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := EdgeOrbitsDigraph(Group((1, 3), (1, 2)(3, 4)),\n> \ [[1, 2], [4, 5]], 5);\n\ngap> OutNeighbours(digraph);\n[ [ 2, 4, 5 ], [ 1, 3, 5 ], [ 2, 4,\ 5 ], [ 1, 3, 5 ], [ ] ]\ngap> RepresentativeOutNeighbours(digraph);\n[ [ 2, \ 4, 5 ], [ ] ]\n", count := [ 3, 1, 10, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 6 ], mathmode := "Text", name := "Example", next := 33842, root := ~, start := 33490, stop := 33841 ), rec( content := "\n ", count := [ 3, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 10, 3 ], mathmode := "Text", name := "Description", next := 33859, root := ~, start := 32861, stop := 33858 ) ], count := [ 3, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 33873, root := ~, start := 32740, stop := 33872 ), rec( content := "\n\n ", count := [ 3, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]list", Name := "DigraphByInNeig\ hbours" ), content := 0, count := [ 3, 1, 11, 2 ], mathmode := "Text", name := "Oper", next := 33949, root := ~, start := 33894, stop := 33948 ), rec( attributes := rec( Arg := "[filt, ]list", Name := "DigraphByInNeig\ hbors" ), content := 0, count := [ 3, 1, 11, 4 ], mathmode := "Text", name := "Oper", next := 34006, root := ~, start := 33952, stop := 34005 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Returns", next := 34038, root := ~, start := 34009, stop := 34037 ), rec( attributes := rec( ), content := [ rec( content := "\n If th\ e optional first argument ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "A", next := 34099, root := ~, start := 34088, stop := 34098 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph be\ ing created will\n belong to. For example, if ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "A", next := 34241, root := ~, start := 34230, stop := 34240 ), rec( content := " is ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Ref", next := 34275, root := ~, start := 34245, stop := 34274 ), rec( content := ",\n the\ n the digraph being created will be mutable, if ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "A", next := 34341, root := ~, start := 34330, stop := 34340 ), rec( content := " is ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Ref", next := 34379, root := ~, start := 34345, stop := 34378 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "A", next := 34462, root := ~, start := 34451, stop := 34461 ), rec( content := " is not \ present, then ", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Ref", next := 34518, root := ~, start := 34484, stop := 34517 ), rec( content := " is used\ by default.", count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "P", next := 34542, root := ~, start := 34538, stop := 34541 ), rec( content := "\n\n If\ ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "A", next := 34560, root := ~, start := 34549, stop := 34559 ), rec( content := " is a li\ st of lists of positive integers list the range\n ", count := [ 3, 1, 11, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. Length(", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "list", count := [ 3, 1, 11, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := \ "Text", name := "A", next := 34645, root := ~, start := 34634, stop := 34644 ) , rec( content := "\ )]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34651, root := ~, start := 34618, stop := 34650 ), rec( content := ", then t\ his function returns the\n digraph with vertices ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ ^0=", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "M", next := 34719, root := ~, start := 34708, stop := 34718 ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. Length(", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "list", count := [ 3, 1, 11, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := \ "Text", name := "A", next := 34746, root := ~, start := 34735, stop := 34745 ) , rec( content := "\ )]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34752, root := ~, start := 34719, stop := 34751 ), rec( content := ", and\n \ edges corresponding to the entries of ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "A", next := 34809, root := ~, start := 34798, stop := 34808 ), rec( content := ". More p\ recisely, there is an\n edge with source vertex ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34873, root := ~, start := 34865, stop := 34872 ), rec( content := " and ran\ ge vertex ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34899, root := ~, start := 34891, stop := 34898 ), rec( content := " if ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34911, root := ~, start := 34903, stop := 34910 ), rec( content := " is list\ \n ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "list", count := [ 3, 1, 11, 6 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := \ "Text", name := "A", next := 34936, root := ~, start := 34925, stop := 34935 ) , rec( content := "\ [j]", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "C", next := 34943, root := ~, start := 34922, stop := 34942 ), rec( content := ". ", count := [ 3, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "P", next := 34949, root := ~, start := 34945, stop := 34948 ), rec( content := "\n\n If\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 34964, root := ~, start := 34956, stop := 34963 ), rec( content := " occurs \ list ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "list", count := [ 3, 1, 11, 7 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := \ "Text", name := "A", next := 34991, root := ~, start := 34980, stop := 34990 ) , rec( content := "\ [j]", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 34998, root := ~, start := 34977, stop := 34997 ), rec( content := " with mu\ ltiplicity ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 35025, root := ~, start := 35017, stop := 35024 ), rec( content := ",\n the\ n there are ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 35052, root := ~, start := 35044, stop := 35051 ), rec( content := " multipl\ e edges from ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 35081, root := ~, start := 35073, stop := 35080 ), rec( content := " to ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "C", next := 35093, root := ~, start := 35085, stop := 35092 ), rec( content := ". ", count := [ 3, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 11, 8 ], mathmode := "Text", name := "P", next := 35099, root := ~, start := 35095, stop := 35098 ), rec( content := "\n\n Se\ e ", count := [ 3, 1, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "InNeigh\ bours" ), content := 0, count := [ 3, 1, 11, 8 ], mathmode := "Text", name := "Ref", next := 35133, root := ~, start := 35107, stop := 35132 ), rec( content := ".\n\n \ ", count := [ 3, 1, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphByInNeighbours([\n> [2, 5, 8, 10], [2, 3, 4, 5, 6, 8, 9, 10]\ ,\n> [1], [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4],\n> [1, 5, 9],\ [1, 2, 7, 8], [3, 5]]);\n\ngap>\ D := DigraphByInNeighbours([[2, 3, 2], [1], [1, 2, 3]]);\n\ngap> D := DigraphByInNeighbours(IsMutableDigra\ ph, \n> [[2, 3, 2], [1], [1, 2, 3]]);\n\n", count := [ 3, 1, 11, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 11, 9 ], mathmode := "Text", name := "Example", next := 35669, root := ~, start := 35140, stop := 35668 ), rec( content := "\n ", count := [ 3, 1, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 11, 5 ], mathmode := "Text", name := "Description", next := 35686, root := ~, start := 34041, stop := 35685 ) ], count := [ 3, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 35700, root := ~, start := 33879, stop := 35699 ), rec( content := "\n\n ", count := [ 3, 1, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "G[, gens]", Name := "CayleyDigraph" ), content := 0, count := [ 3, 1, 12, 2 ], mathmode := "Text", name := "Oper", next := 35765, root := ~, start := 35721, stop := 35764 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble digraph.", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "Returns", next := 35808, root := ~, start := 35768, stop := 35807 ), rec( attributes := rec( ), content := [ rec( content := "\n Let\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 35841, root := ~, start := 35833, stop := 35840 ), rec( content := " be any \ group and let ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 35874, root := ~, start := 35863, stop := 35873 ), rec( content := " be a li\ st of elements of\n ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 35912, root := ~, start := 35904, stop := 35911 ), rec( content := ". This o\ peration returns an immutable digraph that corresponds to\n the Cayley grap\ h of ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 36010, root := ~, start := 36002, stop := 36009 ), rec( content := " with re\ spect\n ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 36039, root := ~, start := 36028, stop := 36038 ), rec( content := ". The ve\ rtices are the elements of ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 36082, root := ~, start := 36074, stop := 36081 ), rec( content := ". There \ exists an edge\n from the vertex ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "C", next := 36133, root := ~, start := 36125, stop := 36132 ), rec( content := " to the \ vertex ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "C", next := 36156, root := ~, start := 36148, stop := 36155 ), rec( content := " if and \ only if there exists\n a generator ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "C", next := 36209, root := ~, start := 36201, stop := 36208 ), rec( content := " in ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "A", next := 36224, root := ~, start := 36213, stop := 36223 ), rec( content := " such th\ at ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ * g = y", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "C", next := 36251, root := ~, start := 36235, stop := 36250 ), rec( content := ". ", count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "P", next := 36257, root := ~, start := 36253, stop := 36256 ), rec( content := "\n\n \ ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( content := " FIXME W\ hat is the correspondence between vertices and elements? ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "XMLCOMMENT" , root := ~ ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "A", next := 36384, root := ~, start := 36373, stop := 36383 ), rec( content := " is not \ present, then the\n generators of ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "A", next := 36436, root := ~, start := 36428, stop := 36435 ), rec( content := " are use\ d by default.", count := [ 3, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "P", next := 36461, root := ~, start := 36457, stop := 36460 ), rec( content := "\n\n \ The digraph created by this operation belongs to the category ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsCayle\ yDigraph" ), content := 0, count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "Ref", next := 36564, root := ~, start := 36529, stop := 36563 ), rec( content := ", the gr\ oup ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "A", next := 36584, root := ~, start := 36576, stop := 36583 ), rec( content := " can be \ recovered from the\n digraph using ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "GroupOf\ CayleyDigraph" ), content := 0, count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "Ref", next := 36663, root := ~, start := 36629, stop := 36662 ), rec( content := ", and th\ e generators\n ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "A", next := 36699, root := ~, start := 36688, stop := 36698 ), rec( content := " can be \ obtained using ", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Generat\ orsOfCayleyDigraph" ), content := 0, count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "Ref", next := 36761, root := ~, start := 36722, stop := 36760 ), rec( content := ".", count := [ 3, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 1, 12, 6 ], mathmode := "Text", name := "P", next := 36766, root := ~, start := 36762, stop := 36765 ), rec( content := "\n\n \ Note that this function can only return an immutable digraph.\n\n ", count := [ 3, 1, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := DihedralGroup(8);\n\ngap> Cay\ leyDigraph(G);\n\ngap> G := Dihed\ ralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> CayleyDigraph(G);\ \n\ngap> digraph := CayleyDigraph\ (G, [()]);\n\ngap> GroupOfCayleyDi\ graph(digraph) = G;\ntrue\ngap> GeneratorsOfCayleyDigraph(digraph);\n[ () ]", count := [ 3, 1, 12, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 1, 12, 7 ], mathmode := "Text", name := "Example", next := 37322, root := ~, start := 36839, stop := 37321 ), rec( content := "\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 37339, root := ~, start := 35811, stop := 37338 ) ], count := [ 3, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 37353, root := ~, start := 35706, stop := 37352 ), rec( content := "\n\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 37367, root := ~, start := 19307, stop := 37366 ), rec( content := "\n\n ", count := [ 3, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Changing represen\ tations", count := [ 3, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 37423, root := ~, start := 37380, stop := 37422 ), rec( content := "\n ", count := [ 3, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "AsBinaryRelatio\ n" ), content := 0, count := [ 3, 2, 1, 2 ], mathmode := "Text", name := "Oper", next := 37488, root := ~, start := 37443, stop := 37487 ), rec( attributes := rec( ), content := [ rec( content := "A binary \ relation.", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "Returns", next := 37528, root := ~, start := 37491, stop := 37527 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "A", next := 37566, root := ~, start := 37552, stop := 37565 ), rec( content := " is a di\ graph with a positive number of vertices\n ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "M", next := 37627, root := ~, start := 37619, stop := 37626 ), rec( content := ", and no\ multiple edges, then this operation returns a binary\n relation on the poi\ nts ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1..n]", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "C", next := 37729, root := ~, start := 37716, stop := 37728 ), rec( content := ".\n T\ he pair ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "C", next := 37756, root := ~, start := 37744, stop := 37755 ), rec( content := " is in t\ he binary relation if and only if\n ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "C", next := 37814, root := ~, start := 37802, stop := 37813 ), rec( content := " is an e\ dge in ", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "A", next := 37843, root := ~, start := 37829, stop := 37842 ), rec( content := ".\n " , count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 1, 4 ], mathmode := "Text", name := "P", next := 37853, root := ~, start := 37849, stop := 37852 ), rec( content := "\n", count := [ 3, 2, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 2], [1, 2], [2], [3, 4]]);\n\ngap> AsBinaryRelation(D);\nBinary Relation on 4 points\n", count := [ 3, 2, 1, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 2, 1, 5 ], mathmode := "Text", name := "Example", next := 38035, root := ~, start := 37854, stop := 38034 ), rec( content := "\n ", count := [ 3, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 1, 3 ], mathmode := "Text", name := "Description", next := 38052, root := ~, start := 37531, stop := 38051 ) ], count := [ 3, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 38066, root := ~, start := 37428, stop := 38065 ), rec( content := "\n\n ", count := [ 3, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]trans[, \ n]", Name := "AsDigraph" ), content := 0, count := [ 3, 2, 2, 2 ], mathmode := "Text", name := "Oper", next := 38136, root := ~, start := 38087, stop := 38135 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ , or ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "K", next := 38173, root := ~, start := 38162, stop := 38172 ), rec( content := ".", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 38184, root := ~, start := 38139, stop := 38183 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "A", next := 38247, root := ~, start := 38236, stop := 38246 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "A", next := 38393, root := ~, start := 38382, stop := 38392 ), rec( content := " is ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Ref", next := 38427, root := ~, start := 38397, stop := 38426 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "A", next := 38495, root := ~, start := 38484, stop := 38494 ), rec( content := " is ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Ref", next := 38535, root := ~, start := 38499, stop := 38534 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "A", next := 38620, root := ~, start := 38609, stop := 38619 ), rec( content := " is not \ present, then ", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Ref", next := 38678, root := ~, start := 38642, stop := 38677 ), rec( content := " is used\ by default.", count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "P", next := 38702, root := ~, start := 38698, stop := 38701 ), rec( content := "\n\n \ If ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 38723, root := ~, start := 38711, stop := 38722 ), rec( content := " is a tr\ ansformation, and ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 38757, root := ~, start := 38749, stop := 38756 ), rec( content := " is a no\ n-negative integer\n such that the restriction of ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 38829, root := ~, start := 38817, stop := 38828 ), rec( content := " to ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 4 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := \ "Text", name := "A", next := 38850, root := ~, start := 38842, stop := 38849 ) , rec( content := "\ ]", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 38855, root := ~, start := 38833, stop := 38854 ), rec( content := " defines\ \n a transformation of ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 4 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := \ "Text", name := "A", next := 38905, root := ~, start := 38897, stop := 38904 ) , rec( content := "\ ]", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 38910, root := ~, start := 38888, stop := 38909 ), rec( content := ", then " , count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sDigraph", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "C", next := 38933, root := ~, start := 38917, stop := 38932 ), rec( content := "\n re\ turns the functional digraph with ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 38982, root := ~, start := 38974, stop := 38981 ), rec( content := " vertice\ s defined by\n ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "A", next := 39019, root := ~, start := 39007, stop := 39018 ), rec( content := ". See ", count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsFunct\ ionalDigraph" ), content := 0, count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "Ref", next := 39058, root := ~, start := 39025, stop := 39057 ), rec( content := ".\n " , count := [ 3, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "P", next := 39068, root := ~, start := 39064, stop := 39067 ), rec( content := "\n\n \ Specifically, the digraph returned by ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sDigraph", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 39128, root := ~, start := 39112, stop := 39127 ), rec( content := " has ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "A", next := 39142, root := ~, start := 39134, stop := 39141 ), rec( content := " edges:\\ n for each vertex ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 39178, root := ~, start := 39170, stop := 39177 ), rec( content := " in ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 5 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := \ "Text", name := "A", next := 39199, root := ~, start := 39191, stop := 39198 ) , rec( content := "\ ]", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 39204, root := ~, start := 39182, stop := 39203 ), rec( content := ", there \ is a unique edge\n with source ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 39253, root := ~, start := 39245, stop := 39252 ), rec( content := "; this e\ dge has range ", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ^", count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes :\ = rec( ), content := [ rec( content := "trans", count := [ 3, 2, 2, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 5 ], mathmode := \ "Text", name := "A", next := 39292, root := ~, start := 39280, stop := 39291 ) ], count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "C", next := 39296, root := ~, start := 39275, stop := 39295 ), rec( content := ".\n " , count := [ 3, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "P", next := 39306, root := ~, start := 39302, stop := 39305 ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "A", next := 39352, root := ~, start := 39344, stop := 39351 ), rec( content := " is not \ supplied, then the degree of\n the transformation ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "A", next := 39424, root := ~, start := 39412, stop := 39423 ), rec( content := " is used\ by default. If the restriction of\n ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "A", next := 39484, root := ~, start := 39472, stop := 39483 ), rec( content := " to ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 6 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := \ "Text", name := "A", next := 39505, root := ~, start := 39497, stop := 39504 ) , rec( content := "\ ]", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 39510, root := ~, start := 39488, stop := 39509 ), rec( content := " does no\ t define a transformation of\n ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 6 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := \ "Text", name := "A", next := 39568, root := ~, start := 39560, stop := 39567 ) , rec( content := "\ ]", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 39573, root := ~, start := 39551, stop := 39572 ), rec( content := ", then " , count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sDigraph(", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "trans", count := [ 3, 2, 2, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := \ "Text", name := "A", next := 39605, root := ~, start := 39593, stop := 39604 ) , rec( content := "\ , ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 2, 2, 6 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := \ "Text", name := "A", next := 39615, root := ~, start := 39607, stop := 39614 ) , rec( content := "\ )", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "C", next := 39620, root := ~, start := 39580, stop := 39619 ), rec( content := "\n re\ turns ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "K", next := 39644, root := ~, start := 39633, stop := 39643 ), rec( content := ".\n\n \ ", count := [ 3, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> f := Transformation([4, 3, 3, 1, 7, 9, 10, 4, 2, 3]);\nTransformation( [\ 4, 3, 3, 1, 7, 9, 10, 4, 2, 3 ] )\ngap> AsDigraph(f);\n\ngap> AsDigraph(f, 4);\n\ngap> AsDigraph(f, 5);\nfail\ngap> b := BinaryRelationOnPoi\ nts(\n> [[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\nBinary Relation on 5 po\ ints\ngap> D := AsDigraph(b);\n\n\ ", count := [ 3, 2, 2, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 7 ], mathmode := "Text", name := "Example", next := 40137, root := ~, start := 39651, stop := 40136 ), rec( content := "\n ", count := [ 3, 2, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 40154, root := ~, start := 38187, stop := 40153 ) ], count := [ 3, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 40168, root := ~, start := 38072, stop := 40167 ), rec( content := "\n\n ", count := [ 3, 2, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "Graph" ), content := 0, count := [ 3, 2, 3, 2 ], mathmode := "Text", name := "Oper", next := 40223, root := ~, start := 40189, stop := 40222 ), rec( attributes := rec( ), content := [ rec( content := "A ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 3, 2, 3, 3 ] , name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 3, 2, 3, 3 ], name \ := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ graph.", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Returns", next := 40269, root := ~, start := 40226, stop := 40268 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "A", next := 40307, root := ~, start := 40293, stop := 40306 ), rec( content := " is a mu\ table or immutable digraph without multiple edges, then this operation\n re\ turns a ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 3, 2, 3, 3 ] , name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 3, 2, 3, 3 ], name \ := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ graph that is isomorphic to ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "A", next := 40458, root := ~, start := 40444, stop := 40457 ), rec( content := ". ", count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "P", next := 40464, root := ~, start := 40460, stop := 40463 ), rec( content := "\n\n \ If ", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "A", next := 40487, root := ~, start := 40473, stop := 40486 ), rec( content := " is a mu\ ltidigraph, then since ", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 3, 2, 3, 4 ] , name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 3, 2, 3, 4 ], name \ := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " does no\ t support\n multiple edges, the multiple edges will be reduced to a single \ edge in the\n result. In order words, for a multidigraph this operation wil\ l return the\n same as ", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raph(DigraphRemoveAllMultipleEdges(", count := [ 3, 2, 3, 4 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "C", next := 40755, root := ~, start := 40712, stop := 40754 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "A", next := 40769, root := ~, start := 40755, stop := 40768 ), rec( attributes := rec( ), content := [ rec( content := ")\ )", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "C", next := 40778, root := ~, start := 40769, stop := 40777 ), rec( content := ".\n\n \ ", count := [ 3, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x, y)\ return Intersection(x, y) = []; end);;\ngap> Display(Petersen);\nrec(\n adja\ cencies := [ [ 3, 5, 8 ] ],\n group := \n Group( [ ( 1, 2, 3, 5, 7)( 4, 6, \ 8, 9,10), ( 2, 4)( 6, 9)( 7,10) \n ] ),\n isGraph := true,\n names := [ \ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 3 ], [ 4, 5 ], \n [ 2, 4 ], [ 1, 5 ], \ [ 3, 5 ], [ 1, 4 ], [ 2, 5 ] ],\n order := 10,\n representatives := [ 1 ],\n\ schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 1, 2, 2 ] )\ngap> Digraph(Petersen\ );\n\ngap> Graph(last) = Peterse\ n;\ntrue", count := [ 3, 2, 3, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 5 ], mathmode := "Text", name := "Example", next := 41431, root := ~, start := 40785, stop := 41430 ), rec( content := "\n ", count := [ 3, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 3, 3 ], mathmode := "Text", name := "Description", next := 41448, root := ~, start := 40272, stop := 41447 ) ], count := [ 3, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 41462, root := ~, start := 40174, stop := 41461 ), rec( content := "\n\n ", count := [ 3, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "AsGraph" ), content := 0, count := [ 3, 2, 4, 2 ], mathmode := "Text", name := "Attr", next := 41519, root := ~, start := 41483, stop := 41518 ), rec( attributes := rec( ), content := [ rec( content := "A ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ 3, 2, 4, 3 ] , name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ 3, 2, 4, 3 ], name \ := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " package\ graph.", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 41565, root := ~, start := 41522, stop := 41564 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "A", next := 41603, root := ~, start := 41589, stop := 41602 ), rec( content := " is a di\ graph, then this method returns the same as\n ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Graph" ), content := 0, count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Ref", next := 41678, root := ~, start := 41659, stop := 41677 ), rec( content := ", except\ that if ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "A", next := 41709, root := ~, start := 41695, stop := 41708 ), rec( content := " is immu\ table, then the\n result will be stored as a mutable attribute of ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "A", next := 41799, root := ~, start := 41785, stop := 41798 ), rec( content := ".\n I\ n this latter case, when ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sGraph(", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "C", next := 41846, root := ~, start := 41831, stop := 41845 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "A", next := 41860, root := ~, start := 41846, stop := 41859 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "C", next := 41868, root := ~, start := 41860, stop := 41867 ), rec( content := " is call\ ed\n subsequently, the same ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object \ will be returned as before.\n\n ", count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [3], []]);\n\ngap> G := AsGraph(D);\nrec( adjacencies := [ [ 1, 2 ], [ 3 ], [ ] ], \ group := Group(()), \n isGraph := true, names := [ 1 .. 3 ], order := 3, \n \ representatives := [ 1, 2, 3 ], schreierVector := [ -1, -2, -3 ] )\n", count := [ 3, 2, 4, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 2, 4, 4 ], mathmode := "Text", name := "Example", next := 42279, root := ~, start := 41952, stop := 42278 ), rec( content := "\n ", count := [ 3, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 42296, root := ~, start := 41568, stop := 42295 ) ], count := [ 3, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 42310, root := ~, start := 41468, stop := 42309 ), rec( content := "\n\n ", count := [ 3, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "AsTransformatio\ n" ), content := 0, count := [ 3, 2, 5, 2 ], mathmode := "Text", name := "Attr", next := 42376, root := ~, start := 42331, stop := 42375 ), rec( attributes := rec( ), content := [ rec( content := "A transfo\ rmation, or ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "K", next := 42420, root := ~, start := 42409, stop := 42419 ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Returns", next := 42430, root := ~, start := 42379, stop := 42429 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "A", next := 42468, root := ~, start := 42454, stop := 42467 ), rec( content := " is a fu\ nctional digraph, then ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sTransformation", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "C", next := 42522, root := ~, start := 42499, stop := 42521 ), rec( content := "\n re\ turns the transformation which is defined by ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "A", next := 42588, root := ~, start := 42574, stop := 42587 ), rec( content := ". See " , count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsFunct\ ionalDigraph" ), content := 0, count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Ref", next := 42634, root := ~, start := 42595, stop := 42633 ), rec( content := ". Otherw\ ise,\n ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sTransformation(", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "C", next := 42675, root := ~, start := 42651, stop := 42674 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "A", next := 42689, root := ~, start := 42675, stop := 42688 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "C", next := 42697, root := ~, start := 42689, stop := 42696 ), rec( content := " returns\ ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "K", next := 42717, root := ~, start := 42706, stop := 42716 ), rec( content := ". ", count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "P", next := 42724, root := ~, start := 42720, stop := 42723 ), rec( content := "\n\n \ If ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "A", next := 42747, root := ~, start := 42733, stop := 42746 ), rec( content := " is a fu\ nctional digraph with ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "M", next := 42785, root := ~, start := 42777, stop := 42784 ), rec( content := " vertice\ s, then\n ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sTransformation(", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 42829, root := ~, start := 42805, stop := 42828 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "A", next := 42843, root := ~, start := 42829, stop := 42842 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 42851, root := ~, start := 42843, stop := 42850 ), rec( content := " will re\ turn the\n transformation ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 42895, root := ~, start := 42887, stop := 42894 ), rec( content := " of degr\ ee at most ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "M", next := 42922, root := ~, start := 42914, stop := 42921 ), rec( content := " where f\ or each ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ \n \\leq i \\leq n", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "M", next := 42966, root := ~, start := 42938, stop := 42965 ), rec( content := ", ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ^ f", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 42980, root := ~, start := 42968, stop := 42979 ), rec( content := " is equa\ l to the unique out-neighbour of\n vertex ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "C", next := 43040, root := ~, start := 43032, stop := 43039 ), rec( content := " in ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "A", next := 43058, root := ~, start := 43044, stop := 43057 ), rec( content := ".\n\n \ ", count := [ 3, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1], [3], [2]]);\n\ngap> AsTransformation(D);\nTransformation( [ 1, 3, 2 ] )\ngap> D := Cycl\ eDigraph(IsMutableDigraph, 3);\n\nga\ p> AsTransformation(D);\nTransformation( [ 2, 3, 1 ] )\ngap> AsPermutation(las\ t);\n(1,2,3)\ngap> D := Digraph([[2, 3], [], []]);\n\ngap> AsTransformation(D);\nfail", count := [ 3, 2, 5, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 2, 5, 5 ], mathmode := "Text", name := "Example", next := 43524, root := ~, start := 43065, stop := 43523 ), rec( content := "\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 43541, root := ~, start := 42433, stop := 43540 ) ], count := [ 3, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 43555, root := ~, start := 42316, stop := 43554 ), rec( content := "\n\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 43569, root := ~, start := 37371, stop := 43568 ), rec( content := "\n\n ", count := [ 3, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "New digraphs from\ old", count := [ 3, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 43622, root := ~, start := 43582, stop := 43621 ), rec( content := "\n ", count := [ 3, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphImmutabl\ eCopy" ), content := 0, count := [ 3, 3, 1, 2 ], mathmode := "Text", name := "Oper", next := 43691, root := ~, start := 43642, stop := 43690 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opy" ), content := 0, count := [ 3, 3, 1, 4 ], mathmode := "Text", name := "Oper", next := 43741, root := ~, start := 43694, stop := 43740 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphCopySame\ Mutability" ), content := 0, count := [ 3, 3, 1, 6 ], mathmode := "Text", name := "Oper", next := 43798, root := ~, start := 43744, stop := 43797 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphCopy" ), content := 0, count := [ 3, 3, 1, 8 ], mathmode := "Text", name := "Oper", next := 43841, root := ~, start := 43801, stop := 43840 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "Returns", next := 43873, root := ~, start := 43844, stop := 43872 ), rec( attributes := rec( ), content := [ rec( content := "\n Eac\ h of these operations returns a new copy of ", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "A", next := 43955, root := ~, start := 43941, stop := 43954 ), rec( content := ",\n o\ f the appropriate mutability, retaining none of the attributes or\n propert\ ies of ", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "A", next := 44060, root := ~, start := 44046, stop := 44059 ), rec( content := ". ", count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "P", next := 44066, root := ~, start := 44062, stop := 44065 ), rec( content := "\n\n \ ", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ igraphCopy", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "C", next := 44090, root := ~, start := 44072, stop := 44089 ), rec( content := " is a sy\ nonym for ", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCopySameMutability", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "C", next := 44140, root := ~, start := 44108, stop := 44139 ), rec( content := ".\n", count := [ 3, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(10);\n\ngap>\ DigraphCopy(D) = D;\ntrue\ngap> IsIdenticalObj(DigraphCopy(D), D);\nfalse\nga\ p> DigraphMutableCopy(D);\n\n", count := [ 3, 3, 1, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 1, 11 ], mathmode := "Text", name := "Example", next := 44394, root := ~, start := 44142, stop := 44393 ), rec( content := "\n ", count := [ 3, 3, 1, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 1, 9 ], mathmode := "Text", name := "Description", next := 44411, root := ~, start := 43876, stop := 44410 ) ], count := [ 3, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 44425, root := ~, start := 43627, stop := 44424 ), rec( content := "\n\n ", count := [ 3, 3, 1, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphImmutabl\ eCopyIfImmutable" ), content := 0, count := [ 3, 3, 2, 2 ], mathmode := "Text", name := "Oper", next := 44506, root := ~, start := 44446, stop := 44505 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphImmutabl\ eCopyIfMutable" ), content := 0, count := [ 3, 3, 2, 4 ], mathmode := "Text", name := "Oper", next := 44567, root := ~, start := 44509, stop := 44566 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opyIfMutable" ), content := 0, count := [ 3, 3, 2, 6 ], mathmode := "Text", name := "Oper", next := 44626, root := ~, start := 44570, stop := 44625 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMutableC\ opyIfImmutable" ), content := 0, count := [ 3, 3, 2, 8 ], mathmode := "Text", name := "Oper", next := 44687, root := ~, start := 44629, stop := 44686 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "Returns", next := 44719, root := ~, start := 44690, stop := 44718 ), rec( attributes := rec( ), content := [ rec( content := "\n Eac\ h of these operations returns either the original argument\n ", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "A", next := 44820, root := ~, start := 44806, stop := 44819 ), rec( content := ", or a n\ ew copy of ", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "A", next := 44853, root := ~, start := 44839, stop := 44852 ), rec( content := " of the \ appropriate\n mutability, according to the mutability of ", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "A", next := 44934, root := ~, start := 44920, stop := 44933 ), rec( content := ".\n", count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> C := CycleDigraph(10);\n\ngap>\ D := DigraphImmutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\nfalse\ngap> C = D;\ntrue\ngap> D := D\ igraphImmutableCopyIfMutable(C);\n\n\ gap> IsIdenticalObj(C, D);\ntrue\ngap> C = D;\ntrue\ngap> D := DigraphMutableC\ opyIfMutable(C);\n\ngap> IsMutableDi\ graph(D);\nfalse\ngap> D := DigraphMutableCopyIfImmutable(C);\n\ngap> IsMutableDigraph(D);\ntrue\ngap> C := Cycl\ eDigraph(IsMutableDigraph, 10);\n\ \ngap> D := DigraphImmutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\ntrue\ngap> C = D;\ntrue\ngap> D :\ = DigraphImmutableCopyIfMutable(C);\n\ngap> IsIdenticalObj(C, D);\nfalse\ngap> C = D;\ntrue\ngap> D := Digraph\ MutableCopyIfMutable(C);\n\ngap> I\ sMutableDigraph(D);\ntrue\ngap> D := DigraphMutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\ntrue\ngap>\ IsMutableDigraph(D);\ntrue", count := [ 3, 3, 2, 10 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 10 ], mathmode := "Text", name := "Example", next := 46196, root := ~, start := 44936, stop := 46195 ), rec( content := "\n ", count := [ 3, 3, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 2, 9 ], mathmode := "Text", name := "Description", next := 46213, root := ~, start := 44722, stop := 46212 ) ], count := [ 3, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 46227, root := ~, start := 44431, stop := 46226 ), rec( content := "\n\n ", count := [ 3, 3, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, verts", Name := "InducedSubdigra\ ph" ), content := 0, count := [ 3, 3, 3, 2 ], mathmode := "Text", name := "Oper", next := 46301, root := ~, start := 46248, stop := 46300 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "Returns", next := 46333, root := ~, start := 46304, stop := 46332 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 46371, root := ~, start := 46357, stop := 46370 ), rec( content := " is a di\ graph, and ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 46402, root := ~, start := 46390, stop := 46401 ), rec( content := " is a su\ bset of the\n vertices of ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 46452, root := ~, start := 46438, stop := 46451 ), rec( content := ", then t\ his operation returns a digraph\n constructed from ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 46527, root := ~, start := 46513, stop := 46526 ), rec( content := " by reta\ ining precisely those vertices in\n ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 46585, root := ~, start := 46573, stop := 46584 ), rec( content := ", and th\ ose edges whose source and range vertices are both\n contained in ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "A", next := 46673, root := ~, start := 46661, stop := 46672 ), rec( content := ". ", count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "P", next := 46679, root := ~, start := 46675, stop := 46678 ), rec( content := "\n\n \ The vertices of the induced subdigraph are\n ", count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1..Length(verts)]", count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "C", next := 46757, root := ~, start := 46732, stop := 46756 ), rec( content := " but the\ original vertex labels can be\n accessed via ", count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "Ref", next := 46846, root := ~, start := 46813, stop := 46845 ), rec( content := ". ", count := [ 3, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "P", next := 46852, root := ~, start := 46848, stop := 46851 ), rec( content := "\n\n \ If ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "A", next := 46875, root := ~, start := 46861, stop := 46874 ), rec( content := " belongs\ to ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "Ref", next := 46917, root := ~, start := 46887, stop := 46916 ), rec( content := ", then\n\ ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "A", next := 46942, root := ~, start := 46928, stop := 46941 ), rec( content := " is modi\ fied in place. If ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "A", next := 46982, root := ~, start := 46968, stop := 46981 ), rec( content := " belongs\ to ", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "Ref", next := 47032, root := ~, start := 46994, stop := 47031 ), rec( content := ", a new \ immutable digraph containing the\n appropriate\n vertices and edges is r\ eturned.", count := [ 3, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 3, 6 ], mathmode := "Text", name := "P", next := 47128, root := ~, start := 47124, stop := 47127 ), rec( content := "\n\n \ ", count := [ 3, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 1, 2, 3, 4, 4], [1, 3, 4], [3, 1], [1, 1]]);\n\ngap> InducedSubdigraph(D, [1, 3, 4\ ]);\n\ngap> DigraphVertices(l\ ast);\n[ 1 .. 3 ]\ngap> D := DigraphMutableCopy(D);\n\ngap> new := InducedSubdigraph(D, [1, 3, 4]);\n\ngap> D = new;\ntrue\n", count := [ 3, 3, 3, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 3, 7 ], mathmode := "Text", name := "Example", next := 47606, root := ~, start := 47134, stop := 47605 ), rec( content := "\n ", count := [ 3, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 47623, root := ~, start := 46336, stop := 47622 ) ], count := [ 3, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 47637, root := ~, start := 46233, stop := 47636 ), rec( content := "\n\n ", count := [ 3, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "ReducedDigraph"\ ), content := 0, count := [ 3, 3, 4, 2 ], mathmode := "Text", name := "Oper", next := 47701, root := ~, start := 47658, stop := 47700 ), rec( attributes := rec( Arg := "digraph", Name := "ReducedDigraphA\ ttr" ), content := 0, count := [ 3, 3, 4, 4 ], mathmode := "Text", name := "Attr", next := 47751, root := ~, start := 47704, stop := 47750 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "Returns", next := 47783, root := ~, start := 47754, stop := 47782 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a digraph isomorphic to the subdigraph of\n ", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "A", next := 47886, root := ~, start := 47872, stop := 47885 ), rec( content := " induced\ by the set of non-isolated vertices,\n i.e. the set of those vertices of " , count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "A", next := 47984, root := ~, start := 47970, stop := 47983 ), rec( content := " which a\ re the source or\n range of some edge in ", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "A", next := 48049, root := ~, start := 48035, stop := 48048 ), rec( content := ".\n S\ ee ", count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Induced\ Subdigraph" ), content := 0, count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "Ref", next := 48090, root := ~, start := 48059, stop := 48089 ), rec( content := ".\n " , count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "P", next := 48100, root := ~, start := 48096, stop := 48099 ), rec( content := "\n\n \ The ordering of the remaining vertices of ", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "A", next := 48162, root := ~, start := 48148, stop := 48161 ), rec( content := " is pres\ erved, as are\n the labels of the remaining vertices and edges;\n see ", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "Ref", next := 48277, root := ~, start := 48244, stop := 48276 ), rec( content := " and ", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ EdgeLabels" ), content := 0, count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "Ref", next := 48313, root := ~, start := 48282, stop := 48312 ), rec( content := ".\n T\ his can allow one to match a vertex in the reduced digraph to the\n corresp\ onding vertex in ", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "A", next := 48428, root := ~, start := 48414, stop := 48427 ), rec( content := ".\n " , count := [ 3, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "P", next := 48438, root := ~, start := 48434, stop := 48437 ), rec( content := "\n\n \ If ", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "A", next := 48461, root := ~, start := 48447, stop := 48460 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the isolated \ vertices of the mutable digraph ", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "A", next := 48592, root := ~, start := 48578, stop := 48591 ), rec( content := " are\n \ removed in-place.\n ", count := [ 3, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [], [], [1, 4], []]);\n\ngap> R := ReducedDigraph(D);\n\ngap> OutNeighbours(R);\n[ [ 1, 2 ], [ ], [ 1, 3 ] ]\ngap> Di\ graphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 4, 1 ], [ 4, 4 ] ]\ngap> DigraphEdges(\ R);\n[ [ 1, 1 ], [ 1, 2 ], [ 3, 1 ], [ 3, 3 ] ]\ngap> DigraphVertexLabel(R, 3)\ ;\n4\ngap> DigraphVertexLabel(R, 2);\n2\ngap> D := Digraph(IsMutableDigraph, [\ [], [3], [2]]);\n\ngap> ReducedDigra\ ph(D);\n\ngap> D;\n\n", count := [ 3, 3, 4, 8 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 8 ], mathmode := "Text", name := "Example", next := 49285, root := ~, start := 48623, stop := 49284 ), rec( content := "\n ", count := [ 3, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 4, 5 ], mathmode := "Text", name := "Description", next := 49302, root := ~, start := 47786, stop := 49301 ) ], count := [ 3, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 49316, root := ~, start := 47643, stop := 49315 ), rec( content := "\n\n ", count := [ 3, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraph" ), content := 0, count := [ 3, 3, 5, 2 ], mathmode := "Text", name := "Oper", next := 49392, root := ~, start := 49337, stop := 49391 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphAttr" ), content := 0, count := [ 3, 3, 5, 4 ], mathmode := "Text", name := "Attr", next := 49454, root := ~, start := 49395, stop := 49453 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphWithoutLoops" ), content := 0, count := [ 3, 3, 5, 6 ], mathmode := "Text", name := "Oper", next := 49524, root := ~, start := 49457, stop := 49523 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalSymmetri\ cSubdigraphWithoutLoopsAttr" ), content := 0, count := [ 3, 3, 5, 8 ], mathmode := "Text", name := "Attr", next := 49598, root := ~, start := 49527, stop := 49597 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "Returns", next := 49630, root := ~, start := 49601, stop := 49629 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "A", next := 49668, root := ~, start := 49654, stop := 49667 ), rec( content := " is a di\ graph, then ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximalSymmetricSubdigraph", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "C", next := 49721, root := ~, start := 49688, stop := 49720 ), rec( content := "\n re\ turns a symmetric digraph without multiple edges which has the same\n verte\ x set as ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "A", next := 49828, root := ~, start := 49814, stop := 49827 ), rec( content := ", and wh\ ose edge list is formed from\n ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "A", next := 49883, root := ~, start := 49869, stop := 49882 ), rec( content := " by igno\ ring the multiplicity of edges, and by ignoring\n edges ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u,v]", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "C", next := 49961, root := ~, start := 49949, stop := 49960 ), rec( content := " for whi\ ch there does not exist an edge ", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v,u]", count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "C", next := 50013, root := ~, start := 50001, stop := 50012 ), rec( content := ".\n " , count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 5, 10 ], mathmode := "Text", name := "P", next := 50023, root := ~, start := 50019, stop := 50022 ), rec( content := "\n\n \ The digraph returned by ", count := [ 3, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximalSymmetricSubdigraphWithoutLoops", count := [ 3, 3, 5, 10 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 10 ], mathmode := "Text", name := "C", next := 50098, root := ~, start := 50053, stop := 50097 ), rec( content := " is\n \ the same, except that loops are removed.", count := [ 3, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "P", next := 50150, root := ~, start := 50146, stop := 50149 ), rec( content := "\n \n\ If ", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "A", next := 50177, root := ~, start := 50163, stop := 50176 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "A", next := 50283, root := ~, start := 50269, stop := 50282 ), rec( content := " is chan\ ged in-place into such\n a digraph described above.", count := [ 3, 3, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "P", next := 50348, root := ~, start := 50344, stop := 50347 ), rec( content := "\n\n \ See ", count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "Ref", next := 50390, root := ~, start := 50358, stop := 50389 ), rec( content := ", ", count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "Ref", next := 50420, root := ~, start := 50392, stop := 50419 ), rec( content := ", and\n \ ", count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "Ref", next := 50459, root := ~, start := 50430, stop := 50458 ), rec( content := " for mor\ e information.\n ", count := [ 3, 3, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsSymmetricDigraph(D) and IsMultiDigraph(D);\ \ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]\ngap> \ S := MaximalSymmetricSubdigraph(D);\n\ngap> IsSymmetricDigraph(S) and not IsMultiDigraph(S);\ntrue\nga\ p> OutNeighbours(S);\n[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ]\ngap> D := CycleDigraph(I\ sMutableDigraph, 3);\n\ngap> Maximal\ SymmetricSubdigraph(D);\n\ngap> D;\n\n", count := [ 3, 3, 5, 13 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 5, 13 ], mathmode := "Text", name := "Example", next := 51161, root := ~, start := 50486, stop := 51160 ), rec( content := "\n ", count := [ 3, 3, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 5, 9 ], mathmode := "Text", name := "Description", next := 51178, root := ~, start := 49633, stop := 51177 ) ], count := [ 3, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 51192, root := ~, start := 49322, stop := 51191 ), rec( content := "\n\n ", count := [ 3, 3, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "MaximalAntiSymm\ etricSubdigraph" ), content := 0, count := [ 3, 3, 6, 2 ], mathmode := "Text", name := "Oper", next := 51272, root := ~, start := 51213, stop := 51271 ), rec( attributes := rec( Arg := "digraph", Name := "MaximalAntiSymm\ etricSubdigraphAttr" ), content := 0, count := [ 3, 3, 6, 4 ], mathmode := "Text", name := "Attr", next := 51338, root := ~, start := 51275, stop := 51337 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "Returns", next := 51370, root := ~, start := 51341, stop := 51369 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 51408, root := ~, start := 51394, stop := 51407 ), rec( content := " is a di\ graph, then ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximalAntiSymmetricSubdigraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51465, root := ~, start := 51428, stop := 51464 ), rec( content := "\n re\ turns an anti-symmetric subdigraph of ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 51524, root := ~, start := 51510, stop := 51523 ), rec( content := " formed \ by\n retaining the vertices of ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 51579, root := ~, start := 51565, stop := 51578 ), rec( content := ", discar\ ding any duplicate edges,\n and discarding any edge ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51653, root := ~, start := 51641, stop := 51652 ), rec( content := " of ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 51671, root := ~, start := 51657, stop := 51670 ), rec( content := " where " , count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ > j", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51690, root := ~, start := 51678, stop := 51689 ), rec( content := "\n an\ d the reverse edge ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j,i]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51728, root := ~, start := 51716, stop := 51727 ), rec( content := " is an e\ dge of ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 51757, root := ~, start := 51743, stop := 51756 ), rec( content := ".\n I\ n other words, for every\n symmetric pair of edges ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51829, root := ~, start := 51817, stop := 51828 ), rec( content := " and ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j,i]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51846, root := ~, start := 51834, stop := 51845 ), rec( content := " in ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "A", next := 51864, root := ~, start := 51850, stop := 51863 ), rec( content := ", where \ \n ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51885, root := ~, start := 51877, stop := 51884 ), rec( content := " and ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "C", next := 51898, root := ~, start := 51890, stop := 51897 ), rec( content := " are dis\ tinct, it discards the the edge\n ", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ \\max(i,j),\\min(i,j)]", count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "M", next := 51970, root := ~, start := 51942, stop := 51969 ), rec( content := ".\n " , count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "P", next := 51980, root := ~, start := 51976, stop := 51979 ), rec( content := "\n\n \ If ", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "A", next := 52003, root := ~, start := 51989, stop := 52002 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "A", next := 52109, root := ~, start := 52095, stop := 52108 ), rec( content := " is chan\ ged in-place. ", count := [ 3, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 6, 7 ], mathmode := "Text", name := "P", next := 52135, root := ~, start := 52131, stop := 52134 ), rec( content := "\n\n \ See ", count := [ 3, 3, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsAntis\ ymmetricDigraph" ), content := 0, count := [ 3, 3, 6, 7 ], mathmode := "Text", name := "Ref", next := 52181, root := ~, start := 52145, stop := 52180 ), rec( content := " for mor\ e information.\n ", count := [ 3, 3, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsAntiSymmetricDigraph(D) and IsMultiDigraph\ (D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]\ng\ ap> D := MaximalAntiSymmetricSubdigraph(D);\n\ngap> IsAntiSymmetricDigraph(D) and not IsMultiDigra\ ph(D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 4 ], [ 1 ] ]\ngap> D :\ = Digraph(IsMutableDigraph, [[2], [1]]);\n\ngap> MaximalAntiSymmetricSubdigraph(D);\n\ngap> D;\n\n", count := [ 3, 3, 6, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 6, 8 ], mathmode := "Text", name := "Example", next := 52911, root := ~, start := 52208, stop := 52910 ), rec( content := "\n ", count := [ 3, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 6, 5 ], mathmode := "Text", name := "Description", next := 52928, root := ~, start := 51373, stop := 52927 ) ], count := [ 3, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 52942, root := ~, start := 51198, stop := 52941 ), rec( content := "\n\n ", count := [ 3, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingForest" ), content := 0, count := [ 3, 3, 7, 2 ], mathmode := "Text", name := "Oper", next := 53016, root := ~, start := 52963, stop := 53015 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingForestAttr" ), content := 0, count := [ 3, 3, 7, 4 ], mathmode := "Text", name := "Attr", next := 53076, root := ~, start := 53019, stop := 53075 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingTree" ), content := 0, count := [ 3, 3, 7, 6 ], mathmode := "Text", name := "Oper", next := 53130, root := ~, start := 53079, stop := 53129 ), rec( attributes := rec( Arg := "digraph", Name := "UndirectedSpann\ ingTreeAttr" ), content := 0, count := [ 3, 3, 7, 8 ], mathmode := "Text", name := "Attr", next := 53188, root := ~, start := 53133, stop := 53187 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ , or ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "K", next := 53225, root := ~, start := 53214, stop := 53224 ), rec( content := ".", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Returns", next := 53236, root := ~, start := 53191, stop := 53235 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "A", next := 53274, root := ~, start := 53260, stop := 53273 ), rec( content := " is a di\ graph with at least one vertex, then\n ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedSpanningForest", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "C", next := 53354, root := ~, start := 53323, stop := 53353 ), rec( content := " returns\ an undirected spanning forest of\n ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "A", next := 53414, root := ~, start := 53400, stop := 53413 ), rec( content := ", otherw\ ise this attribute returns ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "K", next := 53460, root := ~, start := 53449, stop := 53459 ), rec( content := ". See ", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsUndir\ ectedSpanningForest" ), content := 0, count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Ref", next := 53513, root := ~, start := 53466, stop := 53512 ), rec( content := " for the\ definition of an undirected\n spanning forest.", count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "P", next := 53574, root := ~, start := 53570, stop := 53573 ), rec( content := "\n\n \ If ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "A", next := 53597, root := ~, start := 53583, stop := 53596 ), rec( content := " is a di\ graph with at least one vertex and whose ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraph" ), content := 0, count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "Ref", next := 53692, root := ~, start := 53646, stop := 53691 ), rec( content := " is conn\ ected (see ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "Ref", next := 53750, root := ~, start := 53711, stop := 53749 ), rec( content := "), then \ ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedSpanningTree", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "C", next := 53787, root := ~, start := 53758, stop := 53786 ), rec( content := " returns\ \n an undirected spanning tree of ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "A", next := 53845, root := ~, start := 53831, stop := 53844 ), rec( content := ", otherw\ ise this attribute\n returns ", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "K", next := 53895, root := ~, start := 53884, stop := 53894 ), rec( content := ". See " , count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsUndir\ ectedSpanningTree" ), content := 0, count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "Ref", next := 53941, root := ~, start := 53902, stop := 53940 ), rec( content := " for the\ \n definition of an undirected spanning tree.", count := [ 3, 3, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "P", next := 54000, root := ~, start := 53996, stop := 53999 ), rec( content := "\n\n \ If ", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "A", next := 54023, root := ~, start := 54009, stop := 54022 ), rec( content := " is immu\ table, then an immutable digraph is returned.\n Otherwise, the mutable digr\ aph ", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "A", next := 54126, root := ~, start := 54112, stop := 54125 ), rec( content := " is chan\ ged in-place into an\n undirected spanning tree of ", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "A", next := 54201, root := ~, start := 54187, stop := 54200 ), rec( content := ".", count := [ 3, 3, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "P", next := 54206, root := ~, start := 54202, stop := 54205 ), rec( content := "\n\n \ Note that for an immutable digraph that has known undirected spanning tree,\n \ the attribute ", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedSpanningTree", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "C", next := 54335, root := ~, start := 54306, stop := 54334 ), rec( content := " returns\ the same digraph as\n the attribute ", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedSpanningForest", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "C", next := 54413, root := ~, start := 54382, stop := 54412 ), rec( content := ".\n\n \ ", count := [ 3, 3, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 1, 3], [1], [4], [3, 4, 3]]);\n\ngap> UndirectedSpanningTree(D);\nfail\ngap> for\ est := UndirectedSpanningForest(D);\n\ngap> OutNeighbours(forest);\n[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ]\nga\ p> IsUndirectedSpanningForest(D, forest);\ntrue\ngap> DigraphConnectedComponen\ ts(forest).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ngap> DigraphConnectedComponents(Max\ imalSymmetricSubdigraph(D)).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ngap> UndirectedSpa\ nningForest(MaximalSymmetricSubdigraph(D))\n> = forest;\ntrue\ngap> D := Compl\ eteDigraph(4);\n\ngap> tree := Und\ irectedSpanningTree(D);\n\ngap> IsUndirectedSpanningTree(D, tree);\ntrue\ngap> tree = UndirectedSpanni\ ngForest(D);\ntrue\ngap> UndirectedSpanningForest(EmptyDigraph(0));\nfail\ngap\ > D := PetersenGraph(IsMutableDigraph);\n\ngap> UndirectedSpanningTree(D);\n\ngap> D;\n\n", count := [ 3, 3, 7, 13 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 7, 13 ], mathmode := "Text", name := "Example", next := 55575, root := ~, start := 54420, stop := 55574 ), rec( content := "\n ", count := [ 3, 3, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 7, 9 ], mathmode := "Text", name := "Description", next := 55592, root := ~, start := 53239, stop := 55591 ) ], count := [ 3, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 55606, root := ~, start := 52948, stop := 55605 ), rec( content := "\n\n ", count := [ 3, 3, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, p", Name := "QuotientDigraph\ " ), content := 0, count := [ 3, 3, 8, 2 ], mathmode := "Text", name := "Oper", next := 55674, root := ~, start := 55627, stop := 55673 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 55706, root := ~, start := 55677, stop := 55705 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 55744, root := ~, start := 55730, stop := 55743 ), rec( content := " is a di\ graph, and ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 55771, root := ~, start := 55763, stop := 55770 ), rec( content := " is a pa\ rtition of the vertices\n of ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 55824, root := ~, start := 55810, stop := 55823 ), rec( content := ", then t\ his operation returns a digraph constructed by\n amalgamating all vertices \ of ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 55926, root := ~, start := 55912, stop := 55925 ), rec( content := " which l\ ie in the same part of\n ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "A", next := 55969, root := ~, start := 55961, stop := 55968 ), rec( content := ".\n " , count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "P", next := 55979, root := ~, start := 55975, stop := 55978 ), rec( content := "\n\n \ A partition of the vertices of ", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "A", next := 56030, root := ~, start := 56016, stop := 56029 ), rec( content := " is a li\ st of non-empty\n disjoint lists, such that the union of all the sub-lists \ is equal\n to vertex set of ", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "A", next := 56159, root := ~, start := 56145, stop := 56158 ), rec( content := ". In par\ ticular, each vertex must appear\n in precisely one sub-list.\n ", count := [ 3, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "P", next := 56239, root := ~, start := 56235, stop := 56238 ), rec( content := "\n\n \ The vertices of ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "A", next := 56275, root := ~, start := 56261, stop := 56274 ), rec( content := " in part\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 56292, root := ~, start := 56284, stop := 56291 ), rec( content := " of ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "A", next := 56304, root := ~, start := 56296, stop := 56303 ), rec( content := " will be\ come\n vertex ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 56336, root := ~, start := 56328, stop := 56335 ), rec( content := " in the \ quotient, and there exists some edge in ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "A", next := 56398, root := ~, start := 56384, stop := 56397 ), rec( content := "\n wi\ th source in part ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 56431, root := ~, start := 56423, stop := 56430 ), rec( content := " and ran\ ge in part ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 56458, root := ~, start := 56450, stop := 56457 ), rec( content := " if and \ only if there\n is an edge from ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 56508, root := ~, start := 56500, stop := 56507 ), rec( content := " to ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "C", next := 56520, root := ~, start := 56512, stop := 56519 ), rec( content := " in the \ quotient.\n In particular, this means that the quotient of a digraph has no\ multiple edges.\n which was a change introduced in version 1.0.0 of the ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package\ .\n ", count := [ 3, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "P", next := 56708, root := ~, start := 56704, stop := 56707 ), rec( content := "\n\n \ If ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "A", next := 56731, root := ~, start := 56717, stop := 56730 ), rec( content := " belongs\ to ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "Ref", next := 56773, root := ~, start := 56743, stop := 56772 ), rec( content := ", then\n\ ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "A", next := 56798, root := ~, start := 56784, stop := 56797 ), rec( content := " is modi\ fied in place. If ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "A", next := 56838, root := ~, start := 56824, stop := 56837 ), rec( content := " belongs\ to ", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "Ref", next := 56888, root := ~, start := 56850, stop := 56887 ), rec( content := ", a new \ immutable digraph with the above\n properties is returned.", count := [ 3, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 8, 7 ], mathmode := "Text", name := "P", next := 56960, root := ~, start := 56956, stop := 56959 ), rec( content := "\n\n \ ", count := [ 3, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 1], [4], [1], [1, 3, 4]]);\n\ngap> DigraphVertices(D);\n[ 1 .. 4 ]\ngap> DigraphEdges(D)\ ;\n[ [ 1, 2 ], [ 1, 1 ], [ 2, 4 ], [ 3, 1 ], [ 4, 1 ], [ 4, 3 ], \n [ 4, 4 ] \ ]\ngap> p := [[1], [2, 4], [3]];\n[ [ 1 ], [ 2, 4 ], [ 3 ] ]\ngap> quo := Quot\ ientDigraph(D, p);\n\ngap> Digraph\ Vertices(quo);\n[ 1 .. 3 ]\ngap> DigraphEdges(quo);\n[ [ 1, 1 ], [ 1, 2 ], [ 2\ , 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> QuotientDigraph(EmptyDigraph(0), [\ ]);\n\n", count := [ 3, 3, 8, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 8, 8 ], mathmode := "Text", name := "Example", next := 57574, root := ~, start := 56966, stop := 57573 ), rec( content := "\n ", count := [ 3, 3, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 57591, root := ~, start := 55709, stop := 57590 ) ], count := [ 3, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 57605, root := ~, start := 55612, stop := 57604 ), rec( content := "\n\n ", count := [ 3, 3, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphReverse"\ ), content := 0, count := [ 3, 3, 9, 2 ], mathmode := "Text", name := "Oper", next := 57669, root := ~, start := 57626, stop := 57668 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReverseA\ ttr" ), content := 0, count := [ 3, 3, 9, 4 ], mathmode := "Text", name := "Attr", next := 57719, root := ~, start := 57672, stop := 57718 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 57751, root := ~, start := 57722, stop := 57750 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ reverse of a digraph is the digraph formed by reversing the orientation\n \ of each of its edges, i.e. for every edge ", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "C", next := 57907, root := ~, start := 57894, stop := 57906 ), rec( content := " of a di\ graph, the\n reverse contains the corresponding edge ", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j, i]", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "C", next := 57983, root := ~, start := 57970, stop := 57982 ), rec( content := ".", count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "P", next := 57988, root := ~, start := 57984, stop := 57987 ), rec( content := "\n\n \ ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphReverse", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "C", next := 58015, root := ~, start := 57994, stop := 58014 ), rec( content := " returns\ the reverse of the digraph ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "A", next := 58065, root := ~, start := 58051, stop := 58064 ), rec( content := ". If\n \ ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "A", next := 58088, root := ~, start := 58074, stop := 58087 ), rec( content := " is immu\ table, then a new immutable digraph is returned. Otherwise,\n the mutable d\ igraph ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "A", next := 58194, root := ~, start := 58180, stop := 58193 ), rec( content := " is chan\ ged in-place into its reverse.\n ", count := [ 3, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> DigraphReverse(D);\n\ngap> OutNeighbours(last);\n[ [ 2, 3, 4 ], [ 4, 5 \ ], [ 1, 2, 5 ], [ 4 ], [ 2, 5 ] ]\ngap> D := Digraph([[2, 4], [1], [4], [3, 4]\ ]);\n\ngap> DigraphEdges(D);\n[ [ \ 1, 2 ], [ 1, 4 ], [ 2, 1 ], [ 3, 4 ], [ 4, 3 ], [ 4, 4 ] ]\ngap> DigraphEdges(\ DigraphReverse(D));\n[ [ 1, 2 ], [ 2, 1 ], [ 3, 4 ], [ 4, 1 ], [ 4, 3 ], [ 4, \ 4 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> Digra\ phReverse(D);\n\ngap> OutNeighbours(\ D);\n[ [ 3 ], [ 1 ], [ 2 ] ]\n", count := [ 3, 3, 9, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 7 ], mathmode := "Text", name := "Example", next := 59058, root := ~, start := 58237, stop := 59057 ), rec( content := "\n ", count := [ 3, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 59075, root := ~, start := 57754, stop := 59074 ) ], count := [ 3, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 59089, root := ~, start := 57611, stop := 59088 ), rec( content := "\n\n ", count := [ 3, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDual" ), content := 0, count := [ 3, 3, 10, 2 ], mathmode := "Text", name := "Oper", next := 59150, root := ~, start := 59110, stop := 59149 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphDualAttr\ " ), content := 0, count := [ 3, 3, 10, 4 ], mathmode := "Text", name := "Attr", next := 59197, root := ~, start := 59153, stop := 59196 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "Returns", next := 59229, root := ~, start := 59200, stop := 59228 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ual", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "E", next := 59265, root := ~, start := 59254, stop := 59264 ), rec( content := " of ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "A", next := 59283, root := ~, start := 59269, stop := 59282 ), rec( content := " has the\ same vertices as ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "A", next := 59323, root := ~, start := 59309, stop := 59322 ), rec( content := ",\n a\ nd there is an edge in the dual from ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "C", next := 59375, root := ~, start := 59367, stop := 59374 ), rec( content := " to ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "C", next := 59387, root := ~, start := 59379, stop := 59386 ), rec( content := "\n wh\ enever there is no edge from ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "C", next := 59431, root := ~, start := 59423, stop := 59430 ), rec( content := " to ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "C", next := 59443, root := ~, start := 59435, stop := 59442 ), rec( content := " in ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "A", next := 59461, root := ~, start := 59447, stop := 59460 ), rec( content := ".\n T\ he ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ual", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "E", next := 59482, root := ~, start := 59471, stop := 59481 ), rec( content := " is some\ times called the ", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omplement", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "E", next := 59524, root := ~, start := 59507, stop := 59523 ), rec( content := ".", count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "P", next := 59529, root := ~, start := 59525, stop := 59528 ), rec( content := "\n\n \ ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ igraphDual", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "C", next := 59553, root := ~, start := 59535, stop := 59552 ), rec( content := " returns\ the dual of the digraph ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "A", next := 59600, root := ~, start := 59586, stop := 59599 ), rec( content := ". If\n \ ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "A", next := 59623, root := ~, start := 59609, stop := 59622 ), rec( content := " is an i\ mmutable digraph, then a new immutable digraph is\n returned. Otherwise, th\ e mutable digraph ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "A", next := 59740, root := ~, start := 59726, stop := 59739 ), rec( content := " is chan\ ged in-place\n into its dual.\n ", count := [ 3, 3, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3], [], [4, 6], [5], [],\n> [7, 8, 9], [], [], []]);\n\ \ngap> DigraphDual(D);\n\ngap> D := CycleDigraph(IsMutableDigraph, \ 3);\n\ngap> DigraphDual(D);\n\ngap> D;\n\n", count := [ 3, 3, 10, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 7 ], mathmode := "Text", name := "Example", next := 60206, root := ~, start := 59784, stop := 60205 ), rec( content := "\n ", count := [ 3, 3, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 10, 5 ], mathmode := "Text", name := "Description", next := 60223, root := ~, start := 59232, stop := 60222 ) ], count := [ 3, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 60237, root := ~, start := 59095, stop := 60236 ), rec( content := "\n\n ", count := [ 3, 3, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSymmetri\ cClosure" ), content := 0, count := [ 3, 3, 11, 2 ], mathmode := "Text", name := "Oper", next := 60310, root := ~, start := 60258, stop := 60309 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphSymmetri\ cClosureAttr" ), content := 0, count := [ 3, 3, 11, 4 ], mathmode := "Text", name := "Attr", next := 60369, root := ~, start := 60313, stop := 60368 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "Returns", next := 60401, root := ~, start := 60372, stop := 60400 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "A", next := 60439, root := ~, start := 60425, stop := 60438 ), rec( content := " is a di\ graph, then this attribute gives the minimal\n symmetric digraph which has \ the same vertices and contains all the edges of\n ", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "A", next := 60590, root := ~, start := 60576, stop := 60589 ), rec( content := ".", count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "P", next := 60595, root := ~, start := 60591, stop := 60594 ), rec( content := "\n \n\ A digraph is ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ymmetric", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "E", next := 60634, root := ~, start := 60618, stop := 60633 ), rec( content := " if its \ adjacency matrix \n ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Adjacen\ cyMatrix" ), content := 0, count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "Ref", next := 60693, root := ~, start := 60664, stop := 60692 ), rec( content := " is symm\ etric. For a digraph with multiple\n edges this means that there are the sa\ me number of edges from a vertex\n ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "C", next := 60823, root := ~, start := 60815, stop := 60822 ), rec( content := " to a ve\ rtex ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "C", next := 60844, root := ~, start := 60836, stop := 60843 ), rec( content := " as ther\ e are from ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "C", next := 60871, root := ~, start := 60863, stop := 60870 ), rec( content := " to ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "C", next := 60883, root := ~, start := 60875, stop := 60882 ), rec( content := "; \n \ see ", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "Ref", next := 60926, root := ~, start := 60894, stop := 60925 ), rec( content := ".", count := [ 3, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "P", next := 60931, root := ~, start := 60927, stop := 60930 ), rec( content := "\n\n \ If ", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "A", next := 60954, root := ~, start := 60940, stop := 60953 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "A", next := 61060, root := ~, start := 61046, stop := 61059 ), rec( content := " is chan\ ged in-place into its\n symmetric closure.\n", count := [ 3, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 3], [2, 4], [1], [3, 4]]);\n\ngap> D := DigraphSymmetricClosure(D);\n\ngap> IsSymmetricDigraph(D);\ntrue\nga\ p> List(OutNeighbours(D), AsSet);\n[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 4 ], [ 2, \ 3, 4 ] ]\ngap> D := Digraph([[2, 2], [1]]);\n\ngap> D := DigraphSymmetricClosure(D);\n\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ \ 1, 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSymmetricClosure(D);\n\ngap> D;\n\n", count := [ 3, 3, 11, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 11, 8 ], mathmode := "Text", name := "Example", next := 61901, root := ~, start := 61113, stop := 61900 ), rec( content := "\n ", count := [ 3, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 61918, root := ~, start := 60404, stop := 61917 ) ], count := [ 3, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 61932, root := ~, start := 60243, stop := 61931 ), rec( content := "\n\n ", count := [ 3, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veClosure" ), content := 0, count := [ 3, 3, 12, 2 ], mathmode := "Text", name := "Oper", next := 62006, root := ~, start := 61953, stop := 62005 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veClosureAttr" ), content := 0, count := [ 3, 3, 12, 4 ], mathmode := "Text", name := "Attr", next := 62066, root := ~, start := 62009, stop := 62065 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveClosure" ), content := 0, count := [ 3, 3, 12, 6 ], mathmode := "Text", name := "Oper", next := 62131, root := ~, start := 62069, stop := 62130 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveClosureAttr" ), content := 0, count := [ 3, 3, 12, 8 ], mathmode := "Text", name := "Attr", next := 62200, root := ~, start := 62134, stop := 62199 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "Returns", next := 62232, root := ~, start := 62203, stop := 62231 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "A", next := 62270, root := ~, start := 62256, stop := 62269 ), rec( content := " is a di\ graph with no multiple edges, then these\n attributes return the (reflexive\ ) transitive closure of ", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "A", next := 62393, root := ~, start := 62379, stop := 62392 ), rec( content := ".", count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "P", next := 62398, root := ~, start := 62394, stop := 62397 ), rec( content := "\n\n \ A digraph is ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ eflexive", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "E", next := 62433, root := ~, start := 62417, stop := 62432 ), rec( content := " if it h\ as a loop at every vertex, and it is\n ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ransitive", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "E", next := 62499, root := ~, start := 62482, stop := 62498 ), rec( content := " if when\ ever ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "C", next := 62524, root := ~, start := 62512, stop := 62523 ), rec( content := " and ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j,k]", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "C", next := 62541, root := ~, start := 62529, stop := 62540 ), rec( content := " are edg\ es of\n ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "A", next := 62573, root := ~, start := 62559, stop := 62572 ), rec( content := ", ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,k]", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "C", next := 62587, root := ~, start := 62575, stop := 62586 ), rec( content := " is also\ an edge. The ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ reflexive)\n transitive closure", count := [ 3, 3, 12, 10 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "E", next := 62653, root := ~, start := 62610, stop := 62652 ), rec( content := " of a di\ graph ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "A", next := 62681, root := ~, start := 62667, stop := 62680 ), rec( content := " is the \ least\n (reflexive and) transitive digraph containing ", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "A", next := 62759, root := ~, start := 62745, stop := 62758 ), rec( content := ".", count := [ 3, 3, 12, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "P", next := 62764, root := ~, start := 62760, stop := 62763 ), rec( content := "\n\n \ If ", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "A", next := 62787, root := ~, start := 62773, stop := 62786 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "A", next := 62893, root := ~, start := 62879, stop := 62892 ), rec( content := " is chan\ ged in-place into its\n (reflexive) transitive closure.", count := [ 3, 3, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "P", next := 62962, root := ~, start := 62958, stop := 62961 ), rec( content := "\n\n \ Let ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "M", next := 62980, root := ~, start := 62972, stop := 62979 ), rec( content := " be the \ number of vertices of ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "A", next := 63024, root := ~, start := 63010, stop := 63023 ), rec( content := ", and le\ t\n ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "M", next := 63046, root := ~, start := 63038, stop := 63045 ), rec( content := " be the \ number of edges. For an arbitrary digraph, these\n attributes will use a v\ ersion of the Floyd-Warshall algorithm, with\n complexity ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (n^3)", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "M", next := 63204, root := ~, start := 63191, stop := 63203 ), rec( content := ".\n\n \ However, for a topologically sortable digraph [see ", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "Ref", next := 63304, root := ~, start := 63262, stop := 63303 ), rec( content := "], these\ attributes will use methods\n with complexity ", count := [ 3, 3, 12, 12 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n + m \\cdot n)", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "M", next := 63388, root := ~, start := 63361, stop := 63387 ), rec( content := " when th\ is is faster.", count := [ 3, 3, 12, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 12, 13 ], mathmode := "Text", name := "P", next := 63413, root := ~, start := 63409, stop := 63412 ), rec( content := "\n\n \ ", count := [ 3, 3, 12, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDiSparse6String(\".H`eOWR`Ul^\");\n\ngap> IsReflexiveDigraph(D) or IsTransitiveDigraph(D);\ \nfalse\ngap> OutNeighbours(D);\n[ [ 4, 6 ], [ 1, 3 ], [ ], [ 5 ], [ ], [ 7,\ 8, 9 ], [ ], [ ], \n [ ] ]\ngap> T := DigraphTransitiveClosure(D);\n\ngap> OutNeighbours(T);\n[\ [ 4, 6, 5, 7, 8, 9 ], [ 1, 3, 4, 5, 6, 7, 8, 9 ], [ ], [ 5 ], \n [ ], [ 7,\ 8, 9 ], [ ], [ ], [ ] ]\ngap> RT := DigraphReflexiveTransitiveClosure(D);\ \n\ngap> OutNeighbours(R\ T);\n[ [ 4, 6, 5, 7, 8, 9, 1 ], [ 1, 3, 4, 5, 6, 7, 8, 9, 2 ], [ 3 ], \n [ 5,\ 4 ], [ 5 ], [ 7, 8, 9, 6 ], [ 7 ], [ 8 ], [ 9 ] ]\ngap> D := CycleDigraph(IsM\ utableDigraph, 3);\n\ngap> DigraphRe\ flexiveTransitiveClosure(D);\n\ngap>\ D;\n\n", count := [ 3, 3, 12, 14 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 12, 14 ], mathmode := "Text", name := "Example", next := 64409, root := ~, start := 63419, stop := 64408 ), rec( content := "\n ", count := [ 3, 3, 12, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 12, 9 ], mathmode := "Text", name := "Description", next := 64426, root := ~, start := 62235, stop := 64425 ) ], count := [ 3, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 64440, root := ~, start := 61938, stop := 64439 ), rec( content := "\n\n ", count := [ 3, 3, 12, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veReduction" ), content := 0, count := [ 3, 3, 13, 2 ], mathmode := "Text", name := "Oper", next := 64516, root := ~, start := 64461, stop := 64515 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphTransiti\ veReductionAttr" ), content := 0, count := [ 3, 3, 13, 4 ], mathmode := "Text", name := "Attr", next := 64578, root := ~, start := 64519, stop := 64577 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveReduction" ), content := 0, count := [ 3, 3, 13, 6 ], mathmode := "Text", name := "Oper", next := 64645, root := ~, start := 64581, stop := 64644 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphReflexiv\ eTransitiveReductionAttr" ), content := 0, count := [ 3, 3, 13, 8 ], mathmode := "Text", name := "Attr", next := 64716, root := ~, start := 64648, stop := 64715 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "Returns", next := 64748, root := ~, start := 64719, stop := 64747 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "A", next := 64786, root := ~, start := 64772, stop := 64785 ), rec( content := " is a to\ pologically sortable digraph\n [see ", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "Ref", next := 64868, root := ~, start := 64832, stop := 64867 ), rec( content := "]\n w\ ith no multiple edges, then these operations return the\n (reflexive) tran\ sitive reduction of ", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "A", next := 64986, root := ~, start := 64972, stop := 64985 ), rec( content := ".", count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "P", next := 64991, root := ~, start := 64987, stop := 64990 ), rec( content := "\n\n \ The (reflexive) transitive reduction of such a digraph is the unique\n leas\ t subgraph such that the (reflexive) transitive closure of the\n subgraph i\ s equal to the (reflexive) transitive closure of ", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "A", next := 65214, root := ~, start := 65200, stop := 65213 ), rec( content := "\n [s\ ee ", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ReflexiveTransitiveClosure" ), content := 0, count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "Ref", next := 65271, root := ~, start := 65224, stop := 65270 ), rec( content := "].\n \ In order words, it is the least subgraph of ", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "A", next := 65336, root := ~, start := 65322, stop := 65335 ), rec( content := " which\n\ retains the same reachability as ", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "A", next := 65394, root := ~, start := 65380, stop := 65393 ), rec( content := ".", count := [ 3, 3, 13, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "P", next := 65399, root := ~, start := 65395, stop := 65398 ), rec( content := "\n\n \ If ", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "A", next := 65422, root := ~, start := 65408, stop := 65421 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "A", next := 65528, root := ~, start := 65514, stop := 65527 ), rec( content := " is chan\ ged in-place into its\n (reflexive) transitive reduction.", count := [ 3, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "P", next := 65599, root := ~, start := 65595, stop := 65598 ), rec( content := "\n\n \ Let ", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "M", next := 65617, root := ~, start := 65609, stop := 65616 ), rec( content := " be the \ number of vertices of an arbitrary digraph, and let\n ", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "M", next := 65693, root := ~, start := 65685, stop := 65692 ), rec( content := " be the \ number of edges.\n Then these operations\n use methods with complexity " , count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n + m \\cdot n)", count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "M", next := 65803, root := ~, start := 65776, stop := 65802 ), rec( content := ".\n " , count := [ 3, 3, 13, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 13, 13 ], mathmode := "Text", name := "P", next := 65813, root := ~, start := 65809, stop := 65812 ), rec( content := "\n ", count := [ 3, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 3], [3], [3]]);;\ngap> DigraphHasLoops(D);\ntrue\ng\ ap> D1 := DigraphReflexiveTransitiveReduction(D);\n\ngap> DigraphHasLoops(D1);\nfalse\ngap> OutNeighbours(D1);\n\ [ [ 2 ], [ 3 ], [ ] ]\ngap> D2 := DigraphTransitiveReduction(D);\n\ngap> DigraphHasLoops(D2);\ntrue\ngap> OutNe\ ighbours(D2);\n[ [ 2, 1 ], [ 3 ], [ 3 ] ]\ngap> DigraphReflexiveTransitiveClos\ ure(D)\n> = DigraphReflexiveTransitiveClosure(D1);\ntrue\ngap> DigraphTransit\ iveClosure(D)\n> = DigraphTransitiveClosure(D2);\ntrue\ngap> D := Digraph(IsM\ utableDigraph, [[1], [1], [1, 2, 3]]);\n\ngap> DigraphReflexiveTransitiveReduction(D);\n\ngap> D;\n\n", count := [ 3, 3, 13, 14 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 13, 14 ], mathmode := "Text", name := "Example", next := 66673, root := ~, start := 65818, stop := 66672 ), rec( content := "\n ", count := [ 3, 3, 13, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 13, 9 ], mathmode := "Text", name := "Description", next := 66690, root := ~, start := 64751, stop := 66689 ) ], count := [ 3, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 66704, root := ~, start := 64446, stop := 66703 ), rec( content := "\n\n ", count := [ 3, 3, 13, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, label \ ]", Name := "DigraphAddVertex" ), content := 0, count := [ 3, 3, 14, 2 ], mathmode := "Text", name := "Oper", next := 66780, root := ~, start := 66725, stop := 66779 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "Returns", next := 66812, root := ~, start := 66783, stop := 66811 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ operation returns a digraph constructed from ", count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "A", next := 66896, root := ~, start := 66882, stop := 66895 ), rec( content := " by addi\ ng a\n single new vertex, and no new edges. ", count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "P", next := 66954, root := ~, start := 66950, stop := 66953 ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abel", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "A", next := 67004, root := ~, start := 66992, stop := 67003 ), rec( content := " is a ", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object,\ \n then the new vertex will be labelled ", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abel", count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "A", next := 67077, root := ~, start := 67065, stop := 67076 ), rec( content := ".\n " , count := [ 3, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "P", next := 67087, root := ~, start := 67083, stop := 67086 ), rec( content := "\n\n \ If ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "A", next := 67110, root := ~, start := 67096, stop := 67109 ), rec( content := " belongs\ to ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "Ref", next := 67152, root := ~, start := 67122, stop := 67151 ), rec( content := ", then t\ he\n vertex is added directly to ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "A", next := 67209, root := ~, start := 67195, stop := 67208 ), rec( content := ". If ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "A", next := 67228, root := ~, start := 67214, stop := 67227 ), rec( content := " belongs\ to\n ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "Ref", next := 67276, root := ~, start := 67244, stop := 67275 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "A", next := 67313, root := ~, start := 67299, stop := 67312 ), rec( content := " with\n \ the additional vertex is returned.", count := [ 3, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 14, 6 ], mathmode := "Text", name := "P", next := 67361, root := ~, start := 67357, stop := 67360 ), rec( content := "\n\n \ ", count := [ 3, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(3);\n\n\ gap> new := DigraphAddVertex(D);\n\ \ngap> D = new;\nfalse\ngap> DigraphVertices(new);\n[ 1 .. 4 ]\ngap> new := Di\ graphAddVertex(D, Group([(1, 2)]));\n\ngap> DigraphVertexLabels(new);\n[ 1, 2, 3, Group([ (1,2) ]) ]\ngap> D := \ CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);\n\ngap> new := DigraphAddVertex(D);\n\ngap> D = new;\ntrue\n", count := [ 3, 3, 14, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 14, 7 ], mathmode := "Text", name := "Example", next := 67966, root := ~, start := 67367, stop := 67965 ), rec( content := "\n ", count := [ 3, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 67983, root := ~, start := 66815, stop := 67982 ) ], count := [ 3, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 67997, root := ~, start := 66710, stop := 67996 ), rec( content := "\n\n ", count := [ 3, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, m", Label := "for a digraph \ and an integer", Name := "DigraphAddVertices" ), content := 0, count := [ 3, 3, 15, 2 ], mathmode := "Text", name := "Oper", next := 68109, root := ~, start := 68018, stop := 68108 ), rec( attributes := rec( Arg := "digraph, labels" , Label := "for a digraph \ and a list of labels", Name := "DigraphAddVertices" ), content := 0, count := [ 3, 3, 15, 4 ], mathmode := "Text", name := "Oper", next := 68214, root := ~, start := 68112, stop := 68213 ), rec( content := " The 3-argument \ version is deliberately undocumented. It is redundant,\n but it is inclu\ ded for backwards compatibility. ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "Returns", next := 68382, root := ~, start := 68353, stop := 68381 ), rec( attributes := rec( ), content := [ rec( content := "\n For\ a non-negative integer ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "A", next := 68438, root := ~, start := 68430, stop := 68437 ), rec( content := ", this o\ peration returns a digraph\n constructed from ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "A", next := 68508, root := ~, start := 68494, stop := 68507 ), rec( content := " by addi\ ng ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "A", next := 68527, root := ~, start := 68519, stop := 68526 ), rec( content := " new ver\ tices.\n ", count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "P", next := 68550, root := ~, start := 68546, stop := 68549 ), rec( content := "\n\n \ Otherwise, if ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "A", next := 68583, root := ~, start := 68570, stop := 68582 ), rec( content := " is a li\ st\n consisting of ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "C", next := 68620, root := ~, start := 68612, stop := 68619 ), rec( content := " ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ , then this operation returns a\n digraph constructed from ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "A", next := 68709, root := ~, start := 68695, stop := 68708 ), rec( content := " by addi\ ng ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "C", next := 68728, root := ~, start := 68720, stop := 68727 ), rec( content := " new ver\ tices,\n which are labelled according to this list.\n ", count := [ 3, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "P", next := 68798, root := ~, start := 68794, stop := 68797 ), rec( content := "\n\n \ If ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 68821, root := ~, start := 68807, stop := 68820 ), rec( content := " belongs\ to ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "Ref", next := 68863, root := ~, start := 68833, stop := 68862 ), rec( content := ", then t\ he\n vertices are added directly to ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 68923, root := ~, start := 68909, stop := 68922 ), rec( content := ", which \ is changed in-place.\n If ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 68973, root := ~, start := 68959, stop := 68972 ), rec( content := " belongs\ to ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "Ref", next := 69017, root := ~, start := 68985, stop := 69016 ), rec( content := ", then\n\ ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 69042, root := ~, start := 69028, stop := 69041 ), rec( content := " itself \ is returned if no vertices are added\n (i.e. ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "m", count := [ 3, 3, 15, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := \ "Text", name := "A", next := 69108, root := ~, start := 69100, stop := 69107 ) , rec( content := "\ =0", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "C", next := 69114, root := ~, start := 69097, stop := 69113 ), rec( content := " or ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "A", next := 69131, root := ~, start := 69118, stop := 69130 ), rec( content := " is empt\ y), otherwise\n the result is a new immutable digraph.\n ", count := [ 3, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 15, 8 ], mathmode := "Text", name := "P", next := 69204, root := ~, start := 69200, stop := 69203 ), rec( content := "\n\n \ ", count := [ 3, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(3);\n\n\ gap> new := DigraphAddVertices(D, 3);\n\ngap> DigraphVertices(new);\n[ 1 .. 6 ]\ngap> new := DigraphAddVertices(\ D, [Group([(1, 2)]), \"d\"]);\n\ng\ ap> DigraphVertexLabels(new);\n[ 1, 2, 3, Group([ (1,2) ]), \"d\" ]\ngap> Digr\ aphAddVertices(D, 0) = D;\ntrue\ngap> D := CompleteBipartiteDigraph(IsMutableD\ igraph, 2, 3);\n\ngap> new := Digra\ phAddVertices(D, 4);\n\ngap> D = ne\ w;\ntrue\n", count := [ 3, 3, 15, 9 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 9 ], mathmode := "Text", name := "Example", next := 69853, root := ~, start := 69210, stop := 69852 ), rec( content := "\n ", count := [ 3, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 15, 5 ], mathmode := "Text", name := "Description", next := 69870, root := ~, start := 68385, stop := 69869 ) ], count := [ 3, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 69884, root := ~, start := 68003, stop := 69883 ), rec( content := "\n\n ", count := [ 3, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edge", Label := "for a digraph \ and an edge", Name := "DigraphAddEdge" ), content := 0, count := [ 3, 3, 16, 2 ], mathmode := "Text", name := "Oper", next := 69992, root := ~, start := 69905, stop := 69991 ), rec( attributes := rec( Arg := "digraph, src, ra\ n", Label := "for a digraph, source, and range", Name := "DigraphAddEdge" ), content := 0, count := [ 3, 3, 16, 4 ], mathmode := "Text", name := "Oper", next := 70093, root := ~, start := 69995, stop := 70092 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "Returns", next := 70125, root := ~, start := 70096, stop := 70124 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70160, root := ~, start := 70149, stop := 70159 ), rec( content := " is a pa\ ir of vertices of ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70200, root := ~, start := 70186, stop := 70199 ), rec( content := ", or ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70215, root := ~, start := 70205, stop := 70214 ), rec( content := " and\n \ ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70234, root := ~, start := 70224, stop := 70233 ), rec( content := " are ver\ tices of ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70265, root := ~, start := 70251, stop := 70264 ), rec( content := ",\n t\ hen this operation returns a digraph constructed from ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70340, root := ~, start := 70326, stop := 70339 ), rec( content := "\n by\ adding a new edge with source ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70389, root := ~, start := 70378, stop := 70388 ), rec( attributes := rec( ), content := [ rec( content := "[\ 1]", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "C", next := 70399, root := ~, start := 70389, stop := 70398 ), rec( content := " [", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70411, root := ~, start := 70401, stop := 70410 ), rec( content := "] and ra\ nge\n ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70438, root := ~, start := 70427, stop := 70437 ), rec( attributes := rec( ), content := [ rec( content := "[\ 2]", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "C", next := 70448, root := ~, start := 70438, stop := 70447 ), rec( content := " [", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "A", next := 70460, root := ~, start := 70450, stop := 70459 ), rec( content := "].\n \ ", count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "P", next := 70471, root := ~, start := 70467, stop := 70470 ), rec( content := "\n \n\ If ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "A", next := 70498, root := ~, start := 70484, stop := 70497 ), rec( content := " belongs\ to ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "Ref", next := 70540, root := ~, start := 70510, stop := 70539 ), rec( content := ", then t\ he edge\n is added directly to ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "A", next := 70595, root := ~, start := 70581, stop := 70594 ), rec( content := ". If ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "A", next := 70614, root := ~, start := 70600, stop := 70613 ), rec( content := " belongs\ to ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "Ref", next := 70664, root := ~, start := 70626, stop := 70663 ), rec( content := ", then a\ n immutable copy of ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "A", next := 70706, root := ~, start := 70692, stop := 70705 ), rec( content := " with th\ e\n additional edge is returned. ", count := [ 3, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 16, 7 ], mathmode := "Text", name := "P", next := 70754, root := ~, start := 70750, stop := 70753 ), rec( content := "\n\n \ ", count := [ 3, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := Digraph([[2], [3], []]);\n\ngap> DigraphEdges(D1);\n[ [ 1, 2 ], [ 2, 3 ] ]\ngap> D2 := DigraphAddEdg\ e(D1, [3, 1]);\n\ngap> DigraphEdge\ s(D2);\n[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D3 := DigraphAddEdge(D2, [2, 3]\ );\n\ngap> DigraphEdges(D3);\ \n[ [ 1, 2 ], [ 2, 3 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D := CycleDigraph(IsMutable\ Digraph, 4);\n\ngap> new := DigraphA\ ddEdge(D, [1, 3]);\n\ngap> DigraphEd\ ges(new);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 3, 4 ], [ 4, 1 ] ]\ngap> D = new;\ \ntrue\n", count := [ 3, 3, 16, 8 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 8 ], mathmode := "Text", name := "Example", next := 71480, root := ~, start := 70760, stop := 71479 ), rec( content := "\n ", count := [ 3, 3, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 16, 5 ], mathmode := "Text", name := "Description", next := 71497, root := ~, start := 70128, stop := 71496 ) ], count := [ 3, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 71511, root := ~, start := 69890, stop := 71510 ), rec( content := "\n\n ", count := [ 3, 3, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edge", Name := "DigraphAddEdgeO\ rbit" ), content := 0, count := [ 3, 3, 17, 2 ], mathmode := "Text", name := "Oper", next := 71586, root := ~, start := 71532, stop := 71585 ), rec( attributes := rec( ), content := [ rec( content := "\n A n\ ew digraph.\n ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "Returns", next := 71630, root := ~, start := 71589, stop := 71629 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns a new digraph with the same vertices and edges as\n ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71742, root := ~, start := 71728, stop := 71741 ), rec( content := " and wit\ h additional edges consisting of the orbit of the\n edge ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71820, root := ~, start := 71809, stop := 71819 ), rec( content := " under t\ he action of the ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "Ref", next := 71871, root := ~, start := 71845, stop := 71870 ), rec( content := " of\n \ ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71893, root := ~, start := 71879, stop := 71892 ), rec( content := ". If ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71910, root := ~, start := 71899, stop := 71909 ), rec( content := " is alre\ ady an edge in ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71947, root := ~, start := 71933, stop := 71946 ), rec( content := ", then\n\ ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 71972, root := ~, start := 71958, stop := 71971 ), rec( content := " is retu\ rned unchanged. The argument ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "A", next := 72023, root := ~, start := 72009, stop := 72022 ), rec( content := " must be\ \n an immutable digraph.\n ", count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "P", next := 72066, root := ~, start := 72062, stop := 72065 ), rec( content := "\n\n \ An edge is simply a pair of vertices of ", count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "A", next := 72126, root := ~, start := 72112, stop := 72125 ), rec( content := ".\n\n \ ", count := [ 3, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := CayleyDigraph(DihedralGroup(8));\n\ngap> gr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);\n\ngap> DigraphEdges(gr1);\n[ [ 1, 2 ], [ 1, 3 ],\ [ 1, 4 ], [ 2, 1 ], [ 2, 5 ], [ 2, 6 ], \n [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 4\ , 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5, 7 ], [ 5, 6 ], [ 5, 8 ], [ 6, 4 ], [ 6, 8 \ ], [ 6, 2 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 ], [ 8, 3 ], [ 8, 2 ], [ 8, 5 ] ]\ \ngap> DigraphEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ],\ [ 2, 5 ], \n [ 2, 6 ], [ 2, 7 ], [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 3, 6 ], \n \ [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 7 ], [ 5, 6 ], \n [ 5, 8 ], [ 5\ , 4 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ], [ 6, 3 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 \ ], [ 7, 2 ], [ 8, 3 ], [ 8, 2 ], \n [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3 := Digraph\ RemoveEdgeOrbit(gr2, [1, 8]);\n\n\ gap> gr3 = gr1;\ntrue\n", count := [ 3, 3, 17, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 5 ], mathmode := "Text", name := "Example", next := 73102, root := ~, start := 72133, stop := 73101 ), rec( content := "\n ", count := [ 3, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 73119, root := ~, start := 71633, stop := 73118 ) ], count := [ 3, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 73133, root := ~, start := 71517, stop := 73132 ), rec( content := "\n\n ", count := [ 3, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edges", Name := "DigraphAddEdges\ " ), content := 0, count := [ 3, 3, 18, 2 ], mathmode := "Text", name := "Oper", next := 73205, root := ~, start := 73154, stop := 73204 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "Returns", next := 73237, root := ~, start := 73208, stop := 73236 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 73273, root := ~, start := 73261, stop := 73272 ), rec( content := " is a (p\ ossibly empty) list of pairs of vertices of\n ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 73343, root := ~, start := 73329, stop := 73342 ), rec( content := ", then t\ his operation returns a digraph constructed from\n ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 73418, root := ~, start := 73404, stop := 73417 ), rec( content := " by addi\ ng the edges specified by ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 73464, root := ~, start := 73452, stop := 73463 ), rec( content := ". More\\ n precisely, for every ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "C", next := 73508, root := ~, start := 73497, stop := 73507 ), rec( content := " in ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "A", next := 73524, root := ~, start := 73512, stop := 73523 ), rec( content := ", a new \ edge\n will be added with source ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge[1]", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "C", next := 73581, root := ~, start := 73567, stop := 73580 ), rec( content := " and ran\ ge ", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges[2]", count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "C", next := 73607, root := ~, start := 73592, stop := 73606 ), rec( content := ".\n " , count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "P", next := 73617, root := ~, start := 73613, stop := 73616 ), rec( content := "\n\n \ If an edge is included in ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "A", next := 73661, root := ~, start := 73649, stop := 73660 ), rec( content := " with mu\ ltiplicity ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "C", next := 73688, root := ~, start := 73680, stop := 73687 ), rec( content := ",\n t\ hen it will be added ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "C", next := 73724, root := ~, start := 73716, stop := 73723 ), rec( content := " times.\\ n\n If ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "A", next := 73754, root := ~, start := 73740, stop := 73753 ), rec( content := " belongs\ to ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "Ref", next := 73796, root := ~, start := 73766, stop := 73795 ), rec( content := ", then t\ he edges\n are added directly to ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "A", next := 73853, root := ~, start := 73839, stop := 73852 ), rec( content := ". If ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "A", next := 73872, root := ~, start := 73858, stop := 73871 ), rec( content := " belongs\ to ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "Ref", next := 73922, root := ~, start := 73884, stop := 73921 ), rec( content := ", then t\ he result is returned as an immutable\n digraph.\n ", count := [ 3, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 18, 5 ], mathmode := "Text", name := "P", next := 73989, root := ~, start := 73985, stop := 73988 ), rec( content := "\n\n \ ", count := [ 3, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> func := function(n)\n> local source, range, i;\n> source := [];\n> ra\ nge := [];\n> for i in [1 .. n - 2] do\n> Add(source, i);\n> Add(range\ , i + 1);\n> od;\n> return Digraph(n, source, range);\n> end;;\ngap> D := fu\ nc(1024);\n\ngap> new := Dig\ raphAddEdges(D,\n> [[1023, 1024], [1, 1024], [1023, 1024], [1024, 1]]);\n\ngap> D = new;\nfalse\ngap>\ D2 := DigraphMutableCopy(func(1024));\n\ngap> new := DigraphAddEdges(D2,\n> [[1023, 1024], [1, 1024], [1023\ , 1024], [1024, 1]]);\n\n\ gap> D2 = new;\ntrue\n", count := [ 3, 3, 18, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 6 ], mathmode := "Text", name := "Example", next := 74726, root := ~, start := 73995, stop := 74725 ), rec( content := "\n ", count := [ 3, 3, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 74743, root := ~, start := 73240, stop := 74742 ) ], count := [ 3, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 74757, root := ~, start := 73139, stop := 74756 ), rec( content := "\n\n ", count := [ 3, 3, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, v", Name := "DigraphRemoveVe\ rtex" ), content := 0, count := [ 3, 3, 19, 2 ], mathmode := "Text", name := "Oper", next := 74829, root := ~, start := 74778, stop := 74828 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "Returns", next := 74861, root := ~, start := 74832, stop := 74860 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 74893, root := ~, start := 74885, stop := 74892 ), rec( content := " is a ve\ rtex of ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 74923, root := ~, start := 74909, stop := 74922 ), rec( content := ", then t\ his operation returns a\n digraph constructed from ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 74998, root := ~, start := 74984, stop := 74997 ), rec( content := " by remo\ ving vertex ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 75026, root := ~, start := 75018, stop := 75025 ), rec( content := ",\n a\ long with any edge whose source or range vertex is ", count := [ 3, 3, 19, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "A", next := 75092, root := ~, start := 75084, stop := 75091 ), rec( content := ".", count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "P", next := 75097, root := ~, start := 75093, stop := 75096 ), rec( content := "\n\n \ If ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "A", next := 75120, root := ~, start := 75106, stop := 75119 ), rec( content := " has ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "C", next := 75133, root := ~, start := 75125, stop := 75132 ), rec( content := " vertice\ s, then the vertices of the returned\n digraph are ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1..n-1]", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "C", next := 75209, root := ~, start := 75194, stop := 75208 ), rec( content := ", but th\ e original labels can be\n accessed via ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "Ref", next := 75292, root := ~, start := 75259, stop := 75291 ), rec( content := ". ", count := [ 3, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "P", next := 75298, root := ~, start := 75294, stop := 75297 ), rec( content := "\n\n \ If ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "A", next := 75321, root := ~, start := 75307, stop := 75320 ), rec( content := " belongs\ to ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "Ref", next := 75363, root := ~, start := 75333, stop := 75362 ), rec( content := ", then t\ he\n vertex is removed directly from ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "A", next := 75424, root := ~, start := 75410, stop := 75423 ), rec( content := ". If ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "A", next := 75443, root := ~, start := 75429, stop := 75442 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "Ref", next := 75491, root := ~, start := 75459, stop := 75490 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "A", next := 75528, root := ~, start := 75514, stop := 75527 ), rec( content := "\n wi\ thout the vertex is returned.", count := [ 3, 3, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 19, 6 ], mathmode := "Text", name := "P", next := 75568, root := ~, start := 75564, stop := 75567 ), rec( content := "\n\n \ ", count := [ 3, 3, 19, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"a\", \ \"b\", \"c\", \"c\"],\n> [\"b\", \"c\", \"a\", \"a\", \"c\"])\ ;\n\ngap> DigraphVertexLabels(D);\ \n[ \"a\", \"b\", \"c\" ]\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 1\ ], [ 3, 1 ], [ 3, 3 ] ]\ngap> new := DigraphRemoveVertex(D, 2);\n\ngap> DigraphVertexLabels(new);\n[ \"a\", \"c\ \" ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\n\ngap> new := DigraphRemoveVertex(D, 1);\n\ngap> DigraphVertexLabels(D);\n[ 2, 3, 4, 5 ]\ngap> D =\ new;\ntrue\n", count := [ 3, 3, 19, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 7 ], mathmode := "Text", name := "Example", next := 76262, root := ~, start := 75574, stop := 76261 ), rec( content := "\n ", count := [ 3, 3, 19, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 19, 3 ], mathmode := "Text", name := "Description", next := 76279, root := ~, start := 74864, stop := 76278 ) ], count := [ 3, 3, 19, 1 ], mathmode := "Text", name := "ManSection", next := 76293, root := ~, start := 74763, stop := 76292 ), rec( content := "\n\n ", count := [ 3, 3, 19, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, verts", Name := "DigraphRemoveVe\ rtices" ), content := 0, count := [ 3, 3, 20, 2 ], mathmode := "Text", name := "Oper", next := 76371, root := ~, start := 76314, stop := 76370 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "Returns", next := 76403, root := ~, start := 76374, stop := 76402 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 76439, root := ~, start := 76427, stop := 76438 ), rec( content := " is a (p\ ossibly empty) duplicate-free list of vertices of\n ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 76515, root := ~, start := 76501, stop := 76514 ), rec( content := ", then t\ his operation returns a digraph constructed from\n ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 76590, root := ~, start := 76576, stop := 76589 ), rec( content := " by remo\ ving every vertex in ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 76631, root := ~, start := 76619, stop := 76630 ), rec( content := ", along \ with any edge\n whose source or range vertex is in ", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "A", next := 76704, root := ~, start := 76692, stop := 76703 ), rec( content := ".", count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "P", next := 76709, root := ~, start := 76705, stop := 76708 ), rec( content := "\n\n \ If ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "A", next := 76732, root := ~, start := 76718, stop := 76731 ), rec( content := " has ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "C", next := 76745, root := ~, start := 76737, stop := 76744 ), rec( content := " vertice\ s, then the vertices of the new\n digraph are ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. n-Length(", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "vert\ s", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := \ "Text", name := "A", next := 76831, root := ~, start := 76819, stop := 76830 ) , rec( content := "\ )]", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "C", next := 76837, root := ~, start := 76801, stop := 76836 ), rec( content := ", but th\ e\n original labels can be accessed via ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "Ref", next := 76920, root := ~, start := 76887, stop := 76919 ), rec( content := ". ", count := [ 3, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "P", next := 76926, root := ~, start := 76922, stop := 76925 ), rec( content := "\n\n \ If ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "A", next := 76949, root := ~, start := 76935, stop := 76948 ), rec( content := " belongs\ to ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "Ref", next := 76991, root := ~, start := 76961, stop := 76990 ), rec( content := ", then t\ he\n vertices are removed directly from ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "A", next := 77055, root := ~, start := 77041, stop := 77054 ), rec( content := ". If ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "A", next := 77074, root := ~, start := 77060, stop := 77073 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "Ref", next := 77122, root := ~, start := 77090, stop := 77121 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "A", next := 77159, root := ~, start := 77145, stop := 77158 ), rec( content := "\n wi\ thout the vertices is returned.", count := [ 3, 3, 20, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 20, 6 ], mathmode := "Text", name := "P", next := 77201, root := ~, start := 77197, stop := 77200 ), rec( content := "\n\n \ ", count := [ 3, 3, 20, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> SetDigraphVertexLabels(D, [\"a\", \"b\ \", \"c\", \"d\", \"e\"]);\ngap> new := DigraphRemoveVertices(D, [2, 4]);\n\ngap> DigraphVertexLabels(new);\n[ \ \"a\", \"c\", \"e\" ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\n\ngap> new := DigraphRemoveVertices(D, [1, 3]\ );\n\ngap> DigraphVertexLabels(D);\n[\ 2, 4, 5 ]\ngap> D = new;\ntrue\n", count := [ 3, 3, 20, 7 ], name := "PCDATA" , root := ~ ) ], count := [ 3, 3, 20, 7 ], mathmode := "Text", name := "Example", next := 77784, root := ~, start := 77207, stop := 77783 ), rec( content := "\n ", count := [ 3, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 20, 3 ], mathmode := "Text", name := "Description", next := 77801, root := ~, start := 76406, stop := 77800 ) ], count := [ 3, 3, 20, 1 ], mathmode := "Text", name := "ManSection", next := 77815, root := ~, start := 76299, stop := 77814 ), rec( content := "\n\n ", count := [ 3, 3, 20, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edge", Label := "for a digraph \ and an edge", Name := "DigraphRemoveEdge" ), content := 0, count := [ 3, 3, 21, 2 ], mathmode := "Text", name := "Oper", next := 77926, root := ~, start := 77836, stop := 77925 ), rec( attributes := rec( Arg := "digraph, src, ra\ n", Label := "for a digraph, source, and range", Name := "DigraphRemoveEdge" ) , content := 0, count := [ 3, 3, 21, 4 ], mathmode := "Text", name := "Oper", next := 78030, root := ~, start := 77929, stop := 78029 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "Returns", next := 78062, root := ~, start := 78033, stop := 78061 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78100, root := ~, start := 78086, stop := 78099 ), rec( content := " is a di\ graph with no multiple edges and ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78152, root := ~, start := 78141, stop := 78151 ), rec( content := " is a\n \ pair of vertices of ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78196, root := ~, start := 78182, stop := 78195 ), rec( content := ", or ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78211, root := ~, start := 78201, stop := 78210 ), rec( content := " and ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78226, root := ~, start := 78216, stop := 78225 ), rec( content := " are\n \ vertices of ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78261, root := ~, start := 78247, stop := 78260 ), rec( content := ", then t\ his operation returns a digraph\n constructed from ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78336, root := ~, start := 78322, stop := 78335 ), rec( content := " by remo\ ving the edge specified by\n ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78386, root := ~, start := 78375, stop := 78385 ), rec( content := " or ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ src, ran]", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "A", next := 78407, root := ~, start := 78390, stop := 78406 ), rec( content := ". ", count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "P", next := 78413, root := ~, start := 78409, stop := 78412 ), rec( content := " \n\n \ If ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "A", next := 78438, root := ~, start := 78424, stop := 78437 ), rec( content := " belongs\ to ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "Ref", next := 78486, root := ~, start := 78450, stop := 78485 ), rec( content := ", then t\ he edge is removed directly from\n ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "A", next := 78545, root := ~, start := 78531, stop := 78544 ), rec( content := ". If ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "A", next := 78564, root := ~, start := 78550, stop := 78563 ), rec( content := " belongs\ to ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "Ref", next := 78614, root := ~, start := 78576, stop := 78613 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "A", next := 78651, root := ~, start := 78637, stop := 78650 ), rec( content := " without\ \n the edge is returned.\n ", count := [ 3, 3, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "P", next := 78694, root := ~, start := 78690, stop := 78693 ), rec( content := "\n\n \ Note that if ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 78727, root := ~, start := 78713, stop := 78726 ), rec( content := " belongs\ to ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "Ref", next := 78771, root := ~, start := 78739, stop := 78770 ), rec( content := ",\n t\ hen a new copy of ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 78810, root := ~, start := 78796, stop := 78809 ), rec( content := " will be\ returned even if ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 78847, root := ~, start := 78836, stop := 78846 ), rec( content := " or \n \ ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ src, ran]", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 78873, root := ~, start := 78856, stop := 78872 ), rec( content := " does no\ t define an edge of ", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "A", next := 78915, root := ~, start := 78901, stop := 78914 ), rec( content := ".", count := [ 3, 3, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 21, 8 ], mathmode := "Text", name := "P", next := 78920, root := ~, start := 78916, stop := 78919 ), rec( content := "\n\n \ ", count := [ 3, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(250000);\n\ngap> D := DigraphRemoveEdge(D, [250000, 1]);\n\ngap> new := DigraphRemoveEdge(D, [25000, 2]);;\nga\ p> new = D;\ntrue\ngap> IsIdenticalObj(new, D);\nfalse\ngap> D := DigraphMutab\ leCopy(D);;\ngap> new := DigraphRemoveEdge(D, 2500, 2);;\ngap> IsIdenticalObj(\ new, D);\ntrue\n", count := [ 3, 3, 21, 9 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 9 ], mathmode := "Text", name := "Example", next := 79350, root := ~, start := 78926, stop := 79349 ), rec( content := "\n ", count := [ 3, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 21, 5 ], mathmode := "Text", name := "Description", next := 79367, root := ~, start := 78065, stop := 79366 ) ], count := [ 3, 3, 21, 1 ], mathmode := "Text", name := "ManSection", next := 79381, root := ~, start := 77821, stop := 79380 ), rec( content := "\n\n ", count := [ 3, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edge", Name := "DigraphRemoveEd\ geOrbit" ), content := 0, count := [ 3, 3, 22, 2 ], mathmode := "Text", name := "Oper", next := 79459, root := ~, start := 79402, stop := 79458 ), rec( attributes := rec( ), content := [ rec( content := "\n A n\ ew digraph.\n ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "Returns", next := 79503, root := ~, start := 79462, stop := 79502 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns a new digraph with the same vertices as\n ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79605, root := ~, start := 79591, stop := 79604 ), rec( content := " and wit\ h the orbit of the edge ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79648, root := ~, start := 79637, stop := 79647 ), rec( content := " (under \ the action\n of the ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "Ref", next := 79704, root := ~, start := 79678, stop := 79703 ), rec( content := " of ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79722, root := ~, start := 79708, stop := 79721 ), rec( content := ") remove\ d.\n If ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79751, root := ~, start := 79740, stop := 79750 ), rec( content := " is not \ an edge in ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79784, root := ~, start := 79770, stop := 79783 ), rec( content := ", then " , count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79805, root := ~, start := 79791, stop := 79804 ), rec( content := " is\n \ returned unchanged. The argument ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "A", next := 79860, root := ~, start := 79846, stop := 79859 ), rec( content := " must be\ an immutable\n digraph.\n ", count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "P", next := 79903, root := ~, start := 79899, stop := 79902 ), rec( content := "\n\n \ An edge is simply a pair of vertices of ", count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "A", next := 79963, root := ~, start := 79949, stop := 79962 ), rec( content := ".\n " , count := [ 3, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := CayleyDigraph(DihedralGroup(8));\n\ngap> gr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);\n\ngap> DigraphEdges(gr1);\n[ [ 1, 2 ], [ 1, 3 ],\ [ 1, 4 ], [ 2, 1 ], [ 2, 5 ], [ 2, 6 ], \n [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 4\ , 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5, 7 ], [ 5, 6 ], [ 5, 8 ], [ 6, 4 ], [ 6, 8 \ ], [ 6, 2 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 ], [ 8, 3 ], [ 8, 2 ], [ 8, 5 ] ]\ \ngap> DigraphEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ],\ [ 2, 5 ], \n [ 2, 6 ], [ 2, 7 ], [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 3, 6 ], \n \ [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 7 ], [ 5, 6 ], \n [ 5, 8 ], [ 5\ , 4 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ], [ 6, 3 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 \ ], [ 7, 2 ], [ 8, 3 ], [ 8, 2 ], \n [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3 := Digraph\ RemoveEdgeOrbit(gr2, [1, 8]);\n\n\ gap> gr3 = gr1;\ntrue\n", count := [ 3, 3, 22, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 5 ], mathmode := "Text", name := "Example", next := 80938, root := ~, start := 79969, stop := 80937 ), rec( content := "\n ", count := [ 3, 3, 22, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 22, 3 ], mathmode := "Text", name := "Description", next := 80955, root := ~, start := 79506, stop := 80954 ) ], count := [ 3, 3, 22, 1 ], mathmode := "Text", name := "ManSection", next := 80969, root := ~, start := 79387, stop := 80968 ), rec( content := "\n\n ", count := [ 3, 3, 22, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edges", Name := "DigraphRemoveEd\ ges" ), content := 0, count := [ 3, 3, 23, 2 ], mathmode := "Text", name := "Oper", next := 81044, root := ~, start := 80990, stop := 81043 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "Returns", next := 81076, root := ~, start := 81047, stop := 81075 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ one of the following holds:\n ", count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 3, 3, 23, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "digraph", count := [ 3, 3, 23, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ], mathmode := "Te\ xt", name := "A", next := 81174, root := ~, start := 81160, stop := 81173 ), rec( cont\ ent := " is a digraph with no multiple edges, and\n ", count := [ 3, 3,\ 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "edges", count := [ 3, 3, 23, 5 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ], math\ mode := "Text", name := "A", next := 81236, root := ~, start := 81224, stop :=\ 81235 ), rec( content := " is a list of pairs of vertices of ", count := [ 3,\ 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "digraph", count := [ 3, 3, 23, 5 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ] , math\ mode := "Text", name := "A", next := 81285, root := ~, start := 81271, stop :=\ 81284 ), rec( content := ", or\n ", count := [ 3, 3, 23, 5 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 5 ], name := "Ite\ m", next := 81303, root := ~, start := 81145, stop := 81302 ), rec( attributes :\ = rec( ), content := [ rec( content := " ", count := [ 3, 3, 23, 7 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "digraph", count := [ 3, 3, 23, 7 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 7 ], mathmode := "Text\ ", name := "A", next := 81331, root := ~, start := 81317, stop := 81330 ), rec( cont\ ent := " is a digraph and ", count := [ 3, 3, 23, 7 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "edges", count := [ 3, 3, 23, 7 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 3, 3, 23, 7 ], mathmode := "Text", name := "A", next\ := 81361, root := ~, start := 81349, stop := 81360 ), rec( content := " is an\ empty list ", count := [ 3, 3, 23, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 3, 3, 23, 7 ], name := "Item", next := 81386, root := ~, start := 813\ 10, stop := 81385 ) ], count := [ 3, 3, 23, 4 ], mathmode := "Text", name := "List", next := 81398, root := ~, start := 81132, stop := 81397 ), rec( content := "\n\n \ then this operation returns a digraph constructed from ", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "A", next := 81473, root := ~, start := 81459, stop := 81472 ), rec( content := "\n by\ removing all of the edges specified by ", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "A", next := 81532, root := ~, start := 81520, stop := 81531 ), rec( content := "\n (s\ ee ", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and an edge", Oper := "DigraphRemoveEdge" ), content := 0, count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "Ref", next := 81607, root := ~, start := 81542, stop := 81606 ), rec( content := ").\n \ ", count := [ 3, 3, 23, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "P", next := 81618, root := ~, start := 81614, stop := 81617 ), rec( content := "\n\n \ If ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "A", next := 81641, root := ~, start := 81627, stop := 81640 ), rec( content := " belongs\ to ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "Ref", next := 81683, root := ~, start := 81653, stop := 81682 ), rec( content := ", then t\ he edge\n is removed directly from ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "A", next := 81742, root := ~, start := 81728, stop := 81741 ), rec( content := ". If ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "A", next := 81761, root := ~, start := 81747, stop := 81760 ), rec( content := " belongs\ to ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "Ref", next := 81811, root := ~, start := 81773, stop := 81810 ), rec( content := ", the ed\ ge is removed from an immutable copy of\n ", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "A", next := 81877, root := ~, start := 81863, stop := 81876 ), rec( content := " and thi\ s new digraph is returned.", count := [ 3, 3, 23, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "P", next := 81915, root := ~, start := 81911, stop := 81914 ), rec( content := "\n\n \ Note that if ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "A", next := 81946, root := ~, start := 81934, stop := 81945 ), rec( content := " is empt\ y, then this operation\n will always return ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "A", next := 82014, root := ~, start := 82000, stop := 82013 ), rec( content := " rather \ than a copy. Also, if any element\n of ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "A", next := 82075, root := ~, start := 82063, stop := 82074 ), rec( content := " is inva\ lid (i.e. does not define an edge of ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "A", next := 82134, root := ~, start := 82120, stop := 82133 ), rec( content := ")\n t\ hen that element will simply be ignored.\n\n ", count := [ 3, 3, 23, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(250000);\n\ngap> D := DigraphRemoveEdges(D, [[250000, 1]]);\n\ngap> D := DigraphMutableCopy(D);\n\ngap> new := DigraphRemoveEdges(D, [[1\ , 2], [2, 3], [3, 100]]);\n\ngap> new = D;\ntrue\n", count := [ 3, 3, 23, 12 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 12 ], mathmode := "Text", name := "Example", next := 82622, root := ~, start := 82187, stop := 82621 ), rec( content := "\n ", count := [ 3, 3, 23, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 23, 3 ], mathmode := "Text", name := "Description", next := 82639, root := ~, start := 81079, stop := 82638 ) ], count := [ 3, 3, 23, 1 ], mathmode := "Text", name := "ManSection", next := 82653, root := ~, start := 80975, stop := 82652 ), rec( content := "\n\n ", count := [ 3, 3, 23, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveLo\ ops" ), content := 0, count := [ 3, 3, 24, 2 ], mathmode := "Text", name := "Oper", next := 82721, root := ~, start := 82674, stop := 82720 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveLo\ opsAttr" ), content := 0, count := [ 3, 3, 24, 4 ], mathmode := "Text", name := "Attr", next := 82775, root := ~, start := 82724, stop := 82774 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "Returns", next := 82807, root := ~, start := 82778, stop := 82806 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "A", next := 82845, root := ~, start := 82831, stop := 82844 ), rec( content := " is a di\ graph, then this operation returns a\n digraph constructed from ", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "A", next := 82934, root := ~, start := 82920, stop := 82933 ), rec( content := " by remo\ ving every loop. A loop\n is an edge with equal source and range.", count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "P", next := 83013, root := ~, start := 83009, stop := 83012 ), rec( content := "\n\n \ If ", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "A", next := 83036, root := ~, start := 83022, stop := 83035 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the loops are\ removed from the mutable digraph ", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "A", next := 83169, root := ~, start := 83155, stop := 83168 ), rec( content := "\n in\ -place.\n ", count := [ 3, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 4], [1, 4], [3, 4], [1, 4, 5], [1, 5]]);\n\ngap> DigraphRemoveLoops(D);\n\ngap> D := Digraph(IsMutableDigraph, [[1, 2]\ , [1]]);\n\ngap> DigraphRemoveLoops(\ D);\n\ngap> D;\n\n", count := [ 3, 3, 24, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 7 ], mathmode := "Text", name := "Example", next := 83623, root := ~, start := 83188, stop := 83622 ), rec( content := "\n ", count := [ 3, 3, 24, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 24, 5 ], mathmode := "Text", name := "Description", next := 83640, root := ~, start := 82810, stop := 83639 ) ], count := [ 3, 3, 24, 1 ], mathmode := "Text", name := "ManSection", next := 83654, root := ~, start := 82659, stop := 83653 ), rec( content := "\n\n ", count := [ 3, 3, 24, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveAl\ lMultipleEdges" ), content := 0, count := [ 3, 3, 25, 2 ], mathmode := "Text", name := "Oper", next := 83733, root := ~, start := 83675, stop := 83732 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphRemoveAl\ lMultipleEdgesAttr" ), content := 0, count := [ 3, 3, 25, 4 ], mathmode := "Text", name := "Attr", next := 83798, root := ~, start := 83736, stop := 83797 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "Returns", next := 83830, root := ~, start := 83801, stop := 83829 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "A", next := 83868, root := ~, start := 83854, stop := 83867 ), rec( content := " is a di\ graph, then this operation returns a\n digraph constructed from ", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "A", next := 83957, root := ~, start := 83943, stop := 83956 ), rec( content := " by remo\ ving all multiple edges.\n The result is the largest subdigraph of ", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "A", next := 84048, root := ~, start := 84034, stop := 84047 ), rec( content := " which d\ oes not\n contain multiple edges.", count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "P", next := 84095, root := ~, start := 84091, stop := 84094 ), rec( content := "\n\n \ If ", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "A", next := 84118, root := ~, start := 84104, stop := 84117 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the multiple \ edges of the mutable digraph ", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "A", next := 84246, root := ~, start := 84232, stop := 84245 ), rec( content := " are\n \ removed in-place.\n ", count := [ 3, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := Digraph([[1, 2, 3, 2], [1, 1, 3], [2, 2, 2]]);\n\ngap> D2 := DigraphRemoveAllMultipleEdges(D1)\ ;\n\ngap> OutNeighbours(D2);\n[ [ \ 1, 2, 3 ], [ 1, 3 ], [ 2 ] ]\ngap> D := Digraph(IsMutableDigraph, [[2, 2], [1]\ ]);\n\ngap> DigraphRemoveAllMul\ tipleEdges(D);\n\ngap> D;\n\n", count := [ 3, 3, 25, 7 ], name := "PCDAT\ A", root := ~ ) ], count := [ 3, 3, 25, 7 ], mathmode := "Text", name := "Example", next := 84799, root := ~, start := 84277, stop := 84798 ), rec( content := "\n ", count := [ 3, 3, 25, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 25, 5 ], mathmode := "Text", name := "Description", next := 84816, root := ~, start := 83833, stop := 84815 ) ], count := [ 3, 3, 25, 1 ], mathmode := "Text", name := "ManSection", next := 84830, root := ~, start := 83660, stop := 84829 ), rec( content := "\n\n ", count := [ 3, 3, 25, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, edges", Label := "for a digraph \ and a list of edges", Name := "DigraphReverseEdges" ), content := 0, count := [ 3, 3, 26, 2 ], mathmode := "Text", name := "Oper", next := 84952, root := ~, start := 84851, stop := 84951 ), rec( attributes := rec( Arg := "digraph, edge", Label := "for a digraph \ and an edge", Name := "DigraphReverseEdge" ), content := 0, count := [ 3, 3, 26, 4 ], mathmode := "Text", name := "Oper", next := 85046, root := ~, start := 84955, stop := 85045 ), rec( attributes := rec( Arg := "digraph, src, ra\ n", Label := "for a digraph, source, and range", Name := "DigraphReverseEdge" ), content := 0, count := [ 3, 3, 26, 6 ], mathmode := "Text", name := "Oper", next := 85151, root := ~, start := 85049, stop := 85150 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "Returns", next := 85183, root := ~, start := 85154, stop := 85182 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 85221, root := ~, start := 85207, stop := 85220 ), rec( content := " is a di\ graph without multiple edges, and ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 85275, root := ~, start := 85263, stop := 85274 ), rec( content := " is a\n \ list of pairs of vertices of ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 85328, root := ~, start := 85314, stop := 85327 ), rec( content := " (the en\ tries of each pair\n corresponding to the source and the range of an edge, \ respectively),\n then ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphReverseEdges", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "C", next := 85463, root := ~, start := 85437, stop := 85462 ), rec( content := " returns\ a digraph constructed from\n ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 85517, root := ~, start := 85503, stop := 85516 ), rec( content := " by reve\ rsing the orientation of every edge specified by\n ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dges", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "A", next := 85590, root := ~, start := 85578, stop := 85589 ), rec( content := ". If onl\ y one edge is to be reversed, then\n ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphReverseEdge", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "C", next := 85662, root := ~, start := 85637, stop := 85661 ), rec( content := " can be \ used instead. In this case, the second\n argument should just be a single v\ ertex-pair, or the second and third\n arguments should be the source and ra\ nge of an edge respectively. ", count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "P", next := 85857, root := ~, start := 85853, stop := 85856 ), rec( content := "\n\n \ Note that even though ", count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "A", next := 85899, root := ~, start := 85885, stop := 85898 ), rec( content := " cannot \ have multiple edges, the\n output may have multiple edges. ", count := [ 3, 3, 26, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "P", next := 85972, root := ~, start := 85968, stop := 85971 ), rec( content := "\n\n \ If ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "A", next := 85995, root := ~, start := 85981, stop := 85994 ), rec( content := " belongs\ to ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "Ref", next := 86037, root := ~, start := 86007, stop := 86036 ), rec( content := ", then t\ he edges\n are reversed in ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "A", next := 86088, root := ~, start := 86074, stop := 86087 ), rec( content := ". If ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "A", next := 86107, root := ~, start := 86093, stop := 86106 ), rec( content := " belongs\ to ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "Ref", next := 86157, root := ~, start := 86119, stop := 86156 ), rec( content := ", an imm\ utable copy of ", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "A", next := 86194, root := ~, start := 86180, stop := 86193 ), rec( content := " with th\ e\n specified edges reversed is returned.", count := [ 3, 3, 26, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 26, 10 ], mathmode := "Text", name := "P", next := 86249, root := ~, start := 86245, stop := 86248 ), rec( content := "\n ", count := [ 3, 3, 26, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDiSparse6String(\".Tg?i@s?t_e?_qEsC\");\n\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 7 ], [\ 1, 8 ], [ 5, 21 ], [ 7, 19 ], [ 9, 1 ], \n [ 11, 2 ], [ 21, 1 ] ]\ngap> new \ := DigraphReverseEdge(D, [7, 19]);\n\ngap> DigraphEdges(new);\n[ [ 1, 2 ], [ 1, 7 ], [ 1, 8 ], [ 5, 21 ], [ 9, \ 1 ], [ 11, 2 ], \n [ 19, 7 ], [ 21, 1 ] ]\ngap> D2 := DigraphMutableCopy(new)\ ;;\ngap> new := DigraphReverseEdges(D2, [[19, 7]]);;\ngap> D2 = new;\ntrue\nga\ p> D = new;\ntrue\n", count := [ 3, 3, 26, 11 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 11 ], mathmode := "Text", name := "Example", next := 86832, root := ~, start := 86254, stop := 86831 ), rec( content := "\n ", count := [ 3, 3, 26, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 26, 7 ], mathmode := "Text", name := "Description", next := 86849, root := ~, start := 85186, stop := 86848 ) ], count := [ 3, 3, 26, 1 ], mathmode := "Text", name := "ManSection", next := 86863, root := ~, start := 84836, stop := 86862 ), rec( content := "\n\n ", count := [ 3, 3, 26, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2, ...", Label := "for an\n ar\ bitrary number of digraphs", Name := "DigraphDisjointUnion" ), content := 0, count := [ 3, 3, 27, 2 ], mathmode := "Text", name := "Func", next := 86985, root := ~, start := 86884, stop := 86984 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphDisjointUnion" ), content := 0, count := [ 3, 3, 27, 4 ], mathmode := "Text", name := "Func", next := 87065, root := ~, start := 86988, stop := 87064 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "Returns", next := 87097, root := ~, start := 87068, stop := 87096 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "A", next := 87149, root := ~, start := 87140, stop := 87148 ), rec( content := ", ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "A", next := 87160, root := ~, start := 87151, stop := 87159 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDisjointUnion", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "C", next := 87217, root := ~, start := 87190, stop := 87216 ), rec( content := " returns\ their disjoint union.\n\n In the second form, if ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "A", next := 87287, root := ~, start := 87276, stop := 87286 ), rec( content := " is a no\ n-empty list of digraphs, then\n ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDisjointUnion", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "C", next := 87357, root := ~, start := 87330, stop := 87356 ), rec( content := " returns\ the disjoint union of the digraphs\n contained in the list. ", count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 27, 6 ], mathmode := "Text", name := "P", next := 87433, root := ~, start := 87429, stop := 87432 ), rec( content := "\n\n \ For a disjoint union of digraphs, the vertex set is the disjoint union of\n \ the vertex sets, and the edge list is the disjoint union of the edge lists.\n\ ", count := [ 3, 3, 27, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "P", next := 87601, root := ~, start := 87597, stop := 87600 ), rec( content := "\n\n \ More specifically, for a collection of digraphs ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87664, root := ~, start := 87655, stop := 87663 ), rec( content := ", ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87675, root := ~, start := 87666, stop := 87674 ), rec( content := ",\n " , count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87691, root := ~, start := 87681, stop := 87690 ), rec( content := ", the di\ sjoint union with have\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87751, root := ~, start := 87726, stop := 87750 ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87760, root := ~, start := 87751, stop := 87759 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87768, root := ~, start := 87760, stop := 87767 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "+\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87777, root := ~, start := 87769, stop := 87776 ), rec( content := "\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87807, root := ~, start := 87782, stop := 87806 ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87816, root := ~, start := 87807, stop := 87815 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87824, root := ~, start := 87816, stop := 87823 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "+\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87833, root := ~, start := 87825, stop := 87832 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87844, root := ~, start := 87834, stop := 87843 ), rec( content := " vertice\ s.\n The edges of ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87881, root := ~, start := 87872, stop := 87880 ), rec( content := " will re\ main unchanged, whilst the edges of the\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87941, root := ~, start := 87933, stop := 87940 ), rec( content := "th digra\ ph, ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 87961, root := ~, start := 87953, stop := 87960 ), rec( attributes := rec( ), content := [ rec( content := "[\ i]", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 87971, root := ~, start := 87961, stop := 87970 ), rec( content := ", will b\ e changed so that they\n belong to the vertices of the disjoint union corre\ sponding to\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 88080, root := ~, start := 88072, stop := 88079 ), rec( attributes := rec( ), content := [ rec( content := "[\ i]", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88090, root := ~, start := 88080, stop := 88089 ), rec( content := ". In pa\ rticular, the edges of ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 88129, root := ~, start := 88121, stop := 88128 ), rec( attributes := rec( ), content := [ rec( content := "[\ i]", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88139, root := ~, start := 88129, stop := 88138 ), rec( content := " will\n \ have their source and range increased by\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88219, root := ~, start := 88194, stop := 88218 ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 88228, root := ~, start := 88219, stop := 88227 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88236, root := ~, start := 88228, stop := 88235 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "+\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88245, root := ~, start := 88237, stop := 88244 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88256, root := ~, start := 88246, stop := 88255 ), rec( content := " ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "+\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88265, root := ~, start := 88257, stop := 88264 ), rec( content := "\n ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88295, root := ~, start := 88270, stop := 88294 ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "A", next := 88303, root := ~, start := 88295, stop := 88302 ), rec( attributes := rec( ), content := [ rec( content := "[\ i-1])", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "C", next := 88316, root := ~, start := 88303, stop := 88315 ), rec( content := ".\n\n \ ", count := [ 3, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 27, 8 ], mathmode := "Text", name := "P", next := 88327, root := ~, start := 88323, stop := 88326 ), rec( content := "\n\n \ Note that previously set ", count := [ 3, 3, 27, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 27, 8 ], mathmode := "Text", name := "Ref", next := 88391, root := ~, start := 88358, stop := 88390 ), rec( content := " will be\ lost.", count := [ 3, 3, 27, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "P", next := 88409, root := ~, start := 88405, stop := 88408 ), rec( content := "\n\n \ If the first digraph ", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 88445, root := ~, start := 88436, stop := 88444 ), rec( content := " [", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 88461, root := ~, start := 88447, stop := 88460 ), rec( content := "] belong\ s to ", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "Ref", next := 88508, root := ~, start := 88474, stop := 88507 ), rec( content := ", then " , count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 88524, root := ~, start := 88515, stop := 88523 ), rec( content := " [", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 88540, root := ~, start := 88526, stop := 88539 ), rec( content := "] is mod\ ified in\n place to contain the appropriate vertices and edges. If ", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "A", next := 88631, root := ~, start := 88617, stop := 88630 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "Ref", next := 88679, root := ~, start := 88647, stop := 88678 ), rec( content := ", a new \ immutable digraph containing the\n appropriate vertices and edges is return\ ed.\n", count := [ 3, 3, 27, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := CycleDigraph(3);\n\ngap> \ OutNeighbours(D1);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> D2 := CompleteDigraph(3);\n<\ immutable complete digraph with 3 vertices>\ngap> OutNeighbours(D2);\n[ [ 2, 3\ ], [ 1, 3 ], [ 1, 2 ] ]\ngap> union := DigraphDisjointUnion(D1, D2);\n\ngap> OutNeighbours(union);\n[ [ 2 ], [ \ 3 ], [ 1 ], [ 5, 6 ], [ 4, 6 ], [ 4, 5 ] ]\n", count := [ 3, 3, 27, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 27, 10 ], mathmode := "Text", name := "Example", next := 89221, root := ~, start := 88768, stop := 89220 ), rec( content := "\n ", count := [ 3, 3, 27, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 27, 5 ], mathmode := "Text", name := "Description", next := 89238, root := ~, start := 87100, stop := 89237 ) ], count := [ 3, 3, 27, 1 ], mathmode := "Text", name := "ManSection", next := 89252, root := ~, start := 86869, stop := 89251 ), rec( content := "\n\n ", count := [ 3, 3, 27, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2, ...", Label := "for a positive\ number of digraphs", Name := "DigraphEdgeUnion" ), content := 0, count := [ 3, 3, 28, 2 ], mathmode := "Text", name := "Func", next := 89368, root := ~, start := 89273, stop := 89367 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphEdgeUnion" ), content := 0, count := [ 3, 3, 28, 4 ], mathmode := "Text", name := "Func", next := 89444, root := ~, start := 89371, stop := 89443 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "Returns", next := 89476, root := ~, start := 89447, stop := 89475 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "A", next := 89528, root := ~, start := 89519, stop := 89527 ), rec( content := ", ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "A", next := 89539, root := ~, start := 89530, stop := 89538 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdgeUnion", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "C", next := 89592, root := ~, start := 89569, stop := 89591 ), rec( content := " returns\ their edge union.\n\n In the second form, if ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "A", next := 89658, root := ~, start := 89647, stop := 89657 ), rec( content := " is a no\ n-empty list of digraphs, then\n ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdgeUnion", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "C", next := 89724, root := ~, start := 89701, stop := 89723 ), rec( content := " returns\ the edge union of the digraphs contained\n in the list. ", count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "P", next := 89796, root := ~, start := 89792, stop := 89795 ), rec( content := "\n\n \ The vertex set of the edge union of a collection of digraphs is the\n ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ nion", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "E", next := 89886, root := ~, start := 89874, stop := 89885 ), rec( content := " of the \ vertex sets, whilst the edge list of the edge union is\n the ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ oncatenation", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "E", next := 89977, root := ~, start := 89957, stop := 89976 ), rec( content := " of the \ edge lists. The number of vertices of the\n edge union is equal to the ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximum", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "E", next := 90073, root := ~, start := 90059, stop := 90072 ), rec( content := " number \ of vertices of one of the\n digraphs, whilst the number of edges of the edg\ e union will equal the\n ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ um", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "E", next := 90195, root := ~, start := 90185, stop := 90194 ), rec( content := " of the \ number of edges of each digraph. ", count := [ 3, 3, 28, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 28, 7 ], mathmode := "Text", name := "P", next := 90240, root := ~, start := 90236, stop := 90239 ), rec( content := "\n\n \ Note that previously set ", count := [ 3, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 28, 7 ], mathmode := "Text", name := "Ref", next := 90304, root := ~, start := 90271, stop := 90303 ), rec( content := " will be\ lost.\n ", count := [ 3, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "P", next := 90327, root := ~, start := 90323, stop := 90326 ), rec( content := "\n\n \ If the first digraph ", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 90363, root := ~, start := 90354, stop := 90362 ), rec( content := " [", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 90379, root := ~, start := 90365, stop := 90378 ), rec( content := "] belong\ s to ", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "Ref", next := 90426, root := ~, start := 90392, stop := 90425 ), rec( content := ", then " , count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 90442, root := ~, start := 90433, stop := 90441 ), rec( content := " [", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 90458, root := ~, start := 90444, stop := 90457 ), rec( content := "] is mod\ ified in\n place to contain the appropriate vertices and edges. If ", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "A", next := 90549, root := ~, start := 90535, stop := 90548 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "Ref", next := 90597, root := ~, start := 90565, stop := 90596 ), rec( content := ", a new \ immutable digraph containing the\n appropriate vertices and edges is return\ ed.\n", count := [ 3, 3, 28, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(10);\n\ngap>\ DigraphEdgeUnion(D, D);\n\ \ngap> D1 := Digraph([[2], [1]]);\n\ngap> D2 := Digraph([[2, 3], [2], [1]]);\n\ngap> union := DigraphEdgeUnion(D1, D2);\n\ngap> OutNeighbours(union);\n[ [ 2, 2, 3 ], [ 1, 2 ],\ [ 1 ] ]\ngap> union = DigraphByEdges(\n> Concatenation(DigraphEdges(D1), Digr\ aphEdges(D2)));\ntrue\n", count := [ 3, 3, 28, 9 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 9 ], mathmode := "Text", name := "Example", next := 91270, root := ~, start := 90686, stop := 91269 ), rec( content := "\n ", count := [ 3, 3, 28, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 28, 5 ], mathmode := "Text", name := "Description", next := 91287, root := ~, start := 89479, stop := 91286 ) ], count := [ 3, 3, 28, 1 ], mathmode := "Text", name := "ManSection", next := 91301, root := ~, start := 89258, stop := 91300 ), rec( content := "\n\n ", count := [ 3, 3, 28, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2, ...", Label := "for a positive\ number of digraphs", Name := "DigraphJoin" ), content := 0, count := [ 3, 3, 29, 2 ], mathmode := "Text", name := "Func", next := 91412, root := ~, start := 91322, stop := 91411 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphJoin" ), content := 0, count := [ 3, 3, 29, 4 ], mathmode := "Text", name := "Func", next := 91483, root := ~, start := 91415, stop := 91482 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "Returns", next := 91515, root := ~, start := 91486, stop := 91514 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "A", next := 91567, root := ~, start := 91558, stop := 91566 ), rec( content := ", ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "A", next := 91578, root := ~, start := 91569, stop := 91577 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphJoin", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "C", next := 91626, root := ~, start := 91608, stop := 91625 ), rec( content := " returns\ their join.\n\n In the second form, if ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "A", next := 91686, root := ~, start := 91675, stop := 91685 ), rec( content := " is a no\ n-empty list of digraphs,\n then ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphJoin", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "C", next := 91747, root := ~, start := 91729, stop := 91746 ), rec( content := " returns\ the join of the digraphs contained in the\n list. ", count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "P", next := 91812, root := ~, start := 91808, stop := 91811 ), rec( content := "\n\n \ The join of a collection of digraphs ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "A", next := 91864, root := ~, start := 91855, stop := 91863 ), rec( content := ", ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "A", next := 91875, root := ~, start := 91866, stop := 91874 ), rec( content := ", ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "C", next := 91887, root := ~, start := 91877, stop := 91886 ), rec( content := " is\n \ formed by first taking the ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ DisjointUnion", Label := "for a list of digraphs" ), content := 0, count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "Ref", next := 91993, root := ~, start := 91922, stop := 91992 ), rec( content := " of the \ collection.\n\n In the disjoint union, if ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ \\neq j", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "M", next := 92059, root := ~, start := 92044, stop := 92058 ), rec( content := " then th\ ere are no edges between\n vertices corresponding to digraphs ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "A", next := 92139, root := ~, start := 92131, stop := 92138 ), rec( attributes := rec( ), content := [ rec( content := "[\ i]", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "C", next := 92149, root := ~, start := 92139, stop := 92148 ), rec( content := " and\n \ ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "A", next := 92166, root := ~, start := 92158, stop := 92165 ), rec( attributes := rec( ), content := [ rec( content := "[\ j]", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "C", next := 92176, root := ~, start := 92166, stop := 92175 ), rec( content := " in the \ collection; the join is created by including\n all such edges.", count := [ 3, 3, 29, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 29, 7 ], mathmode := "Text", name := "P", next := 92252, root := ~, start := 92248, stop := 92251 ), rec( content := "\n\n \ For example, the join of two empty digraphs is a complete bipartite\n digra\ ph.", count := [ 3, 3, 29, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 29, 8 ], mathmode := "Text", name := "P", next := 92342, root := ~, start := 92338, stop := 92341 ), rec( content := "\n\n \ Note that previously set ", count := [ 3, 3, 29, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 29, 8 ], mathmode := "Text", name := "Ref", next := 92406, root := ~, start := 92373, stop := 92405 ), rec( content := " will be\ lost.\n ", count := [ 3, 3, 29, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "P", next := 92429, root := ~, start := 92425, stop := 92428 ), rec( content := "\n\n \ If the first digraph ", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 92465, root := ~, start := 92456, stop := 92464 ), rec( content := " [", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 92481, root := ~, start := 92467, stop := 92480 ), rec( content := "] belong\ s to ", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "Ref", next := 92528, root := ~, start := 92494, stop := 92527 ), rec( content := ", then " , count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 92544, root := ~, start := 92535, stop := 92543 ), rec( content := " [", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[1]", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 92560, root := ~, start := 92546, stop := 92559 ), rec( content := "] is mod\ ified in\n place to contain the appropriate vertices and edges. If ", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "A", next := 92651, root := ~, start := 92637, stop := 92650 ), rec( content := " belongs\ \n to ", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "Ref", next := 92699, root := ~, start := 92667, stop := 92698 ), rec( content := ", a new \ immutable digraph containing the\n appropriate vertices and edges is return\ ed.\n", count := [ 3, 3, 29, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(3);\n\n\ gap> IsCompleteDigraph(DigraphJoin(D, D));\ntrue\ngap> D2 := CycleDigraph(3);\ \n\ngap> DigraphJoin(D, D2);\n\n", count := [ 3, 3, 29, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 29, 10 ], mathmode := "Text", name := "Example", next := 93084, root := ~, start := 92788, stop := 93083 ), rec( content := "\n ", count := [ 3, 3, 29, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 29, 5 ], mathmode := "Text", name := "Description", next := 93101, root := ~, start := 91518, stop := 93100 ) ], count := [ 3, 3, 29, 1 ], mathmode := "Text", name := "ManSection", next := 93115, root := ~, start := 91307, stop := 93114 ), rec( content := "\n\n ", count := [ 3, 3, 29, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "gr1, gr2, ...", Label := "for a positive\ number of digraphs", Name := "DigraphCartesianProduct" ), content := 0, count := [ 3, 3, 30, 2 ], mathmode := "Text", name := "Func", next := 93240, root := ~, start := 93136, stop := 93239 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphCartesianProduct" ), content := 0, count := [ 3, 3, 30, 4 ], mathmode := "Text", name := "Func", next := 93327, root := ~, start := 93243, stop := 93326 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "Returns", next := 93359, root := ~, start := 93330, stop := 93358 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "A", next := 93412, root := ~, start := 93402, stop := 93411 ), rec( content := ", ", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r2", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "A", next := 93424, root := ~, start := 93414, stop := 93423 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProduct", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "C", next := 93484, root := ~, start := 93454, stop := 93483 ), rec( content := " returns\ a digraph isomorphic to their\n Cartesian product.\n ", count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "P", next := 93554, root := ~, start := 93550, stop := 93553 ), rec( content := "\n\n \ In the second form, if ", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "A", next := 93594, root := ~, start := 93583, stop := 93593 ), rec( content := " is a no\ n-empty list of digraphs,\n then ", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProduct", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "C", next := 93667, root := ~, start := 93637, stop := 93666 ), rec( content := " returns\ a digraph isomorphic to the \n Cartesian product of the digraphs contained\ in the list.\n ", count := [ 3, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "P", next := 93774, root := ~, start := 93770, stop := 93773 ), rec( content := "\n \n \ Mathematically, the Cartesian product of two digraphs ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93843, root := ~, start := 93835, stop := 93842 ), rec( content := ", ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93853, root := ~, start := 93845, stop := 93852 ), rec( content := "\n is\ a digraph with vertex set ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ artesian(DigraphVertices(G),\n DigraphVertices(H))", count := [ 3, 3, 30, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93947, root := ~, start := 93887, stop := 93946 ), rec( content := " such th\ at there is an edge from ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, u']", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 93994, root := ~, start := 93980, stop := 93993 ), rec( content := " to\n \ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v, v']", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94016, root := ~, start := 94002, stop := 94015 ), rec( content := " iff ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ u = v ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94035, root := ~, start := 94021, stop := 94034 ), rec( content := " and the\ re is an edge from ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ '", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94071, root := ~, start := 94062, stop := 94070 ), rec( content := " to\n \ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ '", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94088, root := ~, start := 94079, stop := 94087 ), rec( content := " in ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94100, root := ~, start := 94092, stop := 94099 ), rec( content := " or ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ u' = v'", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94119, root := ~, start := 94104, stop := 94118 ), rec( content := " and the\ re is an edge from\n ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94158, root := ~, start := 94150, stop := 94157 ), rec( content := " to ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94170, root := ~, start := 94162, stop := 94169 ), rec( content := " in ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "C", next := 94182, root := ~, start := 94174, stop := 94181 ), rec( content := ".\n " , count := [ 3, 3, 30, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "P", next := 94192, root := ~, start := 94188, stop := 94191 ), rec( content := "\n\n \ Due to technical reasons, the digraph ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94244, root := ~, start := 94236, stop := 94243 ), rec( content := " returne\ d by\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProduct", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94291, root := ~, start := 94261, stop := 94290 ), rec( content := " has ver\ tex set\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. DigraphNrVertices(G)*DigraphNrVertices(H)]", count := [ 3, 3, 30, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94367, root := ~, start := 94311, stop := 94366 ), rec( content := " instead\ , and the\n exact\n method of encoding pairs of vertices into integers \ is implementation\n specific. The original vertex pair can be somewhat reg\ ained by using\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ CartesianProductProjections" ), content := 0, count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "Ref", next := 94595, root := ~, start := 94547, stop := 94594 ), rec( content := ". In add\ ition,\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "Ref", next := 94647, root := ~, start := 94614, stop := 94646 ), rec( content := " are pre\ served: if vertex pair\n ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u,u']", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94695, root := ~, start := 94682, stop := 94694 ), rec( content := " was enc\ oded as ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94719, root := ~, start := 94711, stop := 94718 ), rec( content := " then th\ e vertex label of ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94753, root := ~, start := 94745, stop := 94752 ), rec( content := "\n wi\ ll be the pair of vertex labels of ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94803, root := ~, start := 94795, stop := 94802 ), rec( content := " and ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ '", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94817, root := ~, start := 94808, stop := 94816 ), rec( content := " i.e.\n \ ", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertexLabel(D,i) = [DigraphVertexLabel(G,u),\n DigraphVertexLabel(H\ ,u')]", count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "C", next := 94917, root := ~, start := 94827, stop := 94916 ), rec( content := ".\n " , count := [ 3, 3, 30, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "P", next := 94927, root := ~, start := 94923, stop := 94926 ), rec( content := "\n\n \ As the Cartesian product is associative, the Cartesian product of a collection\ \n of digraphs ", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "A", next := 95039, root := ~, start := 95029, stop := 95038 ), rec( content := ", ", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r2", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "A", next := 95051, root := ~, start := 95041, stop := 95050 ), rec( content := ", ", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "C", next := 95063, root := ~, start := 95053, stop := 95062 ), rec( content := " is comp\ uted in the obvious \n fashion. \n ", count := [ 3, 3, 30, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 30, 10 ], mathmode := "Text", name := "P", next := 95114, root := ~, start := 95110, stop := 95113 ), rec( content := "\n\n", count := [ 3, 3, 30, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := ChainDigraph(4);\n\ngap> \ gr2 := CycleDigraph(3);\n\ngap> gr3 :\ = DigraphCartesianProduct(gr, gr2);\n\ngap> IsIsomorphicDigraph(gr3, \n> Digraph([[2, 5], [3, 6], [4, 7], [8],\ \n> [6, 9], [7, 10], [8, 11], [12],\n> [10, 1], [11, 2], [1\ 2, 3], [4]]));\ntrue\n", count := [ 3, 3, 30, 11 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 11 ], mathmode := "Text", name := "Example", next := 95548, root := ~, start := 95116, stop := 95547 ), rec( content := "\n ", count := [ 3, 3, 30, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 30, 5 ], mathmode := "Text", name := "Description", next := 95565, root := ~, start := 93362, stop := 95564 ) ], count := [ 3, 3, 30, 1 ], mathmode := "Text", name := "ManSection", next := 95579, root := ~, start := 93121, stop := 95578 ), rec( content := "\n\n ", count := [ 3, 3, 30, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "gr1, gr2, ...", Label := "for a positive\ number of digraphs", Name := "DigraphDirectProduct" ), content := 0, count := [ 3, 3, 31, 2 ], mathmode := "Text", name := "Func", next := 95701, root := ~, start := 95600, stop := 95700 ), rec( attributes := rec( Arg := "list", Label := "for a list of \ digraphs", Name := "DigraphDirectProduct" ), content := 0, count := [ 3, 3, 31, 4 ], mathmode := "Text", name := "Func", next := 95785, root := ~, start := 95704, stop := 95784 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "Returns", next := 95817, root := ~, start := 95788, stop := 95816 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, if ", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "A", next := 95870, root := ~, start := 95860, stop := 95869 ), rec( content := ", ", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r2", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "A", next := 95882, root := ~, start := 95872, stop := 95881 ), rec( content := ", etc. a\ re digraphs, then\n ", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProduct", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "C", next := 95939, root := ~, start := 95912, stop := 95938 ), rec( content := " returns\ a digraph isomorphic to their direct\n product.\n ", count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "P", next := 96006, root := ~, start := 96002, stop := 96005 ), rec( content := "\n\n \ In the second form, if ", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "A", next := 96046, root := ~, start := 96035, stop := 96045 ), rec( content := " is a no\ n-empty list of digraphs, then\n ", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProduct", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "C", next := 96116, root := ~, start := 96089, stop := 96115 ), rec( content := " returns\ a digraph isomorphic to the direct\n product of the digraphs contained in \ the list.\n ", count := [ 3, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "P", next := 96219, root := ~, start := 96215, stop := 96218 ), rec( content := "\n\n \ Mathematically, the direct product of two digraphs ", count := [ 3, 3, 31, 7 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96284, root := ~, start := 96276, stop := 96283 ), rec( content := ", ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96294, root := ~, start := 96286, stop := 96293 ), rec( content := " is a\n \ digraph with vertex set ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ artesian(DigraphVertices(G),\n DigraphVertices(H))", count := [ 3, 3, 31, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96388, root := ~, start := 96328, stop := 96387 ), rec( content := " such th\ at there is an edge from ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, u']", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96435, root := ~, start := 96421, stop := 96434 ), rec( content := "\n to\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v, v']", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96457, root := ~, start := 96443, stop := 96456 ), rec( content := " iff the\ re is an edge from ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96492, root := ~, start := 96484, stop := 96491 ), rec( content := " to ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96504, root := ~, start := 96496, stop := 96503 ), rec( content := " in\n \ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96520, root := ~, start := 96512, stop := 96519 ), rec( content := " and an \ edge from ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ '", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96547, root := ~, start := 96538, stop := 96546 ), rec( content := " to ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ '", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96560, root := ~, start := 96551, stop := 96559 ), rec( content := " in ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "C", next := 96572, root := ~, start := 96564, stop := 96571 ), rec( content := ". \n \ ", count := [ 3, 3, 31, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "P", next := 96583, root := ~, start := 96579, stop := 96582 ), rec( content := " \n\n \ Due to technical reasons, the digraph ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 96636, root := ~, start := 96628, stop := 96635 ), rec( content := " returne\ d by\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProduct", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 96680, root := ~, start := 96653, stop := 96679 ), rec( content := " has ver\ tex set\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. DigraphNrVertices(G)*DigraphNrVertices(H)]", count := [ 3, 3, 31, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 96756, root := ~, start := 96700, stop := 96755 ), rec( content := " instead\ , and the\n exact method of encoding pairs of vertices into integers is imp\ lementation\n specific. The original vertex pair can be somewhat regained \ by using\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ DirectProductProjections" ), content := 0, count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "Ref", next := 96976, root := ~, start := 96931, stop := 96975 ), rec( content := ". In add\ ition\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "Ref", next := 97027, root := ~, start := 96994, stop := 97026 ), rec( content := " are pre\ served: if vertex pair\n ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u,u']", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 97075, root := ~, start := 97062, stop := 97074 ), rec( content := " was enc\ oded as ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 97099, root := ~, start := 97091, stop := 97098 ), rec( content := " then th\ e vertex label of ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 97133, root := ~, start := 97125, stop := 97132 ), rec( content := "\n wi\ ll be the pair of vertex labels of ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 97183, root := ~, start := 97175, stop := 97182 ), rec( content := " and ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ '", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 97197, root := ~, start := 97188, stop := 97196 ), rec( content := " i.e.\n \ ", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertexLabel(D,i) = [DigraphVertexLabel(G,u),\n DigraphVertexLabel(H,u\ ')]", count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "C", next := 97295, root := ~, start := 97207, stop := 97294 ), rec( content := ".\n " , count := [ 3, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "P", next := 97305, root := ~, start := 97301, stop := 97304 ), rec( content := "\n\n \ As the direct product is associative, the direct product of a collection \n \ of digraphs ", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "A", next := 97411, root := ~, start := 97401, stop := 97410 ), rec( content := ", ", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r2", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "A", next := 97423, root := ~, start := 97413, stop := 97422 ), rec( content := ", ", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ..", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "C", next := 97435, root := ~, start := 97425, stop := 97434 ), rec( content := " is comp\ uted in the obvious \n fashion. \n ", count := [ 3, 3, 31, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 31, 10 ], mathmode := "Text", name := "P", next := 97486, root := ~, start := 97482, stop := 97485 ), rec( content := "\n\n", count := [ 3, 3, 31, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := ChainDigraph(4);\n\ngap> \ gr2 := CycleDigraph(3);\n\ngap> gr3 :\ = DigraphDirectProduct(gr, gr2);\n\ngap> IsIsomorphicDigraph(gr3, \n> Digraph([[6], [7], [8], [], \n> \ [10], [11], [12], [],\n> [2], [3], [4], []]));\ntrue\n", count := [ 3, 3, 31, 11 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 31, 11 ], mathmode := "Text", name := "Example", next := 97883, root := ~, start := 97488, stop := 97882 ), rec( content := "\n ", count := [ 3, 3, 31, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 31, 5 ], mathmode := "Text", name := "Description", next := 97900, root := ~, start := 95820, stop := 97899 ) ], count := [ 3, 3, 31, 1 ], mathmode := "Text", name := "ManSection", next := 97914, root := ~, start := 95585, stop := 97913 ), rec( content := "\n\n ", count := [ 3, 3, 31, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphCartesia\ nProductProjections" ), content := 0, count := [ 3, 3, 32, 2 ], mathmode := "Text", name := "Attr", next := 97998, root := ~, start := 97935, stop := 97997 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "Returns", next := 98046, root := ~, start := 98001, stop := 98045 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "A", next := 98084, root := ~, start := 98070, stop := 98083 ), rec( content := " is a Ca\ rtesian product digraph, \n ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph = DigraphCartesianProduct(gr_1, gr_2, ... )", count := [ 3, 3, 32, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 98180, root := ~, start := 98122, stop := 98179 ), rec( content := ", then\n\ ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProductProjections", count := [ 3, 3, 32, 3 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 98232, root := ~, start := 98191, stop := 98231 ), rec( content := " returns\ a list ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 98259, root := ~, start := 98248, stop := 98258 ), rec( content := "\n su\ ch that ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj[i]", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 98288, root := ~, start := 98274, stop := 98287 ), rec( content := " is the \ projection onto the ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "C", next := 98324, root := ~, start := 98316, stop := 98323 ), rec( content := "-th\n \ coordinate of the product.\n ", count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "P", next := 98367, root := ~, start := 98363, stop := 98366 ), rec( content := "\n\n \ A projection is an idempotent endomorphism of ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "A", next := 98433, root := ~, start := 98419, stop := 98432 ), rec( content := ". If\n \ ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1, gr2, ...", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "C", next := 98462, root := ~, start := 98442, stop := 98461 ), rec( content := " are all\ loopless digraphs, then the induced \n subdigraph of ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "A", next := 98540, root := ~, start := 98526, stop := 98539 ), rec( content := " on the \ image of ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj[i]", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "C", next := 98571, root := ~, start := 98557, stop := 98570 ), rec( content := " is \n \ isomorphic to ", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r_i", count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "C", next := 98605, root := ~, start := 98594, stop := 98604 ), rec( content := ".\n " , count := [ 3, 3, 32, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "P", next := 98615, root := ~, start := 98611, stop := 98614 ), rec( content := "\n\n \ Currently this attribute is only set upon creating an immutable \n digraph \ via ", count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCartesianProduct", count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "C", next := 98732, root := ~, start := 98702, stop := 98731 ), rec( content := " and the\ re is no way \n of calculating it for other digraphs.\n ", count := [ 3, 3, 32, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 32, 6 ], mathmode := "Text", name := "P", next := 98804, root := ~, start := 98800, stop := 98803 ), rec( content := "\n\n \ For more information see\n ", count := [ 3, 3, 32, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ CartesianProduct", Label := "for a list of digraphs" ), content := 0, count := [ 3, 3, 32, 6 ], mathmode := "Text", name := "Ref", next := 98907, root := ~, start := 98839, stop := 98906 ), rec( content := " \n\n \ ", count := [ 3, 3, 32, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphCartesianProduct(ChainDigraph(3), CycleDigraph(4),\n> Digrap\ h([[2], [2]]));;\ngap> HasDigraphCartesianProductProjections(D);\ntrue\ngap> p\ roj := DigraphCartesianProductProjections(D);; Length(proj);\n3\ngap> IsIdempo\ tent(proj[2]);\ntrue\ngap> RankOfTransformation(proj[3]);\n2\ngap> S := ImageS\ etOfTransformation(proj[2]);;\ngap> IsIsomorphicDigraph(CycleDigraph(4), Induc\ edSubdigraph(D, S));\ntrue\n", count := [ 3, 3, 32, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 7 ], mathmode := "Text", name := "Example", next := 99350, root := ~, start := 98914, stop := 99349 ), rec( content := "\n ", count := [ 3, 3, 32, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 32, 3 ], mathmode := "Text", name := "Description", next := 99367, root := ~, start := 98049, stop := 99366 ) ], count := [ 3, 3, 32, 1 ], mathmode := "Text", name := "ManSection", next := 99381, root := ~, start := 97920, stop := 99380 ), rec( content := "\n\n ", count := [ 3, 3, 32, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDirectPr\ oductProjections" ), content := 0, count := [ 3, 3, 33, 2 ], mathmode := "Text", name := "Attr", next := 99462, root := ~, start := 99402, stop := 99461 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "Returns", next := 99510, root := ~, start := 99465, stop := 99509 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "A", next := 99548, root := ~, start := 99534, stop := 99547 ), rec( content := " is a di\ rect product digraph, \n ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph = DigraphDirectProduct(gr_1, gr_2, ... )", count := [ 3, 3, 33, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 99638, root := ~, start := 99583, stop := 99637 ), rec( content := ", then\n\ ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProductProjections", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 99687, root := ~, start := 99649, stop := 99686 ), rec( content := " returns\ a list ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 99714, root := ~, start := 99703, stop := 99713 ), rec( content := "\n su\ ch that ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj[i]", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 99743, root := ~, start := 99729, stop := 99742 ), rec( content := " is the \ projection onto the ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "C", next := 99779, root := ~, start := 99771, stop := 99778 ), rec( content := "-th\n \ coordinate of the product.\n ", count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "P", next := 99822, root := ~, start := 99818, stop := 99821 ), rec( content := "\n\n \ A projection is an idempotent endomorphism of ", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "A", next := 99888, root := ~, start := 99874, stop := 99887 ), rec( content := ". If\n \ ", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r1, gr2, ...", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "C", next := 99917, root := ~, start := 99897, stop := 99916 ), rec( content := " are all\ loopless digraphs, then the image \n of ", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "A", next := 99982, root := ~, start := 99968, stop := 99981 ), rec( content := " under " , count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roj[i]", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "C", next := 100003, root := ~, start := 99989, stop := 100002 ), rec( content := " is isom\ orphic to ", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ r_i", count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "C", next := 100032, root := ~, start := 100021, stop := 100031 ), rec( content := ".\n " , count := [ 3, 3, 33, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "P", next := 100042, root := ~, start := 100038, stop := 100041 ), rec( content := "\n\n \ Currently this attribute is only set upon creating an immutable\n digraph v\ ia ", count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDirectProduct", count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "C", next := 100155, root := ~, start := 100128, stop := 100154 ), rec( content := " and the\ re is no way of \n calculating it for other digraphs.\n ", count := [ 3, 3, 33, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 33, 6 ], mathmode := "Text", name := "P", next := 100227, root := ~, start := 100223, stop := 100226 ), rec( content := "\n\n \ For more information, see\n ", count := [ 3, 3, 33, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ DirectProduct", Label := "for a list of digraphs" ), content := 0, count := [ 3, 3, 33, 6 ], mathmode := "Text", name := "Ref", next := 100328, root := ~, start := 100263, stop := 100327 ), rec( content := " \n\n \ ", count := [ 3, 3, 33, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphDirectProduct(ChainDigraph(3), CycleDigraph(4),\n> Digraph([\ [2], [2]]));;\ngap> HasDigraphDirectProductProjections(D);\ntrue\ngap> proj :=\ DigraphDirectProductProjections(D);; Length(proj);\n3\ngap> IsIdempotent(proj\ [2]);\ntrue\ngap> RankOfTransformation(proj[3]);\n2\ngap> P := DigraphRemoveAl\ lMultipleEdges(\n> ReducedDigraph(OnDigraphs(D, proj[2])));; \ngap> IsIsomorph\ icDigraph(CycleDigraph(4), P);\ntrue\n", count := [ 3, 3, 33, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 33, 7 ], mathmode := "Text", name := "Example", next := 100780, root := ~, start := 100335, stop := 100779 ), rec( content := "\n ", count := [ 3, 3, 33, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 33, 3 ], mathmode := "Text", name := "Description", next := 100797, root := ~, start := 99513, stop := 100796 ) ], count := [ 3, 3, 33, 1 ], mathmode := "Text", name := "ManSection", next := 100811, root := ~, start := 99387, stop := 100810 ), rec( content := "\n\n ", count := [ 3, 3, 33, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "LineDigraph" ), content := 0, count := [ 3, 3, 34, 2 ], mathmode := "Text", name := "Oper", next := 100872, root := ~, start := 100832, stop := 100871 ), rec( attributes := rec( Arg := "digraph", Name := "EdgeDigraph" ), content := 0, count := [ 3, 3, 34, 4 ], mathmode := "Text", name := "Oper", next := 100915, root := ~, start := 100875, stop := 100914 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "Returns", next := 100947, root := ~, start := 100918, stop := 100946 ), rec( attributes := rec( ), content := [ rec( content := "\n Giv\ en a digraph ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "A", next := 100998, root := ~, start := 100984, stop := 100997 ), rec( content := ", the op\ eration returns the digraph\n obtained by associating a vertex with each ed\ ge of ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "A", next := 101103, root := ~, start := 101089, stop := 101102 ), rec( content := ", and\n \ creating an edge from a vertex ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "K", next := 101152, root := ~, start := 101144, stop := 101151 ), rec( content := " to a ve\ rtex ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "K", next := 101173, root := ~, start := 101165, stop := 101172 ), rec( content := " if and \ only\n if the terminal vertex of the edge associated with ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "K", next := 101249, root := ~, start := 101241, stop := 101248 ), rec( content := " is the \ start\n vertex of the edge associated with ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "K", next := 101310, root := ~, start := 101302, stop := 101309 ), rec( content := ".", count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "P", next := 101315, root := ~, start := 101311, stop := 101314 ), rec( content := "\n\n \ Note that the returned digraph is always a new immutable digraph, and the\n \ argument ", count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "A", next := 101422, root := ~, start := 101408, stop := 101421 ), rec( content := " is neve\ r modified.\n ", count := [ 3, 3, 34, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> LineDigraph(CompleteDigraph(3));\n\ngap> LineDigraph(ChainDigraph(3));\n\n", count := [ 3, 3, 34, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 7 ], mathmode := "Text", name := "Example", next := 101641, root := ~, start := 101446, stop := 101640 ), rec( content := "\n ", count := [ 3, 3, 34, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 34, 5 ], mathmode := "Text", name := "Description", next := 101658, root := ~, start := 100950, stop := 101657 ) ], count := [ 3, 3, 34, 1 ], mathmode := "Text", name := "ManSection", next := 101672, root := ~, start := 100817, stop := 101671 ), rec( content := "\n\n ", count := [ 3, 3, 34, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "LineUndirectedD\ igraph" ), content := 0, count := [ 3, 3, 35, 2 ], mathmode := "Text", name := "Oper", next := 101743, root := ~, start := 101693, stop := 101742 ), rec( attributes := rec( Arg := "digraph", Name := "EdgeUndirectedD\ igraph" ), content := 0, count := [ 3, 3, 35, 4 ], mathmode := "Text", name := "Oper", next := 101796, root := ~, start := 101746, stop := 101795 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "Returns", next := 101828, root := ~, start := 101799, stop := 101827 ), rec( attributes := rec( ), content := [ rec( content := "\n Giv\ en a symmetric digraph ", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "A", next := 101889, root := ~, start := 101875, stop := 101888 ), rec( content := ", the op\ eration returns the\n symmetric digraph obtained by associating a vertex wi\ th each edge of\n ", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "A", next := 102008, root := ~, start := 101994, stop := 102007 ), rec( content := ", ignori\ ng directions and multiplicites, and adding an edge\n between two vertices \ if and only if the corresponding edges have a vertex in\n common.", count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "P", next := 102164, root := ~, start := 102160, stop := 102163 ), rec( content := "\n\n \ Note that the returned digraph is always a new immutable digraph, and the\n \ argument ", count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "A", next := 102271, root := ~, start := 102257, stop := 102270 ), rec( content := " is neve\ r modified.\n ", count := [ 3, 3, 35, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> LineUndirectedDigraph(CompleteDigraph(3));\n\ngap> LineUndirectedDigraph(DigraphSymmetricClosure(ChainDigr\ aph(3)));\n\n", count := [ 3, 3, 35, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 35, 7 ], mathmode := "Text", name := "Example", next := 102535, root := ~, start := 102295, stop := 102534 ), rec( content := "\n ", count := [ 3, 3, 35, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 35, 5 ], mathmode := "Text", name := "Description", next := 102552, root := ~, start := 101831, stop := 102551 ) ], count := [ 3, 3, 35, 1 ], mathmode := "Text", name := "ManSection", next := 102566, root := ~, start := 101678, stop := 102565 ), rec( content := "\n\n ", count := [ 3, 3, 35, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DoubleDigraph" ), content := 0, count := [ 3, 3, 36, 2 ], mathmode := "Text", name := "Oper", next := 102629, root := ~, start := 102587, stop := 102628 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "Returns", next := 102661, root := ~, start := 102632, stop := 102660 ), rec( attributes := rec( ), content := [ rec( content := "\n Let\ ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "A", next := 102700, root := ~, start := 102686, stop := 102699 ), rec( content := " be a di\ graph with vertex set ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "C", next := 102738, root := ~, start := 102730, stop := 102737 ), rec( content := ". This f\ unction\n returns the double digraph of ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "A", next := 102802, root := ~, start := 102788, stop := 102801 ), rec( content := ". The ve\ rtex set of the double\n digraph is the orginal vertex set together with a \ duplicate. The edges are\n ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u_1, v_2]", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "C", next := 102933, root := ~, start := 102916, stop := 102932 ), rec( content := " and ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u_2, v_1]", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "C", next := 102955, root := ~, start := 102938, stop := 102954 ), rec( content := " if and \ only if ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, v]", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "C", next := 102984, root := ~, start := 102971, stop := 102983 ), rec( content := " is an\n\ edge in ", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "A", next := 103017, root := ~, start := 103003, stop := 103016 ), rec( content := ", togeth\ er with the original edges and their\n duplicates.", count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "P", next := 103081, root := ~, start := 103077, stop := 103080 ), rec( content := "\n\n \ If ", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "A", next := 103104, root := ~, start := 103090, stop := 103103 ), rec( content := " is muta\ ble, then ", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "A", next := 103136, root := ~, start := 103122, stop := 103135 ), rec( content := " is modi\ fied in-place. If\n ", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "A", next := 103180, root := ~, start := 103166, stop := 103179 ), rec( content := " is immu\ table, then a new immutable digraph constructed as\n described above is ret\ urned.\n ", count := [ 3, 3, 36, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gamma := Digraph([[2], [3], [1]]);\n\ngap> DoubleDigraph(gamma);\n\n", count := [ 3, 3, 36, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 5 ], mathmode := "Text", name := "Example", next := 103466, root := ~, start := 103276, stop := 103465 ), rec( content := "\n ", count := [ 3, 3, 36, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 36, 3 ], mathmode := "Text", name := "Description", next := 103483, root := ~, start := 102664, stop := 103482 ) ], count := [ 3, 3, 36, 1 ], mathmode := "Text", name := "ManSection", next := 103497, root := ~, start := 102572, stop := 103496 ), rec( content := "\n\n ", count := [ 3, 3, 36, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "BipartiteDouble\ Digraph" ), content := 0, count := [ 3, 3, 37, 2 ], mathmode := "Text", name := "Oper", next := 103569, root := ~, start := 103518, stop := 103568 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "Returns", next := 103601, root := ~, start := 103572, stop := 103600 ), rec( attributes := rec( ), content := [ rec( content := "\n Let\ ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "A", next := 103640, root := ~, start := 103626, stop := 103639 ), rec( content := " be a di\ graph with vertex set ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "C", next := 103678, root := ~, start := 103670, stop := 103677 ), rec( content := ". This f\ unction\n returns the bipartite double digraph of ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "A", next := 103752, root := ~, start := 103738, stop := 103751 ), rec( content := ". The ve\ rtex set of\n the double digraph is the original vertex set together with a\ duplicate. The\n edges are ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u_1, v_2]", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "C", next := 103884, root := ~, start := 103867, stop := 103883 ), rec( content := " and ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u_2, v_1]", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "C", next := 103906, root := ~, start := 103889, stop := 103905 ), rec( content := " if and \ only if ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u,\n v]", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "C", next := 103941, root := ~, start := 103922, stop := 103940 ), rec( content := " is an e\ dge in ", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "A", next := 103970, root := ~, start := 103956, stop := 103969 ), rec( content := ". The re\ sulting graph is bipartite,\n since the orignal edges are not included in t\ he resulting digraph.", count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "P", next := 104080, root := ~, start := 104076, stop := 104079 ), rec( content := "\n\n \ If ", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "A", next := 104103, root := ~, start := 104089, stop := 104102 ), rec( content := " is muta\ ble, then ", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "A", next := 104135, root := ~, start := 104121, stop := 104134 ), rec( content := " is modi\ fied in-place. If\n ", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "A", next := 104179, root := ~, start := 104165, stop := 104178 ), rec( content := " is immu\ table, then a new immutable digraph constructed as\n described above is ret\ urned.\n ", count := [ 3, 3, 37, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gamma := Digraph([[2], [3], [1]]);\n\ngap> BipartiteDoubleDigraph(gamma);\n\n", count := [ 3, 3, 37, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 5 ], mathmode := "Text", name := "Example", next := 104473, root := ~, start := 104275, stop := 104472 ), rec( content := "\n ", count := [ 3, 3, 37, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 37, 3 ], mathmode := "Text", name := "Description", next := 104490, root := ~, start := 103604, stop := 104489 ) ], count := [ 3, 3, 37, 1 ], mathmode := "Text", name := "ManSection", next := 104504, root := ~, start := 103503, stop := 104503 ), rec( content := "\n\n ", count := [ 3, 3, 37, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphAddAllLo\ ops" ), content := 0, count := [ 3, 3, 38, 2 ], mathmode := "Text", name := "Oper", next := 104572, root := ~, start := 104525, stop := 104571 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphAddAllLo\ opsAttr" ), content := 0, count := [ 3, 3, 38, 4 ], mathmode := "Text", name := "Attr", next := 104626, root := ~, start := 104575, stop := 104625 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "Returns", next := 104658, root := ~, start := 104629, stop := 104657 ), rec( attributes := rec( ), content := [ rec( content := "\n For\ a digraph ", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "A", next := 104707, root := ~, start := 104693, stop := 104706 ), rec( content := " this op\ eration returns a new digraph constructed\n from ", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "A", next := 104780, root := ~, start := 104766, stop := 104779 ), rec( content := ", such t\ hat a loop is added for every vertex which did not\n have a loop in ", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "A", next := 104872, root := ~, start := 104858, stop := 104871 ), rec( content := ".", count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "P", next := 104877, root := ~, start := 104873, stop := 104876 ), rec( content := "\n\n \ If ", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "A", next := 104900, root := ~, start := 104886, stop := 104899 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the loops are\ added to the loopless vertices of the mutable digraph\n ", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "A", next := 105058, root := ~, start := 105044, stop := 105057 ), rec( content := " in-plac\ e.\n ", count := [ 3, 3, 38, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := EmptyDigraph(13);\n\ngap>\ D := DigraphAddAllLoops(D);\n\ngap> OutNeighbours(D);\n[ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], \ [ 7 ], [ 8 ], [ 9 ], \n [ 10 ], [ 11 ], [ 12 ], [ 13 ] ]\ngap> D := Digraph([\ [1, 2, 3], [1, 3], [1]]);\n\ngap> \ D := DigraphAddAllLoops(D);\n\ngap> OutNeighbours(D);\n[ [ 1, 2, 3 ], [ 1, 3, 2 ], [ 1, 3 ] ]\ngap> D \ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphAddAllLoops(D);\n\n\ gap> D;\n\n", count := [ 3, 3, 38, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 38, 7 ], mathmode := "Text", name := "Example", next := 105840, root := ~, start := 105073, stop := 105839 ), rec( content := "\n ", count := [ 3, 3, 38, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 38, 5 ], mathmode := "Text", name := "Description", next := 105857, root := ~, start := 104661, stop := 105856 ) ], count := [ 3, 3, 38, 1 ], mathmode := "Text", name := "ManSection", next := 105871, root := ~, start := 104510, stop := 105870 ), rec( content := "\n\n ", count := [ 3, 3, 38, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, i", Label := "for digraph an\ d int", Name := "DistanceDigraph" ), content := 0, count := [ 3, 3, 39, 2 ], mathmode := "Text", name := "Oper", next := 105975, root := ~, start := 105892, stop := 105974 ), rec( attributes := rec( Arg := "digraph, list", Label := "for digraph an\ d list", Name := "DistanceDigraph" ), content := 0, count := [ 3, 3, 39, 4 ], mathmode := "Text", name := "Oper", next := 106065, root := ~, start := 105978, stop := 106064 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "Returns", next := 106097, root := ~, start := 106068, stop := 106096 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ first argument is a digraph, the second argument is a non-negative\n integ\ er or a list of positive integers. This operation returns a digraph on\n th\ e same set of vertices as ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "A", next := 106315, root := ~, start := 106301, stop := 106314 ), rec( content := ", with t\ wo vertices being adjacent\n if and only if the distance between them in ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "A", next := 106412, root := ~, start := 106398, stop := 106411 ), rec( content := " equals \ ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "A", next := 106428, root := ~, start := 106420, stop := 106427 ), rec( content := "\n or\ is a number in ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "A", next := 106462, root := ~, start := 106451, stop := 106461 ), rec( content := ". See " , count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and two vertices", Oper := "DigraphShortestDistance" ), content := 0, count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "Ref", next := 106549, root := ~, start := 106469, stop := 106548 ), rec( content := ". ", count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "P", next := 106555, root := ~, start := 106551, stop := 106554 ), rec( content := "\n\n \ If ", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "A", next := 106578, root := ~, start := 106564, stop := 106577 ), rec( content := " is muta\ ble, then ", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "A", next := 106610, root := ~, start := 106596, stop := 106609 ), rec( content := " is modi\ fied in-place. If\n ", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "A", next := 106654, root := ~, start := 106640, stop := 106653 ), rec( content := " is immu\ table, then a new immutable digraph constructed as\n described above is ret\ urned.\n ", count := [ 3, 3, 39, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := DigraphFromSparse6String(\n> \":]n?AL`BC_DEbEF`GIaGHdIJeGKcKL\ _@McDHfILaBJfHMjKM\");\n\ngap> D\ istanceDigraph(digraph, 1);\n\ng\ ap> DistanceDigraph(digraph, [1, 2]);\n\n", count := [ 3, 3, 39, 7 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 7 ], mathmode := "Text", name := "Example", next := 107094, root := ~, start := 106750, stop := 107093 ), rec( content := "\n ", count := [ 3, 3, 39, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 39, 5 ], mathmode := "Text", name := "Description", next := 107111, root := ~, start := 106100, stop := 107110 ) ], count := [ 3, 3, 39, 1 ], mathmode := "Text", name := "ManSection", next := 107125, root := ~, start := 105877, stop := 107124 ), rec( content := "\n\n ", count := [ 3, 3, 39, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, k", Name := "DigraphClosure"\ ), content := 0, count := [ 3, 3, 40, 2 ], mathmode := "Text", name := "Oper", next := 107192, root := ~, start := 107146, stop := 107191 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Returns", next := 107224, root := ~, start := 107195, stop := 107223 ), rec( attributes := rec( ), content := [ rec( content := "\n Giv\ en a symmetric loopless digraph with no multiple edges ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 107317, root := ~, start := 107303, stop := 107316 ), rec( content := ",\n t\ he ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "k", count := [ 3, 3, 40, 3 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := \ "Text", name := "A", next := 107338, root := ~, start := 107330, stop := 10733\ 7 ), rec( content := "-closure of ", count := [ 3, 3, 40, 3 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "digraph", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 1073\ 64, root := ~, start := 107350, stop := 107363 ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "E", next := 107368, root := ~, start := 107327, stop := 107367 ), rec( content := " is defi\ ned to be the unique\n smallest symmetric loopless digraph ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "C", next := 107445, root := ~, start := 107437, stop := 107444 ), rec( content := " with no\ multiple edges on the\n vertices of ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 107506, root := ~, start := 107492, stop := 107505 ), rec( content := " that co\ ntains all the edges of ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 107552, root := ~, start := 107538, stop := 107551 ), rec( content := "\n an\ d satsifies the property that the sum of the degrees of every two\n non-adj\ acenct vertices in ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "C", next := 107663, root := ~, start := 107655, stop := 107662 ), rec( content := " is less\ than ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "A", next := 107685, root := ~, start := 107677, stop := 107684 ), rec( content := ". See ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Ref", next := 107727, root := ~, start := 107691, stop := 107726 ), rec( content := ", ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Ref", next := 107758, root := ~, start := 107729, stop := 107757 ), rec( content := ", ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Ref", next := 107792, root := ~, start := 107760, stop := 107791 ), rec( content := ", and ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Ref", next := 107829, root := ~, start := 107798, stop := 107828 ), rec( content := ". ", count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "P", next := 107835, root := ~, start := 107831, stop := 107834 ), rec( content := "\n\n \ The operation ", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphClosure", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "C", next := 107876, root := ~, start := 107855, stop := 107875 ), rec( content := " returns\ the ", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "A", next := 107897, root := ~, start := 107889, stop := 107896 ), rec( content := "-closure\ of\n ", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "A", next := 107927, root := ~, start := 107913, stop := 107926 ), rec( content := ".\n " , count := [ 3, 3, 40, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(6);\n\n\ gap> D := DigraphRemoveEdges(D, [[1, 2], [2, 1]]);\n\ngap> closure := DigraphClosure(D, 6);\n\ngap> IsCompleteDigraph(closure);\ntrue\n", count := [ 3, 3, 40, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 40, 5 ], mathmode := "Text", name := "Example", next := 108259, root := ~, start := 107933, stop := 108258 ), rec( content := "\n ", count := [ 3, 3, 40, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 40, 3 ], mathmode := "Text", name := "Description", next := 108276, root := ~, start := 107227, stop := 108275 ) ], count := [ 3, 3, 40, 1 ], mathmode := "Text", name := "ManSection", next := 108290, root := ~, start := 107131, stop := 108289 ), rec( content := "\n\n ", count := [ 3, 3, 40, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphMycielsk\ ian" ), content := 0, count := [ 3, 3, 41, 2 ], mathmode := "Text", name := "Oper", next := 108358, root := ~, start := 108311, stop := 108357 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMycielsk\ ianAttr" ), content := 0, count := [ 3, 3, 41, 4 ], mathmode := "Text", name := "Attr", next := 108412, root := ~, start := 108361, stop := 108411 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "Returns", next := 108444, root := ~, start := 108415, stop := 108443 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "A", next := 108482, root := ~, start := 108468, stop := 108481 ), rec( content := " is a sy\ mmetric digraph, then\n ", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphMycielskian", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "C", next := 108541, root := ~, start := 108516, stop := 108540 ), rec( content := " returns\ the Mycielskian of ", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "A", next := 108583, root := ~, start := 108569, stop := 108582 ), rec( content := ".", count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 41, 6 ], mathmode := "Text", name := "P", next := 108588, root := ~, start := 108584, stop := 108587 ), rec( content := "\n \n\ The Mycielskian of a symmetric digraph is a larger symmetric digraph\n \ constructed from it, which has a larger chromatic number. For further\n inf\ ormation, see ", count := [ 3, 3, 41, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://en.wikipedia.org/wiki/Mycielskian", count := [ 3, 3, 41, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 6 ], mathmode := "Text", name := "URL", next := 108814, root := ~, start := 108762, stop := 108813 ), rec( content := ".", count := [ 3, 3, 41, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "P", next := 108819, root := ~, start := 108815, stop := 108818 ), rec( content := "\n\n \ If ", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "A", next := 108842, root := ~, start := 108828, stop := 108841 ), rec( content := " is immu\ table, then a new immutable digraph is returned.\n Otherwise, the mutable d\ igraph ", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "A", next := 108948, root := ~, start := 108934, stop := 108947 ), rec( content := " is chan\ ged in-place into\n its Mycielskian.", count := [ 3, 3, 41, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 3, 41, 8 ], mathmode := "Text", name := "P", next := 108998, root := ~, start := 108994, stop := 108997 ), rec( content := "\n\n \ ", count := [ 3, 3, 41, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(2);\n\ngap> C\ hromaticNumber(D);\n2\ngap> D := DigraphMycielskian(D);\n\ngap> ChromaticNumber(D);\n3\ngap> D := DigraphMyciel\ skian(D);\n\ngap> ChromaticNumbe\ r(D);\n4\ngap> D := CompleteBipartiteDigraph(IsMutable, 2, 3);\n\ngap> DigraphMycielskian(D);\n\ngap> D;\n\n", count := [ 3, 3, 41, 9 ], name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 9 ], mathmode := "Text", name := "Example", next := 109568, root := ~, start := 109004, stop := 109567 ), rec( content := "\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 41, 5 ], mathmode := "Text", name := "Description", next := 109585, root := ~, start := 108447, stop := 109584 ) ], count := [ 3, 3, 41, 1 ], mathmode := "Text", name := "ManSection", next := 109599, root := ~, start := 108296, stop := 109598 ), rec( content := "\n\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 109613, root := ~, start := 43573, stop := 109612 ), rec( content := "\n\n ", count := [ 3, 3, 41, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Random digraphs", count := [ 3, 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 109660, root := ~, start := 109626, stop := 109659 ), rec( content := "\n ", count := [ 3, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n[, p]", Name := "RandomDigraph" ), content := 0, count := [ 3, 4, 1, 2 ], mathmode := "Text", name := "Oper", next := 109729, root := ~, start := 109680, stop := 109728 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Returns", next := 109761, root := ~, start := 109732, stop := 109760 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "A", next := 109824, root := ~, start := 109813, stop := 109823 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "A", next := 109970, root := ~, start := 109959, stop := 109969 ), rec( content := " is ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Ref", next := 110004, root := ~, start := 109974, stop := 110003 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "A", next := 110072, root := ~, start := 110061, stop := 110071 ), rec( content := " is ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Ref", next := 110112, root := ~, start := 110076, stop := 110111 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "A", next := 110197, root := ~, start := 110186, stop := 110196 ), rec( content := " is not \ present, then ", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Ref", next := 110257, root := ~, start := 110219, stop := 110256 ), rec( content := " is used\ by default.", count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "P", next := 110281, root := ~, start := 110277, stop := 110280 ), rec( content := "\n\n \ If ", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "A", next := 110298, root := ~, start := 110290, stop := 110297 ), rec( content := " is a po\ sitive integer, then this function returns a random\n digraph with ", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "A", next := 110383, root := ~, start := 110375, stop := 110382 ), rec( content := " vertice\ s and without multiple edges. The result\n may or may not have loops.", count := [ 3, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "P", next := 110466, root := ~, start := 110462, stop := 110465 ), rec( content := "\n\n \ If the optional second argument ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "A", next := 110512, root := ~, start := 110504, stop := 110511 ), rec( content := " is a fl\ oat with value\n ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ \\leq ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "M", next := 110553, root := ~, start := 110539, stop := 110552 ), rec( content := " ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ p ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "A", next := 110564, root := ~, start := 110554, stop := 110563 ), rec( content := " ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ \\leq 1", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "M", next := 110579, root := ~, start := 110565, stop := 110578 ), rec( content := ", then a\ n edge will exist between each\n pair of vertices with probability approxim\ ately ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "A", next := 110678, root := ~, start := 110670, stop := 110677 ), rec( content := ".\n I\ f ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "A", next := 110695, root := ~, start := 110687, stop := 110694 ), rec( content := " is not \ specified, then a random probability will be assumed\n (chosen with uniform\ probability).\n ", count := [ 3, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> RandomDigraph(1000);\n\ngap> RandomDigraph(10000, 0.023);\n\ngap> RandomDigraph(IsMutableDigraph, 1000, 1 / 2);\n\n", count := [ 3, 4, 1, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 4, 1, 6 ], mathmode := "Text", name := "Log", next := 111093, root := ~, start := 110799, stop := 111092 ), rec( content := "\n ", count := [ 3, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 111110, root := ~, start := 109764, stop := 111109 ) ], count := [ 3, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 111124, root := ~, start := 109665, stop := 111123 ), rec( content := "\n\n ", count := [ 3, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "n[, m]", Name := "RandomMultiDigr\ aph" ), content := 0, count := [ 3, 4, 2, 2 ], mathmode := "Text", name := "Oper", next := 111191, root := ~, start := 111145, stop := 111190 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "Returns", next := 111223, root := ~, start := 111194, stop := 111222 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 111255, root := ~, start := 111247, stop := 111254 ), rec( content := " is a po\ sitive integer, then this function returns a random\n digraph with ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 111340, root := ~, start := 111332, stop := 111339 ), rec( content := " vertice\ s. If the optional second argument\n ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 111395, root := ~, start := 111387, stop := 111394 ), rec( content := " is a po\ sitive integer, then the digraph will have ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 111454, root := ~, start := 111446, stop := 111453 ), rec( content := "\n ed\ ges. If ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "A", next := 111477, root := ~, start := 111469, stop := 111476 ), rec( content := " is not \ specified, then the number of edges will be\n chosen randomly (with uniform\ probability) from the range\n ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "C", next := 111608, root := ~, start := 111595, stop := 111607 ), rec( attributes := rec( ), content := [ rec( content := "{\ n \\choose 2}", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "M", next := 111628, root := ~, start := 111608, stop := 111627 ), rec( attributes := rec( ), content := [ rec( content := "]\ ", count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "C", next := 111636, root := ~, start := 111628, stop := 111635 ), rec( content := ".\n " , count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 2, 4 ], mathmode := "Text", name := "P", next := 111646, root := ~, start := 111642, stop := 111645 ), rec( content := "\n\n \ The method used by this function chooses each edge from the set of\n all po\ ssible edges with uniform probability. No effort is made to avoid\n creatin\ g multiple edges, so it is possible (but not guaranteed) that the\n result \ will have multiple edges. The result may or may not have loops.", count := [ 3, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 2, 5 ], mathmode := "Text", name := "P", next := 111950, root := ~, start := 111946, stop := 111949 ), rec( content := "\n ", count := [ 3, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> RandomMultiDigraph(1000);\n\ngap> RandomMultiDigraph(1000, 950);\n\n", count := [ 3, 4, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 6 ], mathmode := "Text", name := "Log", next := 112159, root := ~, start := 111955, stop := 112158 ), rec( content := "\n ", count := [ 3, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 112176, root := ~, start := 111226, stop := 112175 ) ], count := [ 3, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 112190, root := ~, start := 111130, stop := 112189 ), rec( content := "\n\n ", count := [ 3, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "RandomTournamen\ t" ), content := 0, count := [ 3, 4, 3, 2 ], mathmode := "Text", name := "Oper", next := 112258, root := ~, start := 112211, stop := 112257 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Returns", next := 112290, root := ~, start := 112261, stop := 112289 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "A", next := 112353, root := ~, start := 112342, stop := 112352 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "A", next := 112499, root := ~, start := 112488, stop := 112498 ), rec( content := " is ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Ref", next := 112533, root := ~, start := 112503, stop := 112532 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "A", next := 112601, root := ~, start := 112590, stop := 112600 ), rec( content := " is ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Ref", next := 112641, root := ~, start := 112605, stop := 112640 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "A", next := 112726, root := ~, start := 112715, stop := 112725 ), rec( content := " is not \ present, then ", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Ref", next := 112786, root := ~, start := 112748, stop := 112785 ), rec( content := " is used\ by default.", count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "P", next := 112810, root := ~, start := 112806, stop := 112809 ), rec( content := "\n\n \ If ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "A", next := 112827, root := ~, start := 112819, stop := 112826 ), rec( content := " is a no\ n-negative integer, this function returns a random\n tournament with ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "A", next := 112914, root := ~, start := 112906, stop := 112913 ), rec( content := " vertice\ s. See ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTourn\ ament" ), content := 0, count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "Ref", next := 112955, root := ~, start := 112929, stop := 112954 ), rec( content := ". ", count := [ 3, 4, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 3, 5 ], mathmode := "Text", name := "P", next := 112961, root := ~, start := 112957, stop := 112960 ), rec( content := "\n ", count := [ 3, 4, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> RandomTournament(10);\n\ngap> Ran\ domTournament(IsMutableDigraph, 10);\n\n", count := [ 3, 4, 3, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 6 ], mathmode := "Text", name := "Log", next := 113153, root := ~, start := 112966, stop := 113152 ), rec( content := "\n ", count := [ 3, 4, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 3, 3 ], mathmode := "Text", name := "Description", next := 113170, root := ~, start := 112293, stop := 113169 ) ], count := [ 3, 4, 3, 1 ], mathmode := "Text", name := "ManSection", next := 113184, root := ~, start := 112196, stop := 113183 ), rec( content := "\n\n ", count := [ 3, 4, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "n", Name := "RandomLattice" ), content := 0, count := [ 3, 4, 4, 2 ], mathmode := "Text", name := "Oper", next := 113241, root := ~, start := 113205, stop := 113240 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Returns", next := 113273, root := ~, start := 113244, stop := 113272 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the optional first argument ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "A", next := 113336, root := ~, start := 113325, stop := 113335 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "A", next := 113482, root := ~, start := 113471, stop := 113481 ), rec( content := " is ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Ref", next := 113516, root := ~, start := 113486, stop := 113515 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "A", next := 113584, root := ~, start := 113573, stop := 113583 ), rec( content := " is ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Ref", next := 113624, root := ~, start := 113588, stop := 113623 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "A", next := 113709, root := ~, start := 113698, stop := 113708 ), rec( content := " is not \ present, then ", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Ref", next := 113769, root := ~, start := 113731, stop := 113768 ), rec( content := " is used\ by default.", count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "P", next := 113793, root := ~, start := 113789, stop := 113792 ), rec( content := "\n\n \ If ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "A", next := 113810, root := ~, start := 113802, stop := 113809 ), rec( content := " is a po\ sitive integer, this function return a random lattice\n with ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "C", next := 113889, root := ~, start := 113881, stop := 113888 ), rec( content := " vertice\ s, where it is guaranteed that ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "C", next := 113936, root := ~, start := 113928, stop := 113935 ), rec( content := " is betw\ een\n ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "A", next := 113960, root := ~, start := 113952, stop := 113959 ), rec( content := " and ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ * ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 4, 4, 4 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 4 ], mathmode := \ "Text", name := "A", next := 113980, root := ~, start := 113972, stop := 11397\ 9 ) ], count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "C", next := 113984, root := ~, start := 113965, stop := 113983 ), rec( content := ". See ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsLatti\ ceDigraph" ), content := 0, count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "Ref", next := 114020, root := ~, start := 113990, stop := 114019 ), rec( content := ". ", count := [ 3, 4, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 4, 4, 5 ], mathmode := "Text", name := "P", next := 114026, root := ~, start := 114022, stop := 114025 ), rec( content := "\n ", count := [ 3, 4, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> RandomLattice(10);\n\ngap> RandomLattice(IsMutableDigraph, 10);\n\n", count := [ 3, 4, 4, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 6 ], mathmode := "Text", name := "Log", next := 114221, root := ~, start := 114031, stop := 114220 ), rec( content := "\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 4, 3 ], mathmode := "Text", name := "Description", next := 114238, root := ~, start := 113276, stop := 114237 ) ], count := [ 3, 4, 4, 1 ], mathmode := "Text", name := "ManSection", next := 114252, root := ~, start := 113190, stop := 114251 ), rec( content := "\n\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 114266, root := ~, start := 109617, stop := 114265 ) , rec( content := "\n\n ", count := [ 3, 4, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Standard examples\ ", count := [ 3, 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 114315, root := ~, start := 114279, stop := 114314 ), rec( content := "\n ", count := [ 3, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "ChainDigraph" ) , content := 0, count := [ 3, 5, 1, 2 ], mathmode := "Text", name := "Oper", next := 114378, root := ~, start := 114335, stop := 114377 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 114410, root := ~, start := 114381, stop := 114409 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "A", next := 114442, root := ~, start := 114434, stop := 114441 ), rec( content := " is a po\ sitive integer, this function returns a chain\n with ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "A", next := 114513, root := ~, start := 114505, stop := 114512 ), rec( content := " vertice\ s and ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "n", count := [ 3, 5, 1, 3 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := \ "Text", name := "A", next := 114538, root := ~, start := 114530, stop := 11453\ 7 ), rec( content := " - 1", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114546, root := ~, start := 114527, stop := 114545 ), rec( content := " edges.\\ n Specifically,\n for each vertex ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114600, root := ~, start := 114592, stop := 114599 ), rec( content := " (with " , count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114615, root := ~, start := 114607, stop := 114614 ), rec( content := " < ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114629, root := ~, start := 114621, stop := 114628 ), rec( content := "), there\ is a directed\n edge with source ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114681, root := ~, start := 114673, stop := 114680 ), rec( content := " and ran\ ge ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ + 1", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "C", next := 114704, root := ~, start := 114692, stop := 114703 ), rec( content := ". ", count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "P", next := 114710, root := ~, start := 114706, stop := 114709 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "A", next := 114758, root := ~, start := 114747, stop := 114757 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "A", next := 114904, root := ~, start := 114893, stop := 114903 ), rec( content := " is ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "Ref", next := 114938, root := ~, start := 114908, stop := 114937 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "A", next := 115006, root := ~, start := 114995, stop := 115005 ), rec( content := " is ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "Ref", next := 115046, root := ~, start := 115010, stop := 115045 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "A", next := 115131, root := ~, start := 115120, stop := 115130 ), rec( content := " is not \ present, then ", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "Ref", next := 115191, root := ~, start := 115153, stop := 115190 ), rec( content := " is used\ by default.", count := [ 3, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 1, 5 ], mathmode := "Text", name := "P", next := 115215, root := ~, start := 115211, stop := 115214 ), rec( content := "\n\n \ The ", count := [ 3, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ReflexiveTransitiveClosure" ), content := 0, count := [ 3, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 115272, root := ~, start := 115225, stop := 115271 ), rec( content := " of a ch\ ain represents a\n total order.\n ", count := [ 3, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ChainDigraph(42);\n\ngap> Chai\ nDigraph(IsMutableDigraph, 10);\n\n\ ", count := [ 3, 5, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 6 ], mathmode := "Text", name := "Example", next := 115501, root := ~, start := 115318, stop := 115500 ), rec( content := "\n ", count := [ 3, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 115518, root := ~, start := 114413, stop := 115517 ) ], count := [ 3, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 115532, root := ~, start := 114320, stop := 115531 ), rec( content := "\n\n ", count := [ 3, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "CompleteDigraph\ " ), content := 0, count := [ 3, 5, 2, 2 ], mathmode := "Text", name := "Oper", next := 115599, root := ~, start := 115553, stop := 115598 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 115631, root := ~, start := 115602, stop := 115630 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "A", next := 115663, root := ~, start := 115655, stop := 115662 ), rec( content := " is a no\ n-negative integer, this function returns the complete\n digraph with ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "A", next := 115751, root := ~, start := 115743, stop := 115750 ), rec( content := " vertice\ s. See ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsCompl\ eteDigraph" ), content := 0, count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 115797, root := ~, start := 115766, stop := 115796 ), rec( content := ". ", count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "P", next := 115803, root := ~, start := 115799, stop := 115802 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "A", next := 115851, root := ~, start := 115840, stop := 115850 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "A", next := 115997, root := ~, start := 115986, stop := 115996 ), rec( content := " is ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 116031, root := ~, start := 116001, stop := 116030 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "A", next := 116099, root := ~, start := 116088, stop := 116098 ), rec( content := " is ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 116139, root := ~, start := 116103, stop := 116138 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "A", next := 116224, root := ~, start := 116213, stop := 116223 ), rec( content := " is not \ present, then ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "Ref", next := 116284, root := ~, start := 116246, stop := 116283 ), rec( content := " is used\ by default.\n ", count := [ 3, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> CompleteDigraph(20);\n\ngap\ > CompleteDigraph(IsMutableDigraph, 10);\n\n", count := [ 3, 5, 2, 5 ], name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 5 ], mathmode := "Text", name := "Example", next := 116502, root := ~, start := 116309, stop := 116501 ), rec( content := "\n ", count := [ 3, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 116519, root := ~, start := 115634, stop := 116518 ) ], count := [ 3, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 116533, root := ~, start := 115538, stop := 116532 ), rec( content := "\n\n ", count := [ 3, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]m, n", Name := "CompleteBiparti\ teDigraph" ), content := 0, count := [ 3, 5, 3, 2 ], mathmode := "Text", name := "Oper", next := 116612, root := ~, start := 116554, stop := 116611 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "Returns", next := 116644, root := ~, start := 116615, stop := 116643 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ omplete bipartite digraph is a digraph whose vertices\n can be partitioned \ into two non-empty vertex sets, such there exists\n a unique edge with sour\ ce ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "C", next := 116833, root := ~, start := 116825, stop := 116832 ), rec( content := " and ran\ ge ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "C", next := 116852, root := ~, start := 116844, stop := 116851 ), rec( content := " if and \ only if\n ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "C", next := 116880, root := ~, start := 116872, stop := 116879 ), rec( content := " and ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "C", next := 116893, root := ~, start := 116885, stop := 116892 ), rec( content := " lie in \ different vertex sets. ", count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "P", next := 116928, root := ~, start := 116924, stop := 116927 ), rec( content := "\n\n \ If ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "A", next := 116945, root := ~, start := 116937, stop := 116944 ), rec( content := " and ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "A", next := 116958, root := ~, start := 116950, stop := 116957 ), rec( content := " are pos\ itive integers, this function\n returns the complete bipartite digraph with\ vertex sets of sizes\n ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "A", next := 117077, root := ~, start := 117069, stop := 117076 ), rec( content := " (contai\ ning the vertices ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. m]", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "C", next := 117118, root := ~, start := 117103, stop := 117117 ), rec( content := ") and\n \ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "A", next := 117136, root := ~, start := 117128, stop := 117135 ), rec( content := " (contai\ ning the vertices ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ m + 1 .. m + n]", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "C", next := 117185, root := ~, start := 117162, stop := 117184 ), rec( content := ").\n \ ", count := [ 3, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "P", next := 117196, root := ~, start := 117192, stop := 117195 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "A", next := 117244, root := ~, start := 117233, stop := 117243 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "A", next := 117390, root := ~, start := 117379, stop := 117389 ), rec( content := " is ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "Ref", next := 117424, root := ~, start := 117394, stop := 117423 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "A", next := 117492, root := ~, start := 117481, stop := 117491 ), rec( content := " is ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "Ref", next := 117532, root := ~, start := 117496, stop := 117531 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "A", next := 117617, root := ~, start := 117606, stop := 117616 ), rec( content := " is not \ present, then ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "Ref", next := 117677, root := ~, start := 117639, stop := 117676 ), rec( content := " is used\ by default.\n\n ", count := [ 3, 5, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> CompleteBipartiteDigraph(2, 3);\n\ngap> CompleteBipartiteDigraph(IsMutableDigraph\ , 3, 2);\n\n", count := [ 3, 5, 3, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 3, 6 ], mathmode := "Text", name := "Example", next := 117941, root := ~, start := 117703, stop := 117940 ), rec( content := "\n ", count := [ 3, 5, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 117958, root := ~, start := 116647, stop := 117957 ) ], count := [ 3, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 117972, root := ~, start := 116539, stop := 117971 ), rec( content := "\n\n ", count := [ 3, 5, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]orders", Name := "CompleteMultipa\ rtiteDigraph" ), content := 0, count := [ 3, 5, 4, 2 ], mathmode := "Text", name := "Oper", next := 118062, root := ~, start := 117997, stop := 118061 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "Returns", next := 118096, root := ~, start := 118067, stop := 118095 ), rec( attributes := rec( ), content := [ rec( content := "\n F\ or a list ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rders", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "A", next := 118145, root := ~, start := 118132, stop := 118144 ), rec( content := " of ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "C", next := 118157, root := ~, start := 118149, stop := 118156 ), rec( content := " positiv\ e integers,\n this function returns the digraph containing ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "C", next := 118236, root := ~, start := 118228, stop := 118235 ), rec( content := " indepen\ dent sets\n of vertices of orders ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "l", count := [ 3, 5, 4, 3 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := \ "Text", name := "A", next := 118294, root := ~, start := 118286, stop := 11829\ 3 ), rec( content := "[1] .. ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := \ "Text", name := "A", next := 118309, root := ~, start := 118301, stop := 11830\ 8 ), rec( content := "[n]]", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "C", next := 118317, root := ~, start := 118282, stop := 118316 ), rec( content := ". Moreov\ er,\n each vertex is adjacent to every other not contained in the\n \ same independent set.\n ", count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "P", next := 118433, root := ~, start := 118429, stop := 118432 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "A", next := 118483, root := ~, start := 118472, stop := 118482 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digrap\ h being created will\n belong to. For example, if ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "A", next := 118633, root := ~, start := 118622, stop := 118632 ), rec( content := " is ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "Ref", next := 118667, root := ~, start := 118637, stop := 118666 ), rec( content := ",\n \ then the digraph being created will be mutable, if ", count := [ 3, 5, 4, 4 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "A", next := 118737, root := ~, start := 118726, stop := 118736 ), rec( content := " is ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "Ref", next := 118779, root := ~, start := 118741, stop := 118778 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "A", next := 118866, root := ~, start := 118855, stop := 118865 ), rec( content := " is not \ present, then ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "Ref", next := 118928, root := ~, start := 118888, stop := 118927 ), rec( content := " is used\ by default.\n ", count := [ 3, 5, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> CompleteMultipartiteDigraph([5, 4, 2]);\n\ngap> CompleteMultipartiteDigraph(IsMuta\ bleDigraph, [5, 4, 2]);\n\n", count := [ 3, 5, 4, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 4, 5 ], mathmode := "Text", name := "Example", next := 119209, root := ~, start := 118955, stop := 119208 ), rec( content := "\n ", count := [ 3, 5, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 4, 3 ], mathmode := "Text", name := "Description", next := 119226, root := ~, start := 118101, stop := 119225 ) ], count := [ 3, 5, 4, 1 ], mathmode := "Text", name := "ManSection", next := 119240, root := ~, start := 117980, stop := 119239 ), rec( content := "\n\n ", count := [ 3, 5, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "CycleDigraph" ) , content := 0, count := [ 3, 5, 5, 2 ], mathmode := "Text", name := "Oper", next := 119304, root := ~, start := 119261, stop := 119303 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "Returns", next := 119336, root := ~, start := 119307, stop := 119335 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "A", next := 119368, root := ~, start := 119360, stop := 119367 ), rec( content := " is a po\ sitive integer, this function returns a ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ycle", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "E", next := 119428, root := ~, start := 119416, stop := 119427 ), rec( content := "\n di\ graph with ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "A", next := 119454, root := ~, start := 119446, stop := 119453 ), rec( content := " vertice\ s and ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "A", next := 119476, root := ~, start := 119468, stop := 119475 ), rec( content := " edges. \ Specifically,\n for each vertex ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119526, root := ~, start := 119518, stop := 119525 ), rec( content := " (with " , count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119541, root := ~, start := 119533, stop := 119540 ), rec( content := " < ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119555, root := ~, start := 119547, stop := 119554 ), rec( content := "), there\ is a directed\n edge with source ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119607, root := ~, start := 119599, stop := 119606 ), rec( content := " and ran\ ge ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ + 1", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119630, root := ~, start := 119618, stop := 119629 ), rec( content := ". In add\ ition, there is\n an edge with source ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119686, root := ~, start := 119678, stop := 119685 ), rec( content := " and ran\ ge ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "C", next := 119705, root := ~, start := 119697, stop := 119704 ), rec( content := ". ", count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "P", next := 119711, root := ~, start := 119707, stop := 119710 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "A", next := 119759, root := ~, start := 119748, stop := 119758 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "A", next := 119905, root := ~, start := 119894, stop := 119904 ), rec( content := " is ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "Ref", next := 119939, root := ~, start := 119909, stop := 119938 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "A", next := 120007, root := ~, start := 119996, stop := 120006 ), rec( content := " is ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "Ref", next := 120047, root := ~, start := 120011, stop := 120046 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "A", next := 120132, root := ~, start := 120121, stop := 120131 ), rec( content := " is not \ present, then ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "Ref", next := 120190, root := ~, start := 120154, stop := 120189 ), rec( content := " is used\ by default.\n ", count := [ 3, 5, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> CycleDigraph(1);\n\ngap> CycleD\ igraph(123);\n\ngap> CycleDigraph(I\ sMutableDigraph, 10);\n\n", count := [ 3, 5, 5, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 5, 5 ], mathmode := "Text", name := "Example", next := 120463, root := ~, start := 120213, stop := 120462 ), rec( content := "\n ", count := [ 3, 5, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 5, 3 ], mathmode := "Text", name := "Description", next := 120480, root := ~, start := 119339, stop := 120479 ) ], count := [ 3, 5, 5, 1 ], mathmode := "Text", name := "ManSection", next := 120494, root := ~, start := 119246, stop := 120493 ), rec( content := "\n\n ", count := [ 3, 5, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n", Name := "EmptyDigraph" ) , content := 0, count := [ 3, 5, 6, 2 ], mathmode := "Text", name := "Oper", next := 120558, root := ~, start := 120515, stop := 120557 ), rec( attributes := rec( Arg := "[filt, ]n", Name := "NullDigraph" ), content := 0, count := [ 3, 5, 6, 4 ], mathmode := "Text", name := "Oper", next := 120603, root := ~, start := 120561, stop := 120602 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "Returns", next := 120635, root := ~, start := 120606, stop := 120634 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "A", next := 120667, root := ~, start := 120659, stop := 120666 ), rec( content := " is a no\ n-negative integer, this function returns the\n ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ mpty", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "E", next := 120737, root := ~, start := 120725, stop := 120736 ), rec( content := " or ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ull", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "E", next := 120752, root := ~, start := 120741, stop := 120751 ), rec( content := " digraph\ with ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "A", next := 120774, root := ~, start := 120766, stop := 120773 ), rec( content := " vertice\ s. An empty\n digraph is one with no edges.\n ", count := [ 3, 5, 6, 5 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "P", next := 120836, root := ~, start := 120832, stop := 120835 ), rec( content := "\n \ \n If the optional first argument ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "A", next := 120890, root := ~, start := 120879, stop := 120889 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "A", next := 121036, root := ~, start := 121025, stop := 121035 ), rec( content := " is ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "Ref", next := 121070, root := ~, start := 121040, stop := 121069 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "A", next := 121138, root := ~, start := 121127, stop := 121137 ), rec( content := " is ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "Ref", next := 121178, root := ~, start := 121142, stop := 121177 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "A", next := 121263, root := ~, start := 121252, stop := 121262 ), rec( content := " is not \ present, then ", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "Ref", next := 121323, root := ~, start := 121285, stop := 121322 ), rec( content := " is used\ by default.", count := [ 3, 5, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "P", next := 121347, root := ~, start := 121343, stop := 121346 ), rec( content := "\n\n \ ", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ ullDigraph", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "C", next := 121371, root := ~, start := 121353, stop := 121370 ), rec( content := " is a sy\ nonym for ", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ mptyDigraph", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "C", next := 121408, root := ~, start := 121389, stop := 121407 ), rec( content := ".\n\n \ ", count := [ 3, 5, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> EmptyDigraph(20);\n\ngap> Null\ Digraph(10);\n\ngap> EmptyDigraph(Is\ MutableDigraph, 10);\n", count := [ 3, 5, 6, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 6, 8 ], mathmode := "Text", name := "Example", next := 121659, root := ~, start := 121415, stop := 121658 ), rec( content := "\n ", count := [ 3, 5, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 6, 5 ], mathmode := "Text", name := "Description", next := 121676, root := ~, start := 120638, stop := 121675 ) ], count := [ 3, 5, 6, 1 ], mathmode := "Text", name := "ManSection", next := 121690, root := ~, start := 120500, stop := 121689 ), rec( content := "\n\n ", count := [ 3, 5, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n, k", Name := "JohnsonDigraph"\ ), content := 0, count := [ 3, 5, 7, 2 ], mathmode := "Text", name := "Oper", next := 121759, root := ~, start := 121711, stop := 121758 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "Returns", next := 121791, root := ~, start := 121762, stop := 121790 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "A", next := 121823, root := ~, start := 121815, stop := 121822 ), rec( content := " and ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "A", next := 121836, root := ~, start := 121828, stop := 121835 ), rec( content := " are non\ -negative integers, then this operation\n returns a symmetric digraph which\ corresponds to the undirected ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ ohnson\n graph", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "E", next := 121976, root := ~, start := 121952, stop := 121975 ), rec( content := " ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ (n, k)", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "M", next := 121991, root := ~, start := 121977, stop := 121990 ), rec( content := ". ", count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "P", next := 121997, root := ~, start := 121993, stop := 121996 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "A", next := 122045, root := ~, start := 122034, stop := 122044 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "A", next := 122191, root := ~, start := 122180, stop := 122190 ), rec( content := " is ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "Ref", next := 122225, root := ~, start := 122195, stop := 122224 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "A", next := 122293, root := ~, start := 122282, stop := 122292 ), rec( content := " is ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "Ref", next := 122333, root := ~, start := 122297, stop := 122332 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "A", next := 122418, root := ~, start := 122407, stop := 122417 ), rec( content := " is not \ present, then ", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "Ref", next := 122478, root := ~, start := 122440, stop := 122477 ), rec( content := " is used\ by default.", count := [ 3, 5, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "P", next := 122502, root := ~, start := 122498, stop := 122501 ), rec( content := "\n\n \ The ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ ohnson graph", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "E", next := 122532, root := ~, start := 122512, stop := 122531 ), rec( content := " ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ (n, k)", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "M", next := 122547, root := ~, start := 122533, stop := 122546 ), rec( content := " has ver\ tices given by all the\n ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "A", next := 122590, root := ~, start := 122582, stop := 122589 ), rec( content := "-subsets\ of the range ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1 .. ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 5, 7, 5 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := \ "Text", name := "A", next := 122629, root := ~, start := 122621, stop := 12262\ 8 ), rec( content := "]", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "C", next := 122634, root := ~, start := 122612, stop := 122633 ), rec( content := ", and tw\ o vertices are\n connected by an edge iff their intersection has size ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "k", count := [ 3, 5, 7, 5 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := \ "Text", name := "A", next := 122727, root := ~, start := 122719, stop := 12272\ 6 ), rec( content := " - 1", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "M", next := 122735, root := ~, start := 122716, stop := 122734 ), rec( content := ".\n\n \ ", count := [ 3, 5, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := JohnsonDigraph(3, 1);\n\ngap> OutNeighbours(gr);\n[ [ 2, 3 ], [ 1, 3 ], [ 1, 2 ] ]\ngap> g\ r := JohnsonDigraph(4, 2);\n\ngap> OutNeighbours(gr);\n[ [ 2, 3, 4, 5 ], [ 1, 3, 4, 6 ], [ 1, 2, 5, 6\ ], [ 1, 2, 5, 6 ], \n [ 1, 3, 4, 6 ], [ 2, 3, 4, 5 ] ]\ngap> JohnsonDigraph(\ 1, 0);\n\ngap> JohnsonDigraph(IsMutable\ Digraph, 1, 0);\n\n", count := [ 3, 5, 7, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 7, 6 ], mathmode := "Text", name := "Example", next := 123284, root := ~, start := 122742, stop := 123283 ), rec( content := "\n ", count := [ 3, 5, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 7, 3 ], mathmode := "Text", name := "Description", next := 123301, root := ~, start := 121794, stop := 123300 ) ], count := [ 3, 5, 7, 1 ], mathmode := "Text", name := "ManSection", next := 123315, root := ~, start := 121696, stop := 123314 ), rec( content := "\n\n ", count := [ 3, 5, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt]", Name := "PetersenGraph" ), content := 0, count := [ 3, 5, 8, 2 ], mathmode := "Text", name := "Oper", next := 123377, root := ~, start := 123336, stop := 123376 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "Returns", next := 123409, root := ~, start := 123380, stop := 123408 ), rec( attributes := rec( ), content := [ rec( content := "\n\n F\ rom ", count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://en.wikipedia.org/wiki/Petersen_graph", count := [ 3, 5, 8, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "URL", next := 123491, root := ~, start := 123436, stop := 123490 ), rec( content := ":", count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 8, 4 ], mathmode := "Text", name := "P", next := 123496, root := ~, start := 123492, stop := 123495 ), rec( content := "\n\n \ ", count := [ 3, 5, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ he Petersen graph is an undirected graph with 10 vertices and 15 edges.\n I\ t is a small graph that serves as a useful example and counterexample for\n \ many problems in graph theory. The Petersen graph is named after Julius\n \ Petersen, who in 1898 constructed it to be the smallest bridgeless cubic\n \ graph with no three-edge-coloring.", count := [ 3, 5, 8, 4 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 4 ], mathmode := "Text", name := "Q", next := 123852, root := ~, start := 123502, stop := 123851 ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "P", next := 123856, root := ~, start := 123852, stop := 123855 ), rec( content := "\n \n\ If the optional first argument ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "A", next := 123908, root := ~, start := 123897, stop := 123907 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "A", next := 124054, root := ~, start := 124043, stop := 124053 ), rec( content := " is ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "Ref", next := 124088, root := ~, start := 124058, stop := 124087 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "A", next := 124156, root := ~, start := 124145, stop := 124155 ), rec( content := " is ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "Ref", next := 124196, root := ~, start := 124160, stop := 124195 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "A", next := 124281, root := ~, start := 124270, stop := 124280 ), rec( content := " is not \ present, then ", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "Ref", next := 124341, root := ~, start := 124303, stop := 124340 ), rec( content := " is used\ by default.", count := [ 3, 5, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 8, 6 ], mathmode := "Text", name := "P", next := 124365, root := ~, start := 124361, stop := 124364 ), rec( content := "\n\n \ See also ", count := [ 3, 5, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "General\ isedPetersenGraph" ), content := 0, count := [ 3, 5, 8, 6 ], mathmode := "Text", name := "Ref", next := 124418, root := ~, start := 124380, stop := 124417 ), rec( content := ".\n\n", count := [ 3, 5, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ChromaticNumber(PetersenGraph());\n3\ngap> PetersenGraph(IsMutableDigrap\ h);\n\n", count := [ 3, 5, 8, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 8, 7 ], mathmode := "Text", name := "Example", next := 124577, root := ~, start := 124421, stop := 124576 ), rec( content := "\n ", count := [ 3, 5, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 8, 3 ], mathmode := "Text", name := "Description", next := 124594, root := ~, start := 123412, stop := 124593 ) ], count := [ 3, 5, 8, 1 ], mathmode := "Text", name := "ManSection", next := 124608, root := ~, start := 123321, stop := 124607 ), rec( content := "\n\n ", count := [ 3, 5, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]n, k", Name := "GeneralisedPete\ rsenGraph" ), content := 0, count := [ 3, 5, 9, 2 ], mathmode := "Text", name := "Oper", next := 124687, root := ~, start := 124629, stop := 124686 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "Returns", next := 124719, root := ~, start := 124690, stop := 124718 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "A", next := 124751, root := ~, start := 124743, stop := 124750 ), rec( content := " is a po\ sitive integer and ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "A", next := 124786, root := ~, start := 124778, stop := 124785 ), rec( content := " is a no\ n-negative\n integer less than ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "n", count := [ 3, 5, 9, 3 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := \ "Text", name := "A", next := 124838, root := ~, start := 124830, stop := 12483\ 7 ), rec( content := " / 2", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "C", next := 124846, root := ~, start := 124827, stop := 124845 ), rec( content := ", then t\ his operation returns\n the ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ eneralised Petersen graph", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "E", next := 124917, root := ~, start := 124884, stop := 124916 ), rec( content := " ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ PG(", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 3, 5, 9, 3 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := \ "Text", name := "A", next := 124933, root := ~, start := 124925, stop := 12493\ 2 ), rec( content := ", k)", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "M", next := 124941, root := ~, start := 124918, stop := 124940 ), rec( content := ". ", count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 9, 4 ], mathmode := "Text", name := "P", next := 124947, root := ~, start := 124943, stop := 124946 ), rec( content := "\n\n \ From\n ", count := [ 3, 5, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttps://en.wikipedia.org/wiki/Generalized_Petersen_graph", count := [ 3, 5, 9, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 4 ], mathmode := "Text", name := "URL", next := 125029, root := ~, start := 124962, stop := 125028 ), rec( content := ":\n " , count := [ 3, 5, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "P", next := 125039, root := ~, start := 125035, stop := 125038 ), rec( content := "\n\n \ ", count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "T\ he generalized Petersen graphs are a family of cubic\n graphs formed by con\ necting the vertices of a regular polygon to the\n corresponding vertices o\ f a star polygon. They include the Petersen graph\n and generalize one of t\ he ways of constructing the Petersen graph. The\n generalized Petersen grap\ h family was introduced in 1950 by H. S. M.\n Coxeter and was given its nam\ e in 1969 by Mark Watkins.", count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "Q", next := 125464, root := ~, start := 125045, stop := 125463 ), rec( content := "\n ", count := [ 3, 5, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "P", next := 125473, root := ~, start := 125469, stop := 125472 ), rec( content := "\n\n \ If the optional first argument ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "A", next := 125521, root := ~, start := 125510, stop := 125520 ), rec( content := " is pres\ ent, then this should\n specify the category or representation the digraph \ being created will\n belong to. For example, if ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "A", next := 125667, root := ~, start := 125656, stop := 125666 ), rec( content := " is ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "Ref", next := 125701, root := ~, start := 125671, stop := 125700 ), rec( content := ",\n t\ hen the digraph being created will be mutable, if ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "A", next := 125769, root := ~, start := 125758, stop := 125768 ), rec( content := " is ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "Ref", next := 125809, root := ~, start := 125773, stop := 125808 ), rec( content := ", then t\ he digraph will be immutable. \n If the optional first argument ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "A", next := 125894, root := ~, start := 125883, stop := 125893 ), rec( content := " is not \ present, then ", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "Ref", next := 125954, root := ~, start := 125916, stop := 125953 ), rec( content := " is used\ by default.", count := [ 3, 5, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 3, 5, 9, 7 ], mathmode := "Text", name := "P", next := 125978, root := ~, start := 125974, stop := 125977 ), rec( content := "\n\n \ See also ", count := [ 3, 5, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Peterse\ nGraph" ), content := 0, count := [ 3, 5, 9, 7 ], mathmode := "Text", name := "Ref", next := 126020, root := ~, start := 125993, stop := 126019 ), rec( content := ".\n\n", count := [ 3, 5, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> GeneralisedPetersenGraph(7, 2);\n\ngap> GeneralisedPetersenGraph(40, 1);\n\ngap> D := GeneralisedPetersenGraph(5, 2\ );\n\ngap> IsIsomorphi\ cDigraph(D, PetersenGraph());\ntrue\ngap> GeneralisedPetersenGraph(IsMutableDi\ graph, 9, 4);\n\n", count := [ 3, 5, 9, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 3, 5, 9, 8 ], mathmode := "Text", name := "Example", next := 126495, root := ~, start := 126023, stop := 126494 ), rec( content := "\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 9, 3 ], mathmode := "Text", name := "Description", next := 126512, root := ~, start := 124722, stop := 126511 ) ], count := [ 3, 5, 9, 1 ], mathmode := "Text", name := "ManSection", next := 126526, root := ~, start := 124614, stop := 126525 ), rec( content := "\n\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 126540, root := ~, start := 114270, stop := 126539 ) , rec( content := "\n\n", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 3, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 126552, root := ~, start := 19173, stop := 126551 ), rec( content := "\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " creating ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 3, 5, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Operators" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Operators", count := [ 4, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 126629, root := ~, start := 126601, stop := 126628 ) , rec( content := "\n\n ", count := [ 4, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Operators for dig\ raphs", count := [ 4, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 126683, root := ~, start := 126642, stop := 126682 ), rec( content := "\n ", count := [ 4, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "digraph1", count := [ 4, 1, 0, 4 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 4 ], mathmode := \ "Text", name := "A", next := 126724, root := ~, start := 126709, stop := 12672\ 3 ), rec( content := " = ", count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "A", next := 1267\ 42, root := ~, start := 126727, stop := 126741 ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "C", next := 126746, root := ~, start := 126706, stop := 126745 ) ], count := [ 4, 1, 0, 4 ], name := "Mark", next := 126753, root := ~, start := 126700, stop := 126752 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "=" ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "=", count := [ 4, 1, 0, 5 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "C", next := 126796, root := ~, start := 126788, stop := 12679\ 5 ), rec( content := " (for digraphs)", count := [ 4, 1, 0, 5 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "Index", next := 126819, root := ~, start := 126773, stop := 126818 ), rec( content := "\n \ returns ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "K", next := 126846, root := ~, start := 126835, stop := 126845 ), rec( content := " if ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "A", next := 126865, root := ~, start := 126850, stop := 126864 ), rec( content := " and ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "A", next := 126885, root := ~, start := 126870, stop := 126884 ), rec( content := " have th\ e same\n vertices, and ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph1", count := [ 4, 1, 0, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "A", next := 126952, root := ~, start := 126937, stop := 12695\ 1 ), rec( content := ") =\n DigraphEdges(", count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph2", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "A", next := 1269\ 93, root := ~, start := 126978, stop := 126992 ), rec( content := ")", count := [ 4, 1, 0, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "C", next := 126998, root := ~, start := 126921, stop := 126997 ), rec( content := ", up to \ some re-ordering of the edge\n lists. ", count := [ 4, 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "P", next := 127053, root := ~, start := 127049, stop := 127052 ), rec( content := "\n\n \ Note that this operator does not compare the vertex labels\n of ", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "A", next := 127146, root := ~, start := 127131, stop := 127145 ), rec( content := " and ", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "A", next := 127166, root := ~, start := 127151, stop := 127165 ), rec( content := ".\n \ ", count := [ 4, 1, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 5 ], name := "Item", next := 127180, root := ~, start := 126759, stop := 127179 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "digraph1", count := [ 4, 1, 0, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 7 ], mathmode := \ "Text", name := "A", next := 127211, root := ~, start := 127196, stop := 12721\ 0 ), rec( content := " < ", count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "A", next := 1272\ 32, root := ~, start := 127217, stop := 127231 ) ], count := [ 4, 1, 0, 7 ], mathmode := "Text", name := "C", next := 127236, root := ~, start := 127193, stop := 127235 ) ], count := [ 4, 1, 0, 7 ], name := "Mark", next := 127243, root := ~, start := 127187, stop := 127242 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "<" ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "<", count := [ 4, 1, 0, 8 ], mathmode \ := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 4, 1, 0, 8 ] , mathmode := \ "Text", name := "C", next := 127292, root := ~, start := 127281, stop := 12729\ 1 ), rec( content := " (for digraphs)", count := [ 4, 1, 0, 8 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "Index", next := 127315, root := ~, start := 127263, stop := 127314 ), rec( content := "\n\n \ This operator returns ", count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "K", next := 127357, root := ~, start := 127346, stop := 127356 ), rec( content := " if one \ of the following holds:\n\n ", count := [ 4, 1, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n The number ", count := [ \ 4, 1, 0, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "n_1", count := [ 4, 1, 0, 10 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 10 ], math\ mode := "Text", name := "M", next := 127452, root := ~, start := 127442, stop \ := 127451 ), rec( content := " of vertices in ", count := [ 4, 1, 0, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "digraph1", count := [ 4, 1, 0, 10 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 10 ], mathmode := "T\ ext", name := "A", next := 127483, root := ~, start := 127468, stop := 127482 ), rec( cont\ ent := " is less than\n the number ", count := [ 4, 1, 0, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "n_2", count := [ 4, 1, 0, 10 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 10 ], mathmode := "Text", name\ := "M", next := 127529, root := ~, start := 127519, stop := 127528 ), rec( cont\ ent := " of vertices in ", count := [ 4, 1, 0, 10 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "digraph2", count := [ 4, 1, 0, 10 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 4, 1, 0, 10 ], mathmode := "Text", name := "A", next\ := 127560, root := ~, start := 127545, stop := 127559 ), rec( content := ";\n\ ", count := [ 4, 1, 0, 10 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 4, 1, 0, 10 ], name := "Item", next := 127578, root := ~, start := 127\ 413, stop := 127577 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n_\ 1 = n_2", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "M", next := \ 127622, root := ~, start := 127606, stop := 127621 ), rec( content := ", and t\ he number ", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "m_1", count\ := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 4, 1, 0, 12 ], mathmode := "Text", name := "M", next := 127649, root :=\ ~, start := 127639, stop := 127648 ), rec( content := " of edges in\n \ ", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph1", coun\ t := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 4, 1, 0, 12 ], mathmode := "Text", name := "A", next := 127688, root :=\ ~, start := 127673, stop := 127687 ), rec( content := " is less than the numb\ er ", count := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := [ rec( content := "m_2", count := \ [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 4, 1, 0, 12 ], mathmode := "Text", name := "M", next := 127723, root := ~, star\ t := 127713, stop := 127722 ), rec( content := " of edges in\n ", coun\ t := [ 4, 1, 0, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "digraph2", count := [ 4, 1, 0\ , 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, \ 0, 12 ], mathmode := "Text", name := "A", next := 127762, root := ~, start := \ 127747, stop := 127761 ), rec( content := ";\n ", count := [ 4, 1, 0, \ 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 12 ], name := "Ite\ m", next := 127780, root := ~, start := 127588, stop := 127779 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 4, 1, 0, 1\ 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "n_1 = n_2", count := [ 4, 1, 0, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 14 ], math\ mode := "Text", name := "M", next := 127824, root := ~, start := 127808, stop \ := 127823 ), rec( content := ", ", count := [ 4, 1, 0, 14 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "m_1 = m_2", count := [ 4, 1, 0, 14 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 14 ], mathmode := "Text", name \ := "M", next := 127842, root := ~, start := 127826, stop := 127841 ), rec( cont\ ent := ", and\n ", count := [ 4, 1, 0, 14 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "DigraphEdges(", count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "digraph1", count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "A", next \ := 127890, root := ~, start := 127875, stop := 127889 ), rec( content := ")", \ count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 4, 1, 0, 14 ], mathmode := "Text", name := "C", next := 127895, root :=\ ~, start := 127859, stop := 127894 ), rec( content := " is less than\n \ ", count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphEdges("\ , count := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ),\ rec( attributes := rec( ), content := [ rec( content := "digraph2", count :=\ [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 4, 1, 0, 14 ], mathmode := "Text", name := "A", next := 127951, root := ~,\ start := 127936, stop := 127950 ), rec( content := ")", count := [ 4, 1, 0, 1\ 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, \ 14 ], mathmode := "Text", name := "C", next := 127956, root := ~, start := 127\ 920, stop := 127955 ), rec( content := " after having both of these sets\n \ have been sorted with respect to the lexicographical order.\n " , coun\ t := [ 4, 1, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 14 ], name := "Ite\ m", next := 128076, root := ~, start := 127790, stop := 128075 ) ], count := [ 4, 1, 0, 9 ], mathmode := "Text", name := "List", next := 128091, root := ~, start := 127397, stop := 128090 ), rec( content := "\n " , count := [ 4, 1, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 8 ], name := "Item", next := 128104, root := ~, start := 127249, stop := 128103 ) ], count := [ 4, 1, 0, 4 ], mathmode := "Text", name := "List", next := 128115, root := ~, start := 126688, stop := 128114 ), rec( content := "\n\n ", count := [ 4, 1, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "super, sub", Name := "IsSubdigraph" ) , content := 0, count := [ 4, 1, 1, 2 ], mathmode := "Text", name := "Oper", next := 128179, root := ~, start := 128135, stop := 128178 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "K", next := 128202, root := ~, start := 128191, stop := 128201 ), rec( content := " or ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "K", next := 128218, root := ~, start := 128206, stop := 128217 ), rec( content := ".", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 128229, root := ~, start := 128182, stop := 128228 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "A", next := 128265, root := ~, start := 128253, stop := 128264 ), rec( content := " and ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "A", next := 128280, root := ~, start := 128270, stop := 128279 ), rec( content := " are dig\ raphs, then this operation returns\n ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "K", next := 128338, root := ~, start := 128327, stop := 128337 ), rec( content := " if ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "A", next := 128352, root := ~, start := 128342, stop := 128351 ), rec( content := " is a su\ bdigraph of ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "A", next := 128384, root := ~, start := 128372, stop := 128383 ), rec( content := ", and ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "K", next := 128402, root := ~, start := 128390, stop := 128401 ), rec( content := "\n if\ it is not. ", count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "P", next := 128425, root := ~, start := 128421, stop := 128424 ), rec( content := "\n\n \ A digraph ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 128451, root := ~, start := 128441, stop := 128450 ), rec( content := " is a ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ubdigraph", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "E", next := 128474, root := ~, start := 128457, stop := 128473 ), rec( content := " of a di\ graph ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 128500, root := ~, start := 128488, stop := 128499 ), rec( content := " if\n \ ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 128518, root := ~, start := 128508, stop := 128517 ), rec( content := " and ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 128535, root := ~, start := 128523, stop := 128534 ), rec( content := " share t\ he same number of vertices, and the\n collection of edges of ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 128618, root := ~, start := 128606, stop := 128617 ), rec( content := " (includ\ ing repeats) contains the\n collection of edges of ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "A", next := 128689, root := ~, start := 128679, stop := 128688 ), rec( content := " (includ\ ing repeats). ", count := [ 4, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "P", next := 128716, root := ~, start := 128712, stop := 128715 ), rec( content := "\n\n \ In other words, ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 128748, root := ~, start := 128738, stop := 128747 ), rec( content := " is a su\ bdigraph of ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 128780, root := ~, start := 128768, stop := 128779 ), rec( content := " if and \ only if\n ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "su\ b", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := \ "Text", name := "A", next := 128831, root := ~, start := 128821, stop := 12883\ 0 ), rec( content := ") = DigraphNrVertices(", count := [ 4, 1, 1, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "super", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 1288\ 65, root := ~, start := 128853, stop := 128864 ), rec( content := ")", count := [ 4, 1, 1, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 128870, root := ~, start := 128800, stop := 128869 ), rec( content := ", and\n \ for each pair of vertices ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 128914, root := ~, start := 128906, stop := 128913 ), rec( content := " and ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 128927, root := ~, start := 128919, stop := 128926 ), rec( content := ", there \ are at least as many\n edges of the form ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "C", next := 128991, root := ~, start := 128978, stop := 128990 ), rec( content := " in ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 129007, root := ~, start := 128995, stop := 129006 ), rec( content := " as ther\ e are in ", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "A", next := 129034, root := ~, start := 129024, stop := 129033 ), rec( content := ".\n " , count := [ 4, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 1, 6 ], mathmode := "Text", name := "P", next := 129044, root := ~, start := 129040, stop := 129043 ), rec( content := "\n\n \ ", count := [ 4, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> g := Digraph([[2, 3], [1], [2, 3]]);\n\ngap> h := Digraph([[2, 3], [], [2]]);\n\ngap> IsSubdigraph(g, h);\ntrue\ngap> IsSubdigraph(h, g);\nf\ alse\ngap> IsSubdigraph(CompleteDigraph(4), CycleDigraph(4));\ntrue\ngap> IsSu\ bdigraph(CycleDigraph(4), ChainDigraph(4));\ntrue\ngap> g := Digraph([[2, 2], \ [1]]);\n\ngap> h := Digraph([\ [2], [1]]);\n\ngap> IsSubdigraph(g\ , h);\ntrue\ngap> IsSubdigraph(h, g);\nfalse", count := [ 4, 1, 1, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 1, 7 ], mathmode := "Text", name := "Example", next := 129652, root := ~, start := 129050, stop := 129651 ), rec( content := "\n ", count := [ 4, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 129669, root := ~, start := 128232, stop := 129668 ) ], count := [ 4, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 129683, root := ~, start := 128120, stop := 129682 ), rec( content := "\n\n ", count := [ 4, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "super, sub", Name := "IsUndirectedSpa\ nningTree" ), content := 0, count := [ 4, 1, 2, 2 ], mathmode := "Text", name := "Oper", next := 129759, root := ~, start := 129703, stop := 129758 ), rec( attributes := rec( Arg := "super, sub", Name := "IsUndirectedSpa\ nningForest" ), content := 0, count := [ 4, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 129820, root := ~, start := 129762, stop := 129819 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "K", next := 129843, root := ~, start := 129832, stop := 129842 ), rec( content := " or ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "K", next := 129859, root := ~, start := 129847, stop := 129858 ), rec( content := ".", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 129870, root := ~, start := 129823, stop := 129869 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ operation ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedSpanningTree", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "C", next := 129936, root := ~, start := 129905, stop := 129935 ), rec( content := " returns\ ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "K", next := 129956, root := ~, start := 129945, stop := 129955 ), rec( content := " if the\\ n digraph ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "A", next := 129986, root := ~, start := 129976, stop := 129985 ), rec( content := " is an u\ ndirected spanning tree of the digraph\n ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "A", next := 130049, root := ~, start := 130037, stop := 130048 ), rec( content := ", and th\ e operation ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedSpanningForest", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "C", next := 130102, root := ~, start := 130069, stop := 130101 ), rec( content := " returns\ \n ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "K", next := 130126, root := ~, start := 130115, stop := 130125 ), rec( content := " if the \ digraph ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ub", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "A", next := 130152, root := ~, start := 130142, stop := 130151 ), rec( content := " is an u\ ndirected spanning forest of\n the digraph ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "A", next := 130217, root := ~, start := 130205, stop := 130216 ), rec( content := ". ", count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "P", next := 130223, root := ~, start := 130219, stop := 130222 ), rec( content := "\n\n \ An ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected spanning tree", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "E", next := 130263, root := ~, start := 130232, stop := 130262 ), rec( content := " of a di\ graph ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "A", next := 130289, root := ~, start := 130277, stop := 130288 ), rec( content := " is a su\ bdigraph\n of ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "A", next := 130325, root := ~, start := 130313, stop := 130324 ), rec( content := " that is\ an undirected tree (see ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsSubdi\ graph" ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 130385, root := ~, start := 130358, stop := 130384 ), rec( content := "\n an\ d ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsUndir\ ectedTree" ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 130425, root := ~, start := 130394, stop := 130424 ), rec( content := "). Note \ that a digraph whose ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraph" ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 130500, root := ~, start := 130454, stop := 130499 ), rec( content := " is not \ connected has no undirected\n spanning trees (see ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "Ref", next := 130592, root := ~, start := 130560, stop := 130591 ), rec( content := "). ", count := [ 4, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "P", next := 130599, root := ~, start := 130595, stop := 130598 ), rec( content := "\n\n \ An ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected spanning forest", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "E", next := 130641, root := ~, start := 130608, stop := 130640 ), rec( content := " of a di\ graph ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 130667, root := ~, start := 130655, stop := 130666 ), rec( content := " is a\n \ subdigraph of ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 130703, root := ~, start := 130691, stop := 130702 ), rec( content := " that is\ an undirected forest (see ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsSubdi\ graph" ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 130770, root := ~, start := 130738, stop := 130769 ), rec( content := " and ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsUndir\ ectedForest" ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 130807, root := ~, start := 130775, stop := 130806 ), rec( content := "), and i\ s not\n contained in any larger such subdigraph of ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 130880, root := ~, start := 130868, stop := 130879 ), rec( content := ". Equiv\ alently, an\n undirected spanning forest is a subdigraph of ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 130962, root := ~, start := 130950, stop := 130961 ), rec( content := " whose c\ onnected\n components coincide with those of the ", count := [ 4, 1, 2, 7 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraph" ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 131067, root := ~, start := 131021, stop := 131066 ), rec( content := " of ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ uper", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "A", next := 131083, root := ~, start := 131071, stop := 131082 ), rec( content := " (see ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 131135, root := ~, start := 131089, stop := 131134 ), rec( content := "). ", count := [ 4, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 4, 1, 2, 8 ], mathmode := "Text", name := "P", next := 131142, root := ~, start := 131138, stop := 131141 ), rec( content := "\n\n \ Note that an undirected spanning tree is an undirected spanning forest that\n \ is connected.\n\n ", count := [ 4, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(4);\n\n\ gap> tree := Digraph([[3], [4], [1, 4], [2, 3]]);\n\ngap> IsSubdigraph(D, tree) and IsUndirectedTree(tree);\ntru\ e\ngap> IsUndirectedSpanningTree(D, tree);\ntrue\ngap> forest := EmptyDigraph(\ 4);\n\ngap> IsSubdigraph(D, forest) a\ nd IsUndirectedForest(forest);\ntrue\ngap> IsUndirectedSpanningForest(D, fores\ t);\nfalse\ngap> IsSubdigraph(tree, forest);\ntrue\ngap> D := DigraphDisjointU\ nion(CycleDigraph(2), CycleDigraph(2));\n\ngap> IsUndirectedTree(D);\nfalse\ngap> IsUndirectedForest(D) and IsUn\ directedSpanningForest(D, D);\ntrue", count := [ 4, 1, 2, 9 ], name := "PCDATA\ ", root := ~ ) ], count := [ 4, 1, 2, 9 ], mathmode := "Text", name := "Example", next := 131994, root := ~, start := 131247, stop := 131993 ), rec( content := "\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 132011, root := ~, start := 129873, stop := 132010 ) ], count := [ 4, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 132025, root := ~, start := 129688, stop := 132024 ), rec( content := "\n\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 132039, root := ~, start := 126633, stop := 132038 ) , rec( content := "\n", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 4, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 132050, root := ~, start := 126574, stop := 132049 ), rec( content := "\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " operators/hashs ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 4, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Attributes and operations" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Attributes and operations\ ", count := [ 5, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 132166, root := ~, start := 132122, stop := 132165 ), rec( content := "\n ", count := [ 5, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Vertices and edge\ s", count := [ 5, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 132215, root := ~, start := 132178, stop := 132214 ), rec( content := "\n ", count := [ 5, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphVertices\ " ), content := 0, count := [ 5, 1, 1, 2 ], mathmode := "Text", name := "Attr", next := 132279, root := ~, start := 132235, stop := 132278 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ integers.", count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 132320, root := ~, start := 132282, stop := 132319 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the vertices of the digraph ", count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "A", next := 132391, root := ~, start := 132377, stop := 132390 ), rec( content := ". ", count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "P", next := 132397, root := ~, start := 132393, stop := 132396 ), rec( content := "\n\n \ Note that the vertices of a digraph are always a range of\n positive intege\ rs from ", count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "C", next := 132496, root := ~, start := 132488, stop := 132495 ), rec( content := " to the \ number of vertices of the graph.\n ", count := [ 5, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \ \"b\"],\n> [\"b\", \"c\", \"a\"]);\n\ngap> DigraphVertices(gr);\n[ 1 .. 3 ]\ngap> gr := Digraph\ ([1, 2, 3, 4, 5, 7],\n> [1, 2, 2, 4, 4],\n> \ [2, 7, 5, 3, 7]);\n\ngap> DigraphV\ ertices(gr);\n[ 1 .. 6 ]\ngap> DigraphVertices(RandomDigraph(100));\n[ 1 .. 10\ 0 ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphVertices(D);\n[ 1 .. 3 ]\n", count := [ 5, 1, 1, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 1, 5 ], mathmode := "Text", name := "Example", next := 133137, root := ~, start := 132541, stop := 133136 ), rec( content := "\n ", count := [ 5, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 133154, root := ~, start := 132323, stop := 133153 ) ], count := [ 5, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 133168, root := ~, start := 132220, stop := 133167 ), rec( content := "\n\n ", count := [ 5, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphNrVertic\ es" ), content := 0, count := [ 5, 1, 2, 2 ], mathmode := "Text", name := "Attr", next := 133235, root := ~, start := 133189, stop := 133234 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "Returns", next := 133268, root := ~, start := 133238, stop := 133267 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the number of vertices of the digraph ", count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "A", next := 133349, root := ~, start := 133335, stop := 133348 ), rec( content := ". ", count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 2, 4 ], mathmode := "Text", name := "P", next := 133355, root := ~, start := 133351, stop := 133354 ), rec( content := "\n\n \ ", count := [ 5, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \ \"b\"],\n> [\"b\", \"c\", \"a\"]);\n\ngap> DigraphNrVertices(gr);\n3\ngap> gr := Digraph([1, 2,\ 3, 4, 5, 7],\n> [1, 2, 2, 4, 4],\n> [2, 7, \ 5, 3, 7]);\n\ngap> DigraphNrVertic\ es(gr);\n6\ngap> DigraphNrVertices(RandomDigraph(100));\n100\ngap> D := CycleD\ igraph(IsMutableDigraph, 3);\n\ngap>\ DigraphNrVertices(D);\n3\n", count := [ 5, 1, 2, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 5 ], mathmode := "Text", name := "Example", next := 133929, root := ~, start := 133361, stop := 133928 ), rec( content := "\n ", count := [ 5, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 133946, root := ~, start := 133271, stop := 133945 ) ], count := [ 5, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 133960, root := ~, start := 133174, stop := 133959 ), rec( content := "\n\n ", count := [ 5, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphEdges" ) , content := 0, count := [ 5, 1, 3, 2 ], mathmode := "Text", name := "Attr", next := 134022, root := ~, start := 133981, stop := 134021 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ lists.", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "Returns", next := 134060, root := ~, start := 134025, stop := 134059 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "C", next := 134100, root := ~, start := 134081, stop := 134099 ), rec( content := " returns\ a list of edges of the digraph ", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "A", next := 134154, root := ~, start := 134140, stop := 134153 ), rec( content := ",\n w\ here each edge is a pair of elements of ", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Vertices" ), content := 0, count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 134230, root := ~, start := 134201, stop := 134229 ), rec( content := " of\n \ the form ", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ source,range]", count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "C", next := 134268, root := ~, start := 134247, stop := 134267 ), rec( content := ".\n " , count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "P", next := 134278, root := ~, start := 134274, stop := 134277 ), rec( content := "\n\n \ The entries of ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "C", next := 134319, root := ~, start := 134299, stop := 134318 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "A", next := 134333, root := ~, start := 134319, stop := 134332 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "C", next := 134341, root := ~, start := 134333, stop := 134340 ), rec( content := " are in \ one-to-one\n corresponence with the edges of ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "A", next := 134410, root := ~, start := 134396, stop := 134409 ), rec( content := ". Hence\ \n ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "C", next := 134443, root := ~, start := 134423, stop := 134442 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "A", next := 134457, root := ~, start := 134443, stop := 134456 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "C", next := 134465, root := ~, start := 134457, stop := 134464 ), rec( content := " is dupl\ icate-free if and only if\n ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "A", next := 134517, root := ~, start := 134503, stop := 134516 ), rec( content := " contain\ s no multiple edges. ", count := [ 5, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "P", next := 134550, root := ~, start := 134546, stop := 134549 ), rec( content := "\n\n \ The entries of ", count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges", count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "C", next := 134590, root := ~, start := 134571, stop := 134589 ), rec( content := " are gua\ ranteed to be sorted by their\n first component (i.e. by the source of each\ edge), but they are not\n necessarily then sorted by the second component.\ \n ", count := [ 5, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := DigraphFromDiSparse6String(\".DaXbOe?EAM@G~\");\n\ngap> edges := ShallowCopy(DigraphEdges(gr))\ ;; Sort(edges);\ngap> edges;\n[ [ 1, 1 ], [ 1, 3 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5\ ], [ 2, 1 ], \n [ 2, 2 ], [ 2, 3 ], [ 2, 5 ], [ 3, 2 ], [ 3, 4 ], [ 3, 5 ], \ \n [ 4, 2 ], [ 4, 4 ], [ 4, 5 ], [ 5, 1 ] ]\ngap> D := CycleDigraph(IsMutable\ Digraph, 3);\n\ngap> DigraphEdges(D)\ ;\n[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ]\n", count := [ 5, 1, 3, 6 ], name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 3, 6 ], mathmode := "Text", name := "Example", next := 135280, root := ~, start := 134757, stop := 135279 ), rec( content := "\n ", count := [ 5, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 135297, root := ~, start := 134063, stop := 135296 ) ], count := [ 5, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 135311, root := ~, start := 133966, stop := 135310 ), rec( content := "\n\n ", count := [ 5, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphNrEdges"\ ), content := 0, count := [ 5, 1, 4, 2 ], mathmode := "Text", name := "Attr", next := 135375, root := ~, start := 135332, stop := 135374 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "Returns", next := 135408, root := ~, start := 135378, stop := 135407 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the number of edges of the digraph ", count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "A", next := 135500, root := ~, start := 135486, stop := 135499 ), rec( content := ".\n " , count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\n> [1, 3, 4, 5], [1, 2, 3, 5], [2, 4, 5], [2, 4, 5], [1]\ ]);;\ngap> DigraphNrEdges(gr);\n15\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\ \n> [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\",\ \"a\"]);\n\ngap> DigraphNrEd\ ges(gr);\n3\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphNrEdges(D);\n3\n", count := [ 5, 1, 4, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 4, 4 ], mathmode := "Text", name := "Example", next := 135947, root := ~, start := 135506, stop := 135946 ), rec( content := "\n ", count := [ 5, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 135964, root := ~, start := 135411, stop := 135963 ) ], count := [ 5, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 135978, root := ~, start := 135317, stop := 135977 ), rec( content := "\n\n ", count := [ 5, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphNrLoops"\ ), content := 0, count := [ 5, 1, 5, 2 ], mathmode := "Text", name := "Attr", next := 136042, root := ~, start := 135999, stop := 136041 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Returns", next := 136075, root := ~, start := 136045, stop := 136074 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the number of loops of the digraph ", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "A", next := 136167, root := ~, start := 136153, stop := 136166 ), rec( content := ". See ", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Ref", next := 136202, root := ~, start := 136173, stop := 136201 ), rec( content := ". ", count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 5, 4 ], mathmode := "Text", name := "P", next := 136208, root := ~, start := 136204, stop := 136207 ), rec( content := "\n ", count := [ 5, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3], [1, 4], [3, 3, 5], [], [2, 5]]);\n\ngap> DigraphNrLoops(D);\n3\ngap> D := Empty\ Digraph(5);\n\ngap> DigraphNrLoops(D)\ ;\n0\ngap> D := CompleteDigraph(5);\n\ngap> DigraphNrLoops(D);\n0\n", count := [ 5, 1, 5, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 5 ], mathmode := "Text", name := "Example", next := 136577, root := ~, start := 136213, stop := 136576 ), rec( content := "\n ", count := [ 5, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 136594, root := ~, start := 136078, stop := 136593 ) ], count := [ 5, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 136608, root := ~, start := 135984, stop := 136607 ), rec( content := "\n\n ", count := [ 5, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSinks" ) , content := 0, count := [ 5, 1, 6, 2 ], mathmode := "Text", name := "Attr", next := 136670, root := ~, start := 136629, stop := 136669 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "Returns", next := 136711, root := ~, start := 136673, stop := 136710 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a list of the sinks of the digraph\n ", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "A", next := 136807, root := ~, start := 136793, stop := 136806 ), rec( content := ".\n A\ sink of a digraph is a vertex with out-degree zero.\n See ", count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "Ref", next := 136906, root := ~, start := 136875, stop := 136905 ), rec( content := ".\n " , count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[3, 5, 2, 2], [3], [], [5, 2, 5, 3], []]);\n\ngap> DigraphSinks(gr);\n[ 3, 5 ]\ngap> D\ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSinks(D);\n[ ]\n", count := [ 5, 1, 6, 4 ], name := "PCDATA\ ", root := ~ ) ], count := [ 5, 1, 6, 4 ], mathmode := "Text", name := "Example", next := 137204, root := ~, start := 136912, stop := 137203 ), rec( content := "\n ", count := [ 5, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 137221, root := ~, start := 136714, stop := 137220 ) ], count := [ 5, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 137235, root := ~, start := 136614, stop := 137234 ), rec( content := "\n\n ", count := [ 5, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSources"\ ), content := 0, count := [ 5, 1, 7, 2 ], mathmode := "Text", name := "Attr", next := 137299, root := ~, start := 137256, stop := 137298 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "Returns", next := 137340, root := ~, start := 137302, stop := 137339 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns an immutable list of the sources of the digraph\n ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "A", next := 137449, root := ~, start := 137435, stop := 137448 ), rec( content := ".\n A\ source of a digraph is a vertex with in-degree zero.\n See ", count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "InDegre\ eOfVertex" ), content := 0, count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "Ref", next := 137548, root := ~, start := 137518, stop := 137547 ), rec( content := ".\n " , count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[3, 5, 2, 2], [3], [], [5, 2, 5, 3], []]);\n\ngap> DigraphSources(gr);\n[ 1, 4 ]\ngap>\ D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSources(D);\n[ ]\n", count := [ 5, 1, 7, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 7, 4 ], mathmode := "Text", name := "Example", next := 137850, root := ~, start := 137554, stop := 137849 ), rec( content := "\n ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 7, 3 ], mathmode := "Text", name := "Description", next := 137867, root := ~, start := 137343, stop := 137866 ) ], count := [ 5, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 137881, root := ~, start := 137241, stop := 137880 ), rec( content := "\n\n ", count := [ 5, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphTopologi\ calSort" ), content := 0, count := [ 5, 1, 8, 2 ], mathmode := "Text", name := "Attr", next := 137953, root := ~, start := 137902, stop := 137952 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ positive integers, or ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "K", next := 138008, root := ~, start := 137997, stop := 138007 ), rec( content := ".", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "Returns", next := 138019, root := ~, start := 137956, stop := 138018 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "A", next := 138057, root := ~, start := 138043, stop := 138056 ), rec( content := " is a di\ graph whose only directed cycles are loops, then\n ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphTopologicalSort", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "C", next := 138147, root := ~, start := 138118, stop := 138146 ), rec( content := " returns\ the vertices of ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "A", next := 138186, root := ~, start := 138172, stop := 138185 ), rec( content := " ordered\ \n so that every edge's source appears no earlier in the list than its rang\ e.\n If the digraph ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "A", next := 138307, root := ~, start := 138293, stop := 138306 ), rec( content := " contain\ s directed cycles of length greater\n than ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "M", next := 138368, root := ~, start := 138360, stop := 138367 ), rec( content := ", then t\ his operation returns ", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "K", next := 138409, root := ~, start := 138398, stop := 138408 ), rec( content := ".\n " , count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 8, 4 ], mathmode := "Text", name := "P", next := 138419, root := ~, start := 138415, stop := 138418 ), rec( content := "\n\n \ See Section ", count := [ 5, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 1, 8, 4 ], mathmode := "Text", name := "Ref", next := 138481, root := ~, start := 138437, stop := 138480 ), rec( content := " for the\ definition\n of a directed cycle, and the definition of a loop.\n\n ", count := [ 5, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "P", next := 138565, root := ~, start := 138561, stop := 138564 ), rec( content := "\n\n \ The method used for this attribute has complexity ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "M", next := 138634, root := ~, start := 138621, stop := 138633 ), rec( content := " where\n\ ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "M", next := 138653, root := ~, start := 138645, stop := 138652 ), rec( content := " is the \ number of edges (counting multiple edges as one) and\n ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "M", next := 138726, root := ~, start := 138718, stop := 138725 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 5, 1, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "P", next := 138773, root := ~, start := 138769, stop := 138772 ), rec( content := "\n ", count := [ 5, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 3], [], [4, 6], [5], [], [7, 8, 9], [], [], []]);\ \n\ngap> DigraphTopologicalSort(D)\ ;\n[ 2, 5, 4, 7, 8, 9, 6, 3, 1 ]\ngap> D := Digraph(IsMutableDigraph, [[2, 3],\ [3], [4], []]);\n\ngap> DigraphTopo\ logicalSort(D);\n[ 4, 3, 2, 1 ]\n", count := [ 5, 1, 8, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 7 ], mathmode := "Text", name := "Example", next := 139144, root := ~, start := 138778, stop := 139143 ), rec( content := "\n ", count := [ 5, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 8, 3 ], mathmode := "Text", name := "Description", next := 139161, root := ~, start := 138022, stop := 139160 ) ], count := [ 5, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 139175, root := ~, start := 137887, stop := 139174 ), rec( content := "\n\n ", count := [ 5, 1, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, i", Name := "DigraphVertexLa\ bel" ), content := 0, count := [ 5, 1, 9, 2 ], mathmode := "Text", name := "Oper", next := 139248, root := ~, start := 139196, stop := 139247 ), rec( attributes := rec( Arg := "digraph, i, obj" , Name := "SetDigraphVerte\ xLabel" ), content := 0, count := [ 5, 1, 9, 4 ], mathmode := "Text", name := "Oper", next := 139310, root := ~, start := 139251, stop := 139309 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 139348, root := ~, start := 139334, stop := 139347 ), rec( content := " is a di\ graph, then the first operation returns the label\n of the vertex ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 139432, root := ~, start := 139424, stop := 139431 ), rec( content := ". The se\ cond operation can be used to set the label\n of the vertex ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 139510, root := ~, start := 139502, stop := 139509 ), rec( content := " in ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 139528, root := ~, start := 139514, stop := 139527 ), rec( content := " to the \ arbitrary ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object\\ n ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ bj", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "A", next := 139573, root := ~, start := 139563, stop := 139572 ), rec( content := ". ", count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "P", next := 139580, root := ~, start := 139576, stop := 139579 ), rec( content := "\n\n \ The label of a vertex can be changed an arbitrary number of times. If no\n \ label has been set for the vertex ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "A", next := 139705, root := ~, start := 139697, stop := 139704 ), rec( content := ", then t\ he default value is\n ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "A", next := 139745, root := ~, start := 139737, stop := 139744 ), rec( content := ". ", count := [ 5, 1, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "P", next := 139751, root := ~, start := 139747, stop := 139750 ), rec( content := "\n\n \ If ", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "A", next := 139774, root := ~, start := 139760, stop := 139773 ), rec( content := " is a di\ graph created from a record with a component\n ", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertices", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "C", next := 139846, root := ~, start := 139831, stop := 139845 ), rec( content := ", then t\ he labels of the vertices are set to the value of\n this component.", count := [ 5, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 9, 8 ], mathmode := "Text", name := "P", next := 139927, root := ~, start := 139923, stop := 139926 ), rec( content := "\n\n \ Induced subdigraphs, and some other operations which create new digraphs from\ \n old ones, inherit their labels from their parents.\n\n ", count := [ 5, 1, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphVertexLabel(D, 3);\n3\ngap> D := Digraph([\"a\ \", \"b\", \"c\"], [], []);\n\ngap> D\ igraphVertexLabel(D, 2);\n\"b\"\ngap> SetDigraphVertexLabel(D, 2, \"d\");\ngap\ > DigraphVertexLabel(D, 2);\n\"d\"\ngap> D := InducedSubdigraph(D, [1, 2]);\n<\ immutable empty digraph with 2 vertices>\ngap> DigraphVertexLabel(D, 2);\n\"d\ \"\ngap> D := Digraph(IsMutableDigraph, [\"e\", \"f\", \"g\"], [], []);\n\ngap> DigraphVertexLabel(D, 1);\n\"e\"\ngap\ > SetDigraphVertexLabel(D, 1, \"h\");\ngap> DigraphVertexLabel(D, 1);\n\"h\"\n\ gap> InducedSubdigraph(D, [1, 2]);\n\ng\ ap> DigraphVertexLabel(D, 1);\n\"h\"\n", count := [ 5, 1, 9, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 9, 9 ], mathmode := "Text", name := "Example", next := 140863, root := ~, start := 140071, stop := 140862 ), rec( content := "\n ", count := [ 5, 1, 9, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 9, 5 ], mathmode := "Text", name := "Description", next := 140880, root := ~, start := 139313, stop := 140879 ) ], count := [ 5, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 140894, root := ~, start := 139181, stop := 140893 ), rec( content := "\n\n ", count := [ 5, 1, 9, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphVertexLa\ bels" ), content := 0, count := [ 5, 1, 10, 2 ], mathmode := "Text", name := "Oper", next := 140963, root := ~, start := 140915, stop := 140962 ), rec( attributes := rec( Arg := "digraph, list", Name := "SetDigraphVerte\ xLabels" ), content := 0, count := [ 5, 1, 10, 4 ], mathmode := "Text", name := "Oper", next := 141023, root := ~, start := 140966, stop := 141022 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 141061, root := ~, start := 141047, stop := 141060 ), rec( content := " is a di\ graph, then ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphVertexLabels", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "C", next := 141107, root := ~, start := 141081, stop := 141106 ), rec( content := " returns\ a\n copy of the labels of the vertices in ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 141174, root := ~, start := 141160, stop := 141173 ), rec( content := ".\n " , count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etDigraphVertexLabels", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "C", next := 141209, root := ~, start := 141180, stop := 141208 ), rec( content := " can be \ used to set the labels of the vertices\n in ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 141277, root := ~, start := 141263, stop := 141276 ), rec( content := " to the \ list of\n arbitrary ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "A", next := 141332, root := ~, start := 141321, stop := 141331 ), rec( content := ". ", count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "P", next := 141338, root := ~, start := 141334, stop := 141337 ), rec( content := "\n\n \ The label of a vertex can be changed an arbitrary number of times. If no\n \ label has been set for the vertex ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "A", next := 141463, root := ~, start := 141455, stop := 141462 ), rec( content := ", then t\ he default value is\n ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "A", next := 141503, root := ~, start := 141495, stop := 141502 ), rec( content := ". ", count := [ 5, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "P", next := 141509, root := ~, start := 141505, stop := 141508 ), rec( content := "\n\n \ If ", count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "A", next := 141532, root := ~, start := 141518, stop := 141531 ), rec( content := " is a di\ graph created from a record with a component\n ", count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertices", count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "C", next := 141604, root := ~, start := 141589, stop := 141603 ), rec( content := ", then t\ he labels of the vertices are set to the value of\n this component.", count := [ 5, 1, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "P", next := 141685, root := ~, start := 141681, stop := 141684 ), rec( content := "\n\n \ Induced subdigraphs, and other operations which create new digraphs from\n \ old ones, inherit their labels from their parents.\n\n ", count := [ 5, 1, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphVertexLabels(D);\n[ 1 .. 5 ]\ngap> D := Digra\ ph([\"a\", \"b\", \"c\"], [], []);\n\ \ngap> DigraphVertexLabels(D);\n[ \"a\", \"b\", \"c\" ]\ngap> SetDigraphVertex\ Label(D, 2, \"d\");\ngap> DigraphVertexLabels(D);\n[ \"a\", \"d\", \"c\" ]\nga\ p> D := InducedSubdigraph(D, [1, 3]);\n\ngap> DigraphVertexLabels(D);\n[ \"a\", \"c\" ]\ngap> D := Digraph(IsMutabl\ eDigraph, [\"e\", \"f\", \"g\"], [], []);\n\ngap> SetDigraphVertexLabels(D, [\"h\", \"i\", \"j\"]);\ngap> DigraphVert\ exLabels(D);\n[ \"h\", \"i\", \"j\" ]\ngap> InducedSubdigraph(D, [1, 3]);\n\ngap> DigraphVertexLabels(D);\n[ \"h\", \ \"j\" ]\n", count := [ 5, 1, 10, 9 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 9 ], mathmode := "Text", name := "Example", next := 142648, root := ~, start := 141824, stop := 142647 ), rec( content := "\n ", count := [ 5, 1, 10, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 10, 5 ], mathmode := "Text", name := "Description", next := 142665, root := ~, start := 141026, stop := 142664 ) ], count := [ 5, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 142679, root := ~, start := 140900, stop := 142678 ), rec( content := "\n\n ", count := [ 5, 1, 10, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, i, j", Name := "DigraphEdgeLabe\ l" ), content := 0, count := [ 5, 1, 11, 2 ], mathmode := "Text", name := "Oper", next := 142753, root := ~, start := 142700, stop := 142752 ), rec( attributes := rec( Arg := "digraph, i, j, o\ bj", Name := "SetDigraphEdgeLabel" ), content := 0, count := [ 5, 1, 11, 4 ], mathmode := "Text", name := "Oper", next := 142817, root := ~, start := 142756, stop := 142816 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "A", next := 142855, root := ~, start := 142841, stop := 142854 ), rec( content := " is a di\ graph without multiple edges, then the first\n operation returns the label \ of the edge from vertex ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "A", next := 142972, root := ~, start := 142964, stop := 142971 ), rec( content := " to vert\ ex\n ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "A", next := 142995, root := ~, start := 142987, stop := 142994 ), rec( content := ". The s\ econd operation can be used to set the label of the edge\n between vertex " , count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "A", next := 143087, root := ~, start := 143079, stop := 143086 ), rec( content := " and ver\ tex ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "A", next := 143107, root := ~, start := 143099, stop := 143106 ), rec( content := " to the \ arbitrary ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " object\\ n ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ bj", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "A", next := 143152, root := ~, start := 143142, stop := 143151 ), rec( content := ". ", count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 11, 6 ], mathmode := "Text", name := "P", next := 143158, root := ~, start := 143154, stop := 143157 ), rec( content := "\n\n \ The label of an edge can be changed an arbitrary number of times. If no\n l\ abel has been set for the edge, then the default value is ", count := [ 5, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 6 ], mathmode := "Text", name := "A", next := 143307, root := ~, start := 143299, stop := 143306 ), rec( content := ". ", count := [ 5, 1, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "P", next := 143313, root := ~, start := 143309, stop := 143312 ), rec( content := "\n\n \ Induced subdigraphs, and some other operations which create new digraphs from\ \n old ones, inherit their edge labels from their parents.\n\n See also \ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Oper := "Digraph\ EdgeLabels" ), content := 0, count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "Ref", next := 143502, root := ~, start := 143471, stop := 143501 ), rec( content := ".\n\n \ ", count := [ 5, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphEdgeLabel(D, 3, 1);\n1\ngap> SetDigraphEdgeLa\ bel(D, 2, 5, [42]);\ngap> DigraphEdgeLabel(D, 2, 5);\n[ 42 ]\ngap> D := Induce\ dSubdigraph(D, [2, 5]);\n\ngap> Di\ graphEdgeLabel(D, 1, 2);\n[ 42 ]\ngap> D := ChainDigraph(IsMutableDigraph, 5);\ \n\ngap> DigraphEdgeLabel(D, 2, 3);\ \n1\ngap> SetDigraphEdgeLabel(D, 4, 5, [1729]);\ngap> DigraphEdgeLabel(D, 4, 5\ );\n[ 1729 ]\ngap> InducedSubdigraph(D, [4, 5]);\n\ngap> DigraphEdgeLabel(D, 1, 2);\n[ 1729 ]\n", count := [ 5, 1, 11, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 11, 8 ], mathmode := "Text", name := "Example", next := 144197, root := ~, start := 143509, stop := 144196 ), rec( content := "\n ", count := [ 5, 1, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 11, 5 ], mathmode := "Text", name := "Description", next := 144214, root := ~, start := 142820, stop := 144213 ) ], count := [ 5, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 144228, root := ~, start := 142685, stop := 144227 ), rec( content := "\n\n ", count := [ 5, 1, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphEdgeLabe\ ls" ), content := 0, count := [ 5, 1, 12, 2 ], mathmode := "Text", name := "Oper", next := 144295, root := ~, start := 144249, stop := 144294 ), rec( attributes := rec( Arg := "digraph, labels" , Label := "for a digraph \ and a list of lists", Name := "SetDigraphEdgeLabels" ), content := 0, count := [ 5, 1, 12, 4 ], mathmode := "Text", name := "Oper", next := 144401, root := ~, start := 144298, stop := 144400 ), rec( attributes := rec( Arg := "digraph, func", Label := "for a digraph \ and a function", Name := "SetDigraphEdgeLabels" ), content := 0, count := [ 5, 1, 12, 6 ], mathmode := "Text", name := "Oper", next := 144500, root := ~, start := 144404, stop := 144499 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "A", next := 144538, root := ~, start := 144524, stop := 144537 ), rec( content := " is a di\ graph without multiple edges, then\n ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdgeLabels", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 144609, root := ~, start := 144585, stop := 144608 ), rec( content := " returns\ a copy of the labels of the edges\n in ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "A", next := 144673, root := ~, start := 144659, stop := 144672 ), rec( content := " as a li\ st of lists ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 144706, root := ~, start := 144693, stop := 144705 ), rec( content := " such th\ at\n ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels[i][j]", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 144740, root := ~, start := 144721, stop := 144739 ), rec( content := " is the \ label on the edge from vertex ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 144786, root := ~, start := 144778, stop := 144785 ), rec( content := "\n to\ vertex ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours(digraph)[i][j]", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 144836, root := ~, start := 144801, stop := 144835 ), rec( content := ".\n\n \ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etDigraphEdgeLabels", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 144870, root := ~, start := 144843, stop := 144869 ), rec( content := " can be \ used to set the labels of the edges in\n ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "A", next := 144935, root := ~, start := 144921, stop := 144934 ), rec( content := " without\ multiple edges to the list ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ abels", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "A", next := 144984, root := ~, start := 144971, stop := 144983 ), rec( content := " of list\ s of\n arbitrary ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ such that ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[i][j]", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 145052, root := ~, start := 145035, stop := 145051 ), rec( content := " is the \ label on the edge\n from vertex ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 145102, root := ~, start := 145094, stop := 145101 ), rec( content := " to the \ vertex ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours(digraph>[i][j]", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 145152, root := ~, start := 145117, stop := 145151 ), rec( content := ".\n\n \ Alternatively ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etDigraphEdgeLabels", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 145200, root := ~, start := 145173, stop := 145199 ), rec( content := " can be \ called with binary function\n ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "A", next := 145251, root := ~, start := 145240, stop := 145250 ), rec( content := " that as\ its second argument that when passed two vertices ", count := [ 5, 1, 12, 7 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 145318, root := ~, start := 145310, stop := 145317 ), rec( content := "\n an\ d ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 145335, root := ~, start := 145327, stop := 145334 ), rec( content := " returns\ the label for the edge between vertex ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 145390, root := ~, start := 145382, stop := 145389 ), rec( content := " and ver\ tex\n ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "C", next := 145414, root := ~, start := 145406, stop := 145413 ), rec( content := ". ", count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 12, 8 ], mathmode := "Text", name := "P", next := 145420, root := ~, start := 145416, stop := 145419 ), rec( content := "\n\n \ The label of an edge can be changed an arbitrary number of times. If no\n l\ abel has been set for an edge, then the default value is ", count := [ 5, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 8 ], mathmode := "Text", name := "C", next := 145568, root := ~, start := 145560, stop := 145567 ), rec( content := ". ", count := [ 5, 1, 12, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 12, 9 ], mathmode := "Text", name := "P", next := 145574, root := ~, start := 145570, stop := 145573 ), rec( content := "\n\n \ Induced subdigraphs, and some other operations which create new digraphs\n \ from old ones, inherit their labels from their parents.\n\n ", count := [ 5, 1, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&DHUEe_\");\n\ngap> DigraphEdgeLabels(D);\n[ [ 1 ], [ 1, 1, 1 ], [ 1 ], \ [ 1, 1, 1 ], [ 1, 1, 1 ] ]\ngap> SetDigraphEdgeLabel(D, 2, 1, \"d\");\ngap> Di\ graphEdgeLabels(D);\n[ [ 1 ], [ \"d\", 1, 1 ], [ 1 ], [ 1, 1, 1 ], [ 1, 1, 1 ]\ ]\ngap> D := InducedSubdigraph(D, [1, 2, 3]);\n\ngap> DigraphEdgeLabels(D);\n[ [ 1 ], [ \"d\", 1 ], [ 1 ] ]\nga\ p> OutNeighbours(D);\n[ [ 3 ], [ 1, 3 ], [ 1 ] ]\ngap> D := CompleteBipartiteD\ igraph(IsMutableDigraph, 2, 3);\n\n\ gap> DigraphEdgeLabels(D);\n[ [ 1, 1, 1 ], [ 1, 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ \ 1, 1 ] ]\ngap> SetDigraphEdgeLabel(D, 2, 4, \"a\");\ngap> DigraphEdgeLabels(D)\ ;\n[ [ 1, 1, 1 ], [ 1, \"a\", 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ] ]\ngap> Induce\ dSubdigraph(D, [1, 2, 3, 4]);\n\ngap\ > DigraphEdgeLabels(D);\n[ [ 1, 1 ], [ 1, \"a\" ], [ 1, 1 ], [ 1, 1 ] ]\ngap> \ OutNeighbors(D);\n[ [ 3, 4 ], [ 3, 4 ], [ 1, 2 ], [ 1, 2 ] ]\n", count := [ 5, 1, 12, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 12, 10 ], mathmode := "Text", name := "Example", next := 146785, root := ~, start := 145718, stop := 146784 ), rec( content := "\n ", count := [ 5, 1, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 12, 7 ], mathmode := "Text", name := "Description", next := 146802, root := ~, start := 144503, stop := 146801 ) ], count := [ 5, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 146816, root := ~, start := 144234, stop := 146815 ), rec( content := "\n\n ", count := [ 5, 1, 12, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphInEdges"\ ), content := 0, count := [ 5, 1, 13, 2 ], mathmode := "Text", name := "Oper", next := 146888, root := ~, start := 146837, stop := 146887 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ edges.", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "Returns", next := 146926, root := ~, start := 146891, stop := 146925 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphInEdges", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "C", next := 146968, root := ~, start := 146947, stop := 146967 ), rec( content := " returns\ the list of all edges of ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "A", next := 147016, root := ~, start := 147002, stop := 147015 ), rec( content := "\n wh\ ich have ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "A", next := 147045, root := ~, start := 147032, stop := 147044 ), rec( content := " as thei\ r range.\n ", count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [3, 3], [4, 4], [1, 1]]);\n\ngap> DigraphInEdges(D, 2);\n[ [ 1, 2 ], [ 1, 2 ] ]\ \n", count := [ 5, 1, 13, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 4 ], mathmode := "Text", name := "Example", next := 147251, root := ~, start := 147066, stop := 147250 ), rec( content := "\n ", count := [ 5, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 13, 3 ], mathmode := "Text", name := "Description", next := 147268, root := ~, start := 146929, stop := 147267 ) ], count := [ 5, 1, 13, 1 ], mathmode := "Text", name := "ManSection", next := 147282, root := ~, start := 146822, stop := 147281 ), rec( content := "\n\n ", count := [ 5, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphOutEdges\ " ), content := 0, count := [ 5, 1, 14, 2 ], mathmode := "Text", name := "Oper", next := 147355, root := ~, start := 147303, stop := 147354 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ edges.", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "Returns", next := 147393, root := ~, start := 147358, stop := 147392 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphOutEdges", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "C", next := 147436, root := ~, start := 147414, stop := 147435 ), rec( content := " returns\ the list of all edges of ", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "A", next := 147484, root := ~, start := 147470, stop := 147483 ), rec( content := "\n wh\ ich have ", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "A", next := 147513, root := ~, start := 147500, stop := 147512 ), rec( content := " as thei\ r source.\n ", count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [3, 3], [4, 4], [1, 1]]);\n\ngap> DigraphOutEdges(D, 2);\n[ [ 2, 3 ], [ 2, 3 ] \ ]\n", count := [ 5, 1, 14, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 4 ], mathmode := "Text", name := "Example", next := 147721, root := ~, start := 147535, stop := 147720 ), rec( content := "\n ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 14, 3 ], mathmode := "Text", name := "Description", next := 147738, root := ~, start := 147396, stop := 147737 ) ], count := [ 5, 1, 14, 1 ], mathmode := "Text", name := "ManSection", next := 147752, root := ~, start := 147288, stop := 147751 ), rec( content := "\n\n ", count := [ 5, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, list", Label := "for digraph an\ d list", Name := "IsDigraphEdge" ), content := 0, count := [ 5, 1, 15, 2 ], mathmode := "Text", name := "Oper", next := 147858, root := ~, start := 147773, stop := 147857 ), rec( attributes := rec( Arg := "digraph, u, v", Label := "for digraph an\ d two pos ints", Name := "IsDigraphEdge" ), content := 0, count := [ 5, 1, 15, 4 ], mathmode := "Text", name := "Oper", next := 147954, root := ~, start := 147861, stop := 147953 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "K", next := 147977, root := ~, start := 147966, stop := 147976 ), rec( content := " or ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "K", next := 147993, root := ~, start := 147981, stop := 147992 ), rec( content := ".", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "Returns", next := 148004, root := ~, start := 147957, stop := 148003 ), rec( attributes := rec( ), content := [ rec( content := "\n In \ the first form, this function returns ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "K", next := 148077, root := ~, start := 148066, stop := 148076 ), rec( content := " if and \ only if the list\n ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "A", next := 148117, root := ~, start := 148106, stop := 148116 ), rec( content := " specifi\ es an edge in the digraph ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "A", next := 148165, root := ~, start := 148151, stop := 148164 ), rec( content := ". Speci\ fically,\n this operation returns ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "K", next := 148220, root := ~, start := 148209, stop := 148219 ), rec( content := " if ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "A", next := 148235, root := ~, start := 148224, stop := 148234 ), rec( content := " is a pa\ ir of positive\n integers where ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "A", next := 148288, root := ~, start := 148277, stop := 148287 ), rec( attributes := rec( ), content := [ rec( content := "[\ 1]", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "C", next := 148298, root := ~, start := 148288, stop := 148297 ), rec( content := " is the \ source and ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "A", next := 148328, root := ~, start := 148317, stop := 148327 ), rec( attributes := rec( ), content := [ rec( content := "[\ 2]", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "C", next := 148338, root := ~, start := 148328, stop := 148337 ), rec( content := "\n is\ the range of an edge in ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "A", next := 148384, root := ~, start := 148370, stop := 148383 ), rec( content := ", and ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "K", next := 148402, root := ~, start := 148390, stop := 148401 ), rec( content := " otherwi\ se. ", count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "P", next := 148418, root := ~, start := 148414, stop := 148417 ), rec( content := "\n\n \ The second form simply returns ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "K", next := 148466, root := ~, start := 148455, stop := 148465 ), rec( content := " if ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes :\ = rec( ), content := [ rec( content := "u", count := [ 5, 1, 15, 6 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 6 ], mathmode := \ "Text", name := "A", next := 148482, root := ~, start := 148474, stop := 14848\ 1 ), rec( content := ", ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 6 ], mathmode := \ "Text", name := "A", next := 148492, root := ~, start := 148484, stop := 14849\ 1 ), rec( content := "]", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "C", next := 148497, root := ~, start := 148470, stop := 148496 ), rec( content := " is\n \ an edge in ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "A", next := 148530, root := ~, start := 148516, stop := 148529 ), rec( content := ", and ", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "K", next := 148548, root := ~, start := 148536, stop := 148547 ), rec( content := " otherwi\ se.\n\n", count := [ 5, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 2], [6], [], [3], [], [1]]);\n\ngap> IsDigraphEdge(D, [1, 1]);\nfalse\ngap> IsDigra\ phEdge(D, [1, 2]);\ntrue\ngap> IsDigraphEdge(D, [1, 8]);\nfalse\n", count := [ 5, 1, 15, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 15, 7 ], mathmode := "Text", name := "Example", next := 148805, root := ~, start := 148561, stop := 148804 ), rec( content := "\n ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 15, 5 ], mathmode := "Text", name := "Description", next := 148822, root := ~, start := 148007, stop := 148821 ) ], count := [ 5, 1, 15, 1 ], mathmode := "Text", name := "ManSection", next := 148836, root := ~, start := 147758, stop := 148835 ), rec( content := "\n\n ", count := [ 5, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, list", Name := "IsMatching" ), content := 0, count := [ 5, 1, 16, 2 ], mathmode := "Text", name := "Oper", next := 148902, root := ~, start := 148857, stop := 148901 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsMaximalMatchi\ ng" ), content := 0, count := [ 5, 1, 16, 4 ], mathmode := "Text", name := "Oper", next := 148957, root := ~, start := 148905, stop := 148956 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsMaximumMatchi\ ng" ), content := 0, count := [ 5, 1, 16, 6 ], mathmode := "Text", name := "Oper", next := 149012, root := ~, start := 148960, stop := 149011 ), rec( attributes := rec( Arg := "digraph, list", Name := "IsPerfectMatchi\ ng" ), content := 0, count := [ 5, 1, 16, 8 ], mathmode := "Text", name := "Oper", next := 149067, root := ~, start := 149015, stop := 149066 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 149090, root := ~, start := 149079, stop := 149089 ), rec( content := " or ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 149106, root := ~, start := 149094, stop := 149105 ), rec( content := ".", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "Returns", next := 149117, root := ~, start := 149070, stop := 149116 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 149155, root := ~, start := 149141, stop := 149154 ), rec( content := " is a di\ graph and ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 149184, root := ~, start := 149173, stop := 149183 ), rec( content := " is a li\ st of pairs of\n vertices of ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 149237, root := ~, start := 149223, stop := 149236 ), rec( content := ", then " , count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMatching", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "C", next := 149261, root := ~, start := 149244, stop := 149260 ), rec( content := " returns\ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 149281, root := ~, start := 149270, stop := 149280 ), rec( content := " if\n \ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 149300, root := ~, start := 149289, stop := 149299 ), rec( content := " is a ma\ tching of ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 149332, root := ~, start := 149318, stop := 149331 ), rec( content := ". The o\ peration\n ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximalMatching", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "C", next := 149377, root := ~, start := 149353, stop := 149376 ), rec( content := " returns\ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 149397, root := ~, start := 149386, stop := 149396 ), rec( content := " if ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 149412, root := ~, start := 149401, stop := 149411 ), rec( content := " is a ma\ ximal \n matching, ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximumMatching", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "C", next := 149465, root := ~, start := 149441, stop := 149464 ), rec( content := " returns\ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 149485, root := ~, start := 149474, stop := 149484 ), rec( content := " if ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 149500, root := ~, start := 149489, stop := 149499 ), rec( content := " is a\n \ maximum matching and ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPerfectMatching", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "C", next := 149555, root := ~, start := 149531, stop := 149554 ), rec( content := " returns\ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 149575, root := ~, start := 149564, stop := 149574 ), rec( content := " if \n \ ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 149595, root := ~, start := 149584, stop := 149594 ), rec( content := " is a pe\ rfect, matching of ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "A", next := 149636, root := ~, start := 149622, stop := 149635 ), rec( content := ", respec\ tively. \n Otherwise, each of these operations return ", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "K", next := 149713, root := ~, start := 149701, stop := 149712 ), rec( content := ".\n " , count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "P", next := 149723, root := ~, start := 149719, stop := 149722 ), rec( content := "\n\n \ A ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ atching", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "E", next := 149746, root := ~, start := 149731, stop := 149745 ), rec( content := " ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "C", next := 149755, root := ~, start := 149747, stop := 149754 ), rec( content := " of a di\ graph ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "A", next := 149783, root := ~, start := 149769, stop := 149782 ), rec( content := " is a su\ bset of the\n edges of ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "A", next := 149830, root := ~, start := 149816, stop := 149829 ), rec( content := ", i.e. " , count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEdges(", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 5, 1, 16, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := \ "Text", name := "A", next := 149867, root := ~, start := 149853, stop := 14986\ 6 ), rec( content := ")", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "C", next := 149872, root := ~, start := 149837, stop := 149871 ), rec( content := ", such\n\ that no pair of distinct edges in ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "C", next := 149925, root := ~, start := 149917, stop := 149924 ), rec( content := " are inc\ ident to the same vertex\n of ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "A", next := 149979, root := ~, start := 149965, stop := 149978 ), rec( content := ". Note \ that this definition allows a matching to contain\n loops. See ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "Ref", next := 150083, root := ~, start := 150053, stop := 150082 ), rec( content := ". The m\ atching ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "C", next := 150107, root := ~, start := 150099, stop := 150106 ), rec( content := " is\n \ ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "E", next := 150129, root := ~, start := 150115, stop := 150128 ), rec( content := " if it i\ s contained in no larger matching of the digraph,\n is ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximum", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "E", next := 150208, root := ~, start := 150194, stop := 150207 ), rec( content := " if it h\ as the greatest cardinality among all matchings and\n is ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erfect", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "E", next := 150289, root := ~, start := 150275, stop := 150288 ), rec( content := " if ever\ y vertex of the digraph is incident to an edge in\n the matching. Every ma\ ximum or perfect matching is maximal. Note, however,\n that not every perfe\ ct matching of digraphs with loops is maximum.\n\n ", count := [ 5, 1, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [1, 2], [2, 3, 4], [3, 5], [1]]);\n\ngap> IsMatching(D, [[2, 1], [3, 2]]);\nfalse\n\ gap> edges := [[3, 2]];;\ngap> IsMatching(D, edges);\ntrue\ngap> IsMaximalMatc\ hing(D, edges);\nfalse\ngap> edges := [[2, 1], [3, 4]];;\ngap> IsMaximalMatchi\ ng(D, edges);\ntrue\ngap> IsPerfectMatching(D, edges);\nfalse\ngap> edges := [\ [1, 2], [3, 3], [4, 5]];;\ngap> IsPerfectMatching(D, edges);\ntrue\ngap> IsMax\ imumMatching(D, edges);\nfalse\ngap> edges := [[1, 1], [2, 2], [3, 3], [4, 5]]\ ;;\ngap> IsMaximumMatching(D, edges);\ntrue\n", count := [ 5, 1, 16, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 16, 11 ], mathmode := "Text", name := "Example", next := 151103, root := ~, start := 150503, stop := 151102 ), rec( content := "\n ", count := [ 5, 1, 16, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 16, 9 ], mathmode := "Text", name := "Description", next := 151120, root := ~, start := 149120, stop := 151119 ) ], count := [ 5, 1, 16, 1 ], mathmode := "Text", name := "ManSection", next := 151134, root := ~, start := 148842, stop := 151133 ), rec( content := "\n\n ", count := [ 5, 1, 16, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalM\ atching" ), content := 0, count := [ 5, 1, 17, 2 ], mathmode := "Text", name := "Attr", next := 151206, root := ~, start := 151155, stop := 151205 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ pairs of vertices.", count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "Returns", next := 151256, root := ~, start := 151209, stop := 151255 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a maximal matching of the digraph ", count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "A", next := 151347, root := ~, start := 151333, stop := 151346 ), rec( content := ".\n " , count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "P", next := 151357, root := ~, start := 151353, stop := 151356 ), rec( content := "\n\n \ For the definition of a maximal matching, see ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alMatching" ), content := 0, count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "Ref", next := 151440, root := ~, start := 151409, stop := 151439 ), rec( content := ". \n\n \ ", count := [ 5, 1, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDiSparse6String(\".IeAoXCJU@|SHAe?d\");\n\ngap> M := DigraphMaximalMatching(D);; IsMaxim\ alMatching(D, M);\ntrue\ngap> D := RandomDigraph(100);;\ngap> IsMaximalMatchin\ g(D, DigraphMaximalMatching(D));\ntrue\ngap> D := GeneralisedPetersenGraph(IsM\ utableDigraph, 9, 2);\n\ngap> IsMa\ ximalMatching(D, DigraphMaximalMatching(D));\ntrue\n", count := [ 5, 1, 17, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 17, 5 ], mathmode := "Text", name := "Example", next := 151908, root := ~, start := 151448, stop := 151907 ), rec( content := "\n ", count := [ 5, 1, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 17, 3 ], mathmode := "Text", name := "Description", next := 151925, root := ~, start := 151259, stop := 151924 ) ], count := [ 5, 1, 17, 1 ], mathmode := "Text", name := "ManSection", next := 151939, root := ~, start := 151140, stop := 151938 ), rec( content := "\n\n ", count := [ 5, 1, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximumM\ atching" ), content := 0, count := [ 5, 1, 18, 2 ], mathmode := "Text", name := "Attr", next := 152011, root := ~, start := 151960, stop := 152010 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ pairs of vertices.", count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "Returns", next := 152061, root := ~, start := 152014, stop := 152060 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns a maximum matching of the digraph ", count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "A", next := 152152, root := ~, start := 152138, stop := 152151 ), rec( content := ".\n " , count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "P", next := 152162, root := ~, start := 152158, stop := 152161 ), rec( content := "\n\n \ For the definition of a maximum matching, see ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ umMatching" ), content := 0, count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "Ref", next := 152245, root := ~, start := 152214, stop := 152244 ), rec( content := ".\n I\ f ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "A", next := 152268, root := ~, start := 152254, stop := 152267 ), rec( content := " is bipa\ rtite (see ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsBipar\ titeDigraph" ), content := 0, count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "Ref", next := 152319, root := ~, start := 152287, stop := 152318 ), rec( content := "), then \ \n the algorithm used has complexity ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m*sqrt(n))", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "C", next := 152385, root := ~, start := 152366, stop := 152384 ), rec( content := ". Otherw\ ise for general \n graphs the complexity is ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m*n*log(n))", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "C", next := 152459, root := ~, start := 152439, stop := 152458 ), rec( content := ". Here " , count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "C", next := 152474, root := ~, start := 152466, stop := 152473 ), rec( content := " is the \ number of vertices \n and ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "C", next := 152518, root := ~, start := 152510, stop := 152517 ), rec( content := " is the \ number of edges.\n\n ", count := [ 5, 1, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&I@EA_A?AdDp[_c??OO\");\n\ngap> M := DigraphMaximumMatching(D);; IsMaxi\ malMatching(D, M);\ntrue\ngap> Length(M);\n5\ngap> D := Digraph([[5, 6, 7, 8],\ [6, 7, 8], [7, 8], [8], \n> [], [], [], []]);;\ngap> M := Di\ graphMaximumMatching(D);\n[ [ 1, 5 ], [ 2, 6 ], [ 3, 7 ], [ 4, 8 ] ]\ngap> D :\ = GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> M := DigraphMaximumMatching(D);;\ngap> IsMaximalMatc\ hing(D, M);\ntrue\ngap> Length(M);\n9\n", count := [ 5, 1, 18, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 1, 18, 5 ], mathmode := "Text", name := "Example", next := 153145, root := ~, start := 152548, stop := 153144 ), rec( content := "\n ", count := [ 5, 1, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 18, 3 ], mathmode := "Text", name := "Description", next := 153162, root := ~, start := 152064, stop := 153161 ) ], count := [ 5, 1, 18, 1 ], mathmode := "Text", name := "ManSection", next := 153176, root := ~, start := 151945, stop := 153175 ), rec( content := "\n\n ", count := [ 5, 1, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 153190, root := ~, start := 132169, stop := 153189 ) , rec( content := "\n\n ", count := [ 5, 1, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Neighbours and de\ gree", count := [ 5, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 153243, root := ~, start := 153203, stop := 153242 ), rec( content := "\n ", count := [ 5, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "AdjacencyMatrix\ " ), content := 0, count := [ 5, 2, 1, 2 ], mathmode := "Text", name := "Attr", next := 153307, root := ~, start := 153263, stop := 153306 ), rec( attributes := rec( Arg := "digraph", Name := "AdjacencyMatrix\ MutableCopy" ), content := 0, count := [ 5, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 153365, root := ~, start := 153310, stop := 153364 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix of non-negative integers.", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 153428, root := ~, start := 153368, stop := 153427 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the adjacency matrix ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "C", next := 153502, root := ~, start := 153492, stop := 153501 ), rec( content := " of the \ digraph\n ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "A", next := 153536, root := ~, start := 153522, stop := 153535 ), rec( content := ".\n T\ he value of the matrix entry ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "C", next := 153588, root := ~, start := 153572, stop := 153587 ), rec( content := " is the \ number of edges\n in ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "A", next := 153633, root := ~, start := 153619, stop := 153632 ), rec( content := " with so\ urce ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "C", next := 153654, root := ~, start := 153646, stop := 153653 ), rec( content := " and ran\ ge ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "C", next := 153673, root := ~, start := 153665, stop := 153672 ), rec( content := ". If ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "A", next := 153692, root := ~, start := 153678, stop := 153691 ), rec( content := "\n ha\ s no vertices, then the empty list is returned. ", count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "P", next := 153751, root := ~, start := 153747, stop := 153750 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ djacencyMatrix", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "C", next := 153792, root := ~, start := 153770, stop := 153791 ), rec( content := " returns\ an immutable list of\n lists, whereas the function ", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ djacencyMatrixMutableCopy", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "C", next := 153887, root := ~, start := 153854, stop := 153886 ), rec( content := " returns\ a copy\n of ", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ djacencyMatrix", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "C", next := 153932, root := ~, start := 153910, stop := 153931 ), rec( content := " that is\ a mutable list of mutable lists. ", count := [ 5, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 1, 7 ], mathmode := "Text", name := "P", next := 153978, root := ~, start := 153974, stop := 153977 ), rec( content := "\n\n \ ", count := [ 5, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\n> [2, 2, 2], [1, 3, 6, 8, 9, 10], [4, 6, 8],\n> [1, 2, \ 3, 9], [3, 3], [3, 5, 6, 10], [1, 2, 7],\n> [1, 2, 3, 10, 5, 6, 10], [1, 3, 4,\ 5, 8, 10],\n> [2, 3, 4, 6, 7, 10]]);\n\ngap> mat := AdjacencyMatrix(gr);;\ngap> Display(mat);\n[ [ 0, \ 3, 0, 0, 0, 0, 0, 0, 0, 0 ],\n [ 1, 0, 1, 0, 0, 1, 0, 1, 1, \ 1 ],\n [ 0, 0, 0, 1, 0, 1, 0, 1, 0, 0 ],\n [ 1, 1, 1, 0, 0, \ 0, 0, 0, 1, 0 ],\n [ 0, 0, 2, 0, 0, 0, 0, 0, 0, 0 ],\n [ 0, \ 0, 1, 0, 1, 1, 0, 0, 0, 1 ],\n [ 1, 1, 0, 0, 0, 0, 1, 0, 0,\ 0 ],\n [ 1, 1, 1, 0, 1, 1, 0, 0, 0, 2 ],\n [ 1, 0, 1, 1, 1,\ 0, 0, 1, 0, 1 ],\n [ 0, 1, 1, 1, 0, 1, 1, 0, 0, 1 ] ]\ngap> D\ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> Display(AdjacencyMatrix(D));\n[ [ 0, 1, 0 ],\n [ 0, 0, 1 ],\ \n [ 1, 0, 0 ] ]\n", count := [ 5, 2, 1, 8 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 8 ], mathmode := "Text", name := "Example", next := 154948, root := ~, start := 153984, stop := 154947 ), rec( content := "\n ", count := [ 5, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 154965, root := ~, start := 153431, stop := 154964 ) ], count := [ 5, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 154979, root := ~, start := 153248, stop := 154978 ), rec( content := "\n\n ", count := [ 5, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "CharacteristicP\ olynomial" ), content := 0, count := [ 5, 2, 2, 2 ], mathmode := "Text", name := "Attr", next := 155053, root := ~, start := 155000, stop := 155052 ), rec( attributes := rec( ), content := [ rec( content := "A polynom\ ial with integer coefficients.", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 155114, root := ~, start := 155056, stop := 155113 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the characteristic polynomial of the digraph\n ", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "A", next := 155221, root := ~, start := 155207, stop := 155220 ), rec( content := ". That i\ s it returns the characteristic polynomial\n of the adjacency matrix of t\ he digraph ", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "A", next := 155331, root := ~, start := 155317, stop := 155330 ), rec( content := "\n ", count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 2], [1, 3, 6, 8, 9, 10], [4, 6, 8],\n> [1, 2, 3\ , 9], [3, 3], [3, 5, 6, 10], [1, 2, 7],\n> [1, 2, 3, 10, 5, 6, 10], [1, 3, 4, \ 5, 8, 10],\n> [2, 3, 4, 6, 7, 10]]);\n\ngap> CharacteristicPolynomial(D);\nx_1^10-3*x_1^9-7*x_1^8-x_1^7+1\ 4*x_1^6+x_1^5-26*x_1^4+51*x_1^3-10*x_1^2\\\n+18*x_1-30\ngap> D := CompleteDigr\ aph(5);\n\ngap> CharacteristicPoly\ nomial(D);\nx_1^5-10*x_1^3-20*x_1^2-15*x_1-4\ngap> D := CycleDigraph(IsMutable\ Digraph, 3);\n\ngap> CharacteristicP\ olynomial(D);\nx_1^3-1\n", count := [ 5, 2, 2, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 4 ], mathmode := "Text", name := "Example", next := 155997, root := ~, start := 155336, stop := 155996 ), rec( content := "\n ", count := [ 5, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 156014, root := ~, start := 155117, stop := 156013 ) ], count := [ 5, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 156028, root := ~, start := 154985, stop := 156027 ), rec( content := "\n\n ", count := [ 5, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "BooleanAdjacenc\ yMatrix" ), content := 0, count := [ 5, 2, 3, 2 ], mathmode := "Text", name := "Attr", next := 156100, root := ~, start := 156049, stop := 156099 ), rec( attributes := rec( Arg := "digraph", Name := "BooleanAdjacenc\ yMatrixMutableCopy" ), content := 0, count := [ 5, 2, 3, 4 ], mathmode := "Text", name := "Oper", next := 156165, root := ~, start := 156103, stop := 156164 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix of booleans.", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "Returns", next := 156215, root := ~, start := 156168, stop := 156214 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 156253, root := ~, start := 156239, stop := 156252 ), rec( content := " is a di\ graph with a positive number of vertices\n ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 156314, root := ~, start := 156306, stop := 156313 ), rec( content := ", then " , count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ooleanAdjacencyMatrix(", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 156351, root := ~, start := 156321, stop := 156350 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 156365, root := ~, start := 156351, stop := 156364 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 156373, root := ~, start := 156365, stop := 156372 ), rec( content := "\n re\ turns the boolean adjacency matrix ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 156425, root := ~, start := 156415, stop := 156424 ), rec( content := " of ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 156443, root := ~, start := 156429, stop := 156442 ), rec( content := ". The\n\ value of the matrix entry ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[j][i]", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 156496, root := ~, start := 156480, stop := 156495 ), rec( content := " is ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "K", next := 156511, root := ~, start := 156500, stop := 156510 ), rec( content := " if and \ only if\n there exists an edge in ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 156569, root := ~, start := 156555, stop := 156568 ), rec( content := " with so\ urce ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 156590, root := ~, start := 156582, stop := 156589 ), rec( content := " and ran\ ge\n ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "C", next := 156613, root := ~, start := 156605, stop := 156612 ), rec( content := ". If ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "A", next := 156633, root := ~, start := 156619, stop := 156632 ), rec( content := " has no \ vertices, then the empty list is\n returned. ", count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 3, 6 ], mathmode := "Text", name := "P", next := 156692, root := ~, start := 156688, stop := 156691 ), rec( content := "\n\n \ Note that the boolean adjacency matrix loses information about multiple\n e\ dges. ", count := [ 5, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "P", next := 156786, root := ~, start := 156782, stop := 156785 ), rec( content := "\n\n \ The attribute ", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ooleanAdjacencyMatrix", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "C", next := 156835, root := ~, start := 156806, stop := 156834 ), rec( content := " returns\ an immutable list of\n immutable lists, whereas the function\n ", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ooleanAdjacencyMatrixMutableCopy", count := [ 5, 2, 3, 7 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "C", next := 156951, root := ~, start := 156911, stop := 156950 ), rec( content := " returns\ a copy of the\n ", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ooleanAdjacencyMatrix", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "C", next := 157007, root := ~, start := 156978, stop := 157006 ), rec( content := " that is\ a mutable list of mutable lists. ", count := [ 5, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 3, 8 ], mathmode := "Text", name := "P", next := 157053, root := ~, start := 157049, stop := 157052 ), rec( content := "\n ", count := [ 5, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[3, 4], [2, 3], [1, 2, 4], [4]]);\n\ngap> PrintArray(BooleanAdjacencyMatrix(gr));\n[ [ fal\ se, false, true, true ],\n [ false, true, true, false ],\n [ tr\ ue, true, false, true ],\n [ false, false, false, true ] ]\ngap> gr\ := CycleDigraph(4);;\ngap> PrintArray(BooleanAdjacencyMatrix(gr));\n[ [ fals\ e, true, false, false ],\n [ false, false, true, false ],\n [ fals\ e, false, false, true ],\n [ true, false, false, false ] ]\ngap> Boo\ leanAdjacencyMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDigraph(IsMutableD\ igraph, 3);\n\ngap> PrintArray(Boole\ anAdjacencyMatrix(D));\n[ [ false, true, false ],\n [ false, false, t\ rue ],\n [ true, false, false ] ]\n", count := [ 5, 2, 3, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 3, 9 ], mathmode := "Text", name := "Example", next := 157888, root := ~, start := 157058, stop := 157887 ), rec( content := "\n ", count := [ 5, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 3, 5 ], mathmode := "Text", name := "Description", next := 157905, root := ~, start := 156218, stop := 157904 ) ], count := [ 5, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 157919, root := ~, start := 156034, stop := 157918 ), rec( content := "\n\n ", count := [ 5, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphAdjacenc\ yFunction" ), content := 0, count := [ 5, 2, 4, 2 ], mathmode := "Text", name := "Attr", next := 157993, root := ~, start := 157940, stop := 157992 ), rec( attributes := rec( ), content := [ rec( content := "A functio\ n.", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 158026, root := ~, start := 157996, stop := 158025 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "A", next := 158064, root := ~, start := 158050, stop := 158063 ), rec( content := " is a di\ graph, then ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAdjacencyFunction", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "C", next := 158115, root := ~, start := 158084, stop := 158114 ), rec( content := " returns\ \n a function which takes two integer parameters ", count := [ 5, 2, 4, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ , y", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "C", next := 158185, root := ~, start := 158174, stop := 158184 ), rec( content := " and ret\ urns\n ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "K", next := 158213, root := ~, start := 158202, stop := 158212 ), rec( content := " if ther\ e exists an edge from vertex ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "C", next := 158258, root := ~, start := 158250, stop := 158257 ), rec( content := " to vert\ ex ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "y\ ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "C", next := 158277, root := ~, start := 158269, stop := 158276 ), rec( content := "\n in\ ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "A", next := 158299, root := ~, start := 158285, stop := 158298 ), rec( content := " and ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "K", next := 158316, root := ~, start := 158304, stop := 158315 ), rec( content := " if not.\ \n\n ", count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := Digraph([[1, 2], [3], []]);\n\ngap> foo := DigraphAdjacencyFunction(digraph);\nfunction( u, v )\ ... end\ngap> foo(1, 1);\ntrue\ngap> foo(1, 2);\ntrue\ngap> foo(1, 3);\nfalse\ \ngap> foo(3, 1);\nfalse\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> \ [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\", \"a\"]);\n\ \ngap> foo := DigraphAdjacenc\ yFunction(gr);\nfunction( u, v ) ... end\ngap> foo(1, 2);\ntrue\ngap> foo(3, 2\ );\nfalse\ngap> foo(3, 1);\nfalse\ngap> D := CycleDigraph(IsMutableDigraph, 3)\ ;\n\ngap> foo := DigraphAdjacencyFun\ ction(D);\nfunction( u, v ) ... end\ngap> foo(1, 2);\ntrue\ngap> foo(2, 1);\nf\ alse\n", count := [ 5, 2, 4, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 4 ], mathmode := "Text", name := "Example", next := 159097, root := ~, start := 158330, stop := 159096 ), rec( content := "\n ", count := [ 5, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 159114, root := ~, start := 158029, stop := 159113 ) ], count := [ 5, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 159128, root := ~, start := 157925, stop := 159127 ), rec( content := "\n\n ", count := [ 5, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphRange" ) , content := 0, count := [ 5, 2, 5, 2 ], mathmode := "Text", name := "Attr", next := 159190, root := ~, start := 159149, stop := 159189 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphSource" ), content := 0, count := [ 5, 2, 5, 4 ], mathmode := "Text", name := "Attr", next := 159235, root := ~, start := 159193, stop := 159234 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ positive integers.", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "Returns", next := 159285, root := ~, start := 159238, stop := 159284 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphRange", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 159325, root := ~, start := 159306, stop := 159324 ), rec( content := " and ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphSource", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 159350, root := ~, start := 159330, stop := 159349 ), rec( content := " return \ the range and source of\n the digraph ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "A", next := 159412, root := ~, start := 159398, stop := 159411 ), rec( content := ". More p\ recisely, position ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 159447, root := ~, start := 159439, stop := 159446 ), rec( content := " in\n \ ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ igraphRange(", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 5, 2, 5, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := \ "Text", name := "A", next := 159485, root := ~, start := 159471, stop := 15948\ 4 ), rec( content := ")", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 159490, root := ~, start := 159455, stop := 159489 ), rec( content := " is the \ range of the ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "C", next := 159519, root := ~, start := 159511, stop := 159518 ), rec( content := "th edge \ of\n ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "A", next := 159548, root := ~, start := 159534, stop := 159547 ), rec( content := ".\n\n \ ", count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\n> [2, 1, 3, 5], [1, 3, 4], [2, 3], [2], [1, 2, 3, 4]]);\ \n\ngap> DigraphRange(gr);\n[ 2, \ 1, 3, 5, 1, 3, 4, 2, 3, 2, 1, 2, 3, 4 ]\ngap> DigraphSource(gr);\n[ 1, 1, 1, 1\ , 2, 2, 2, 3, 3, 4, 5, 5, 5, 5 ]\ngap> DigraphEdges(gr);\n[ [ 1, 2 ], [ 1, 1 ]\ , [ 1, 3 ], [ 1, 5 ], [ 2, 1 ], [ 2, 3 ], \n [ 2, 4 ], [ 3, 2 ], [ 3, 3 ], [ \ 4, 2 ], [ 5, 1 ], [ 5, 2 ], \n [ 5, 3 ], [ 5, 4 ] ]\ngap> D := CycleDigraph(I\ sMutableDigraph, 3);\n\ngap> Digraph\ Range(D);\n[ 2, 3, 1 ]\ngap> DigraphSource(D);\n[ 1, 2, 3 ]\n", count := [ 5, 2, 5, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 5, 6 ], mathmode := "Text", name := "Example", next := 160176, root := ~, start := 159555, stop := 160175 ), rec( content := "\n ", count := [ 5, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 5, 5 ], mathmode := "Text", name := "Description", next := 160193, root := ~, start := 159288, stop := 160192 ) ], count := [ 5, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 160207, root := ~, start := 159134, stop := 160206 ), rec( content := "\n\n ", count := [ 5, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "OutNeighbours" ), content := 0, count := [ 5, 2, 6, 2 ], mathmode := "Text", name := "Attr", next := 160270, root := ~, start := 160228, stop := 160269 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighbors" ) , content := 0, count := [ 5, 2, 6, 4 ], mathmode := "Text", name := "Attr", next := 160314, root := ~, start := 160273, stop := 160313 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighboursMu\ tableCopy" ), content := 0, count := [ 5, 2, 6, 6 ], mathmode := "Text", name := "Oper", next := 160370, root := ~, start := 160317, stop := 160369 ), rec( attributes := rec( Arg := "digraph", Name := "OutNeighborsMut\ ableCopy" ), content := 0, count := [ 5, 2, 6, 8 ], mathmode := "Text", name := "Oper", next := 160425, root := ~, start := 160373, stop := 160424 ), rec( attributes := rec( ), content := [ rec( content := "The adjac\ encies of a digraph.", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "Returns", next := 160476, root := ~, start := 160428, stop := 160475 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the list ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 160538, root := ~, start := 160528, stop := 160537 ), rec( content := " of out-\ neighbours of each vertex\n of the digraph ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "A", next := 160605, root := ~, start := 160591, stop := 160604 ), rec( content := ".\n M\ ore specifically, a vertex ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 160647, root := ~, start := 160639, stop := 160646 ), rec( content := " appears\ in ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut[i]", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 160672, root := ~, start := 160659, stop := 160671 ), rec( content := " each ti\ me\n there exists an edge with source ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 160728, root := ~, start := 160720, stop := 160727 ), rec( content := " and ran\ ge ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "C", next := 160747, root := ~, start := 160739, stop := 160746 ), rec( content := " in\n \ ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "A", next := 160769, root := ~, start := 160755, stop := 160768 ), rec( content := ". ", count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "P", next := 160775, root := ~, start := 160771, stop := 160774 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "C", next := 160814, root := ~, start := 160794, stop := 160813 ), rec( content := " returns\ an immutable list of\n lists, whereas the function ", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighboursMutableCopy", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "C", next := 160907, root := ~, start := 160876, stop := 160906 ), rec( content := " returns\ a copy\n of ", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utNeighbours", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "C", next := 160950, root := ~, start := 160930, stop := 160949 ), rec( content := " which i\ s a mutable list of mutable lists. ", count := [ 5, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 6, 11 ], mathmode := "Text", name := "P", next := 160997, root := ~, start := 160993, stop := 160996 ), rec( content := "\n\n \ ", count := [ 5, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \ \"b\"],\n> [\"b\", \"a\", \"c\"]);\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ 1, 3 ], [ ] ]\ngap> \ gr := Digraph([[1, 2, 3], [2, 1], [3]]);\n\ngap> OutNeighbours(gr);\n[ [ 1, 2, 3 ], [ 2, 1 ], [ 3 ] ]\ngap> gr :\ = DigraphByAdjacencyMatrix([\n> [1, 2, 1],\n> [1, 1, 0],\n> [0, 0, 1]]);\n<\ immutable multidigraph with 3 vertices, 7 edges>\ngap> OutNeighbours(gr);\n[ [\ 1, 2, 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> OutNeighboursMutableCopy(gr);\n[ [ 1, 2\ , 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 \ ], [ 1 ] ]\n", count := [ 5, 2, 6, 12 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 12 ], mathmode := "Text", name := "Example", next := 161785, root := ~, start := 161003, stop := 161784 ), rec( content := "\n ", count := [ 5, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 6, 9 ], mathmode := "Text", name := "Description", next := 161802, root := ~, start := 160479, stop := 161801 ) ], count := [ 5, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 161816, root := ~, start := 160213, stop := 161815 ), rec( content := "\n\n ", count := [ 5, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "InNeighbours" ) , content := 0, count := [ 5, 2, 7, 2 ], mathmode := "Text", name := "Attr", next := 161878, root := ~, start := 161837, stop := 161877 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighbors" ), content := 0, count := [ 5, 2, 7, 4 ], mathmode := "Text", name := "Attr", next := 161921, root := ~, start := 161881, stop := 161920 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighboursMut\ ableCopy" ), content := 0, count := [ 5, 2, 7, 6 ], mathmode := "Text", name := "Oper", next := 161976, root := ~, start := 161924, stop := 161975 ), rec( attributes := rec( Arg := "digraph", Name := "InNeighborsMuta\ bleCopy" ), content := 0, count := [ 5, 2, 7, 8 ], mathmode := "Text", name := "Oper", next := 162030, root := ~, start := 161979, stop := 162029 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ lists of vertices.", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "Returns", next := 162080, root := ~, start := 162033, stop := 162079 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the list ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 162142, root := ~, start := 162132, stop := 162141 ), rec( content := " of in-n\ eighbours of each vertex\n of the digraph ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "A", next := 162208, root := ~, start := 162194, stop := 162207 ), rec( content := ".\n M\ ore specifically, a vertex ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 162250, root := ~, start := 162242, stop := 162249 ), rec( content := " appears\ in ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn[i]", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 162275, root := ~, start := 162262, stop := 162274 ), rec( content := " each ti\ me\n there exists an edge with source ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 162331, root := ~, start := 162323, stop := 162330 ), rec( content := " and ran\ ge ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "C", next := 162350, root := ~, start := 162342, stop := 162349 ), rec( content := " in\n \ ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "A", next := 162372, root := ~, start := 162358, stop := 162371 ), rec( content := ". ", count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "P", next := 162378, root := ~, start := 162374, stop := 162377 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nNeighbours", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "C", next := 162416, root := ~, start := 162397, stop := 162415 ), rec( content := " returns\ an immutable list of\n lists, whereas the function ", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nNeighboursMutableCopy", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "C", next := 162508, root := ~, start := 162478, stop := 162507 ), rec( content := " returns\ a copy\n of ", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nNeighbours", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "C", next := 162550, root := ~, start := 162531, stop := 162549 ), rec( content := " which i\ s a mutable list of mutable lists. ", count := [ 5, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "P", next := 162597, root := ~, start := 162593, stop := 162596 ), rec( content := "\n\n \ Note that each entry of ", count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "C", next := 162637, root := ~, start := 162627, stop := 162636 ), rec( content := " is sort\ ed into ascending order. ", count := [ 5, 2, 7, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 7, 12 ], mathmode := "Text", name := "P", next := 162674, root := ~, start := 162670, stop := 162673 ), rec( content := "\n\n \ ", count := [ 5, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \ \"b\"],\n> [\"b\", \"a\", \"c\"]);\n\ngap> InNeighbours(gr);\n[ [ 2 ], [ 1 ], [ 2 ] ]\ngap> gr \ := Digraph([[1, 2, 3], [2, 1], [3]]);\n\ngap> InNeighbours(gr);\n[ [ 1, 2 ], [ 1, 2 ], [ 1, 3 ] ]\ngap> gr := Di\ graphByAdjacencyMatrix([\n> [1, 2, 1],\n> [1, 1, 0],\n> [0, 0, 1]]);\n\ngap> InNeighbours(gr);\n[ [ 1, 2\ ], [ 1, 1, 2 ], [ 1, 3 ] ]\ngap> InNeighboursMutableCopy(gr);\n[ [ 1, 2 ], [ \ 1, 1, 2 ], [ 1, 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> InNeighbours(D);\n[ [ 3 ], [ 1 ], [ 2 \ ] ]\n", count := [ 5, 2, 7, 13 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 13 ], mathmode := "Text", name := "Example", next := 163455, root := ~, start := 162680, stop := 163454 ), rec( content := "\n ", count := [ 5, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 7, 9 ], mathmode := "Text", name := "Description", next := 163472, root := ~, start := 162083, stop := 163471 ) ], count := [ 5, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 163486, root := ~, start := 161822, stop := 163485 ), rec( content := "\n\n ", count := [ 5, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "OutDegrees" ), content := 0, count := [ 5, 2, 8, 2 ], mathmode := "Text", name := "Attr", next := 163546, root := ~, start := 163507, stop := 163545 ), rec( attributes := rec( Arg := "digraph", Name := "OutDegreeSequen\ ce" ), content := 0, count := [ 5, 2, 8, 4 ], mathmode := "Text", name := "Attr", next := 163595, root := ~, start := 163549, stop := 163594 ), rec( attributes := rec( Arg := "digraph", Name := "OutDegreeSet" ) , content := 0, count := [ 5, 2, 8, 6 ], mathmode := "Text", name := "Attr", next := 163639, root := ~, start := 163598, stop := 163638 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ non-negative integers.", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "Returns", next := 163693, root := ~, start := 163642, stop := 163692 ), rec( attributes := rec( ), content := [ rec( content := "\n\n G\ iven a digraph ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "A", next := 163745, root := ~, start := 163731, stop := 163744 ), rec( content := " with ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "M", next := 163759, root := ~, start := 163751, stop := 163758 ), rec( content := " vertice\ s, the function\n ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utDegrees", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 163804, root := ~, start := 163787, stop := 163803 ), rec( content := " returns\ an immutable list ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 163841, root := ~, start := 163831, stop := 163840 ), rec( content := " of leng\ th ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "M", next := 163860, root := ~, start := 163852, stop := 163859 ), rec( content := ", such t\ hat\n for a vertex ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 163897, root := ~, start := 163889, stop := 163896 ), rec( content := " in ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "A", next := 163915, root := ~, start := 163901, stop := 163914 ), rec( content := ", the va\ lue of ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut[i]", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 163943, root := ~, start := 163930, stop := 163942 ), rec( content := " is the\\ n out-degree of vertex ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "C", next := 163984, root := ~, start := 163976, stop := 163983 ), rec( content := ".\n S\ ee ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "Ref", next := 164025, root := ~, start := 163994, stop := 164024 ), rec( content := ". ", count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "P", next := 164031, root := ~, start := 164027, stop := 164030 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utDegreeSequence", count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "C", next := 164074, root := ~, start := 164050, stop := 164073 ), rec( content := " returns\ the same list,\n after it has been sorted into non-increasing order. ", count := [ 5, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "P", next := 164158, root := ~, start := 164154, stop := 164157 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ utDegreeSet", count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "C", next := 164196, root := ~, start := 164177, stop := 164195 ), rec( content := " returns\ the same list, sorted into\n increasing order with duplicate entries remov\ ed. ", count := [ 5, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 8, 10 ], mathmode := "Text", name := "P", next := 164289, root := ~, start := 164285, stop := 164288 ), rec( content := "\n\n \ ", count := [ 5, 2, 8, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3, 2, 2], [], [2, 1], []]);\n\ngap> OutDegrees(D);\n[ 4, 0, 2, 0 ]\ngap> OutDegreeS\ equence(D);\n[ 4, 2, 0, 0 ]\ngap> OutDegreeSet(D);\n[ 0, 2, 4 ]\ngap> D := Cyc\ leDigraph(IsMutableDigraph, 3);\n\ng\ ap> OutDegrees(D);\n[ 1, 1, 1 ]\n", count := [ 5, 2, 8, 11 ], name := "PCDATA" , root := ~ ) ], count := [ 5, 2, 8, 11 ], mathmode := "Text", name := "Example", next := 164659, root := ~, start := 164295, stop := 164658 ), rec( content := "\n ", count := [ 5, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 8, 7 ], mathmode := "Text", name := "Description", next := 164676, root := ~, start := 163696, stop := 164675 ) ], count := [ 5, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 164690, root := ~, start := 163492, stop := 164689 ), rec( content := "\n\n ", count := [ 5, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "InDegrees" ), content := 0, count := [ 5, 2, 9, 2 ], mathmode := "Text", name := "Attr", next := 164749, root := ~, start := 164711, stop := 164748 ), rec( attributes := rec( Arg := "digraph", Name := "InDegreeSequenc\ e" ), content := 0, count := [ 5, 2, 9, 4 ], mathmode := "Text", name := "Attr", next := 164797, root := ~, start := 164752, stop := 164796 ), rec( attributes := rec( Arg := "digraph", Name := "InDegreeSet" ), content := 0, count := [ 5, 2, 9, 6 ], mathmode := "Text", name := "Attr", next := 164840, root := ~, start := 164800, stop := 164839 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ non-negative integers.", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "Returns", next := 164894, root := ~, start := 164843, stop := 164893 ), rec( attributes := rec( ), content := [ rec( content := "\n\n G\ iven a digraph ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "A", next := 164946, root := ~, start := 164932, stop := 164945 ), rec( content := " with ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "M", next := 164960, root := ~, start := 164952, stop := 164959 ), rec( content := " vertice\ s, the function\n ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nDegrees", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 165004, root := ~, start := 164988, stop := 165003 ), rec( content := " returns\ an immutable list ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 165041, root := ~, start := 165031, stop := 165040 ), rec( content := " of leng\ th ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "M", next := 165060, root := ~, start := 165052, stop := 165059 ), rec( content := ", such t\ hat\n for a vertex ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 165097, root := ~, start := 165089, stop := 165096 ), rec( content := " in ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "A", next := 165115, root := ~, start := 165101, stop := 165114 ), rec( content := ", the va\ lue of ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn[i]", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 165143, root := ~, start := 165130, stop := 165142 ), rec( content := " is\n \ the in-degree of vertex ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "C", next := 165183, root := ~, start := 165175, stop := 165182 ), rec( content := ".\n S\ ee ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "InDegre\ eOfVertex" ), content := 0, count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "Ref", next := 165223, root := ~, start := 165193, stop := 165222 ), rec( content := ". ", count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "P", next := 165229, root := ~, start := 165225, stop := 165228 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nDegreeSequence", count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "C", next := 165271, root := ~, start := 165248, stop := 165270 ), rec( content := " returns\ the same list,\n after it has been sorted into non-increasing order. ", count := [ 5, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "P", next := 165355, root := ~, start := 165351, stop := 165354 ), rec( content := "\n\n \ The function ", count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nDegreeSet", count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "C", next := 165392, root := ~, start := 165374, stop := 165391 ), rec( content := " returns\ the same list, sorted into\n increasing order with duplicate entries remov\ ed. ", count := [ 5, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 9, 10 ], mathmode := "Text", name := "P", next := 165485, root := ~, start := 165481, stop := 165484 ), rec( content := "\n\n \ ", count := [ 5, 2, 9, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3, 2, 2, 4], [], [2, 1, 4], []]);\n\ngap> InDegrees(D);\n[ 2, 3, 1, 2 ]\ngap> InDeg\ reeSequence(D);\n[ 3, 2, 2, 1 ]\ngap> InDegreeSet(D);\n[ 1, 2, 3 ]\ngap> D := \ CycleDigraph(IsMutableDigraph, 3);\n\ \ngap> InDegrees(D);\n[ 1, 1, 1 ]\n", count := [ 5, 2, 9, 11 ], name := "PCDAT\ A", root := ~ ) ], count := [ 5, 2, 9, 11 ], mathmode := "Text", name := "Example", next := 165857, root := ~, start := 165491, stop := 165856 ), rec( content := "\n ", count := [ 5, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 9, 7 ], mathmode := "Text", name := "Description", next := 165874, root := ~, start := 164897, stop := 165873 ) ], count := [ 5, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 165888, root := ~, start := 164696, stop := 165887 ), rec( content := "\n\n ", count := [ 5, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "OutDegreeOfVert\ ex" ), content := 0, count := [ 5, 2, 10, 2 ], mathmode := "Text", name := "Oper", next := 165963, root := ~, start := 165909, stop := 165962 ), rec( attributes := rec( ), content := [ rec( content := "The non-n\ egative integer.", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "Returns", next := 166010, root := ~, start := 165966, stop := 166009 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns the out-degree of the vertex ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 166096, root := ~, start := 166083, stop := 166095 ), rec( content := " in the\\ n digraph ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 166130, root := ~, start := 166116, stop := 166129 ), rec( content := ".\n T\ he out-degree of ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 166167, root := ~, start := 166154, stop := 166166 ), rec( content := " is the \ number of edges in ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 166208, root := ~, start := 166194, stop := 166207 ), rec( content := "\n wh\ ose source is ", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "A", next := 166242, root := ~, start := 166229, stop := 166241 ), rec( content := ".\n " , count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 10, 4 ], mathmode := "Text", name := "P", next := 166252, root := ~, start := 166248, stop := 166251 ), rec( content := "\n ", count := [ 5, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 1], [1, 4], [2, 2, 4, 2], [1, 1, 2, 2, 1, 2, 2]\ ]);\n\ngap> OutDegreeOfVerte\ x(D, 1);\n3\ngap> OutDegreeOfVertex(D, 2);\n2\ngap> OutDegreeOfVertex(D, 3);\n\ 4\ngap> OutDegreeOfVertex(D, 4);\n7\n", count := [ 5, 2, 10, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 10, 5 ], mathmode := "Text", name := "Example", next := 166548, root := ~, start := 166257, stop := 166547 ), rec( content := "\n ", count := [ 5, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 166565, root := ~, start := 166013, stop := 166564 ) ], count := [ 5, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 166579, root := ~, start := 165894, stop := 166578 ), rec( content := "\n\n ", count := [ 5, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "OutNeighboursOf\ Vertex" ), content := 0, count := [ 5, 2, 11, 2 ], mathmode := "Text", name := "Oper", next := 166658, root := ~, start := 166600, stop := 166657 ), rec( attributes := rec( Arg := "digraph, vertex" , Name := "OutNeighborsOfV\ ertex" ), content := 0, count := [ 5, 2, 11, 4 ], mathmode := "Text", name := "Oper", next := 166718, root := ~, start := 166661, stop := 166717 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "Returns", next := 166759, root := ~, start := 166721, stop := 166758 ), rec( attributes := rec( ), content := [ rec( content := "\n\n T\ his operation returns the list ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 166823, root := ~, start := 166813, stop := 166822 ), rec( content := " of vert\ ices of the digraph\n ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "A", next := 166869, root := ~, start := 166855, stop := 166868 ), rec( content := ".\n A\ vertex ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 166892, root := ~, start := 166884, stop := 166891 ), rec( content := " appears\ in the list ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 166923, root := ~, start := 166913, stop := 166922 ), rec( content := " each ti\ me there exists an\n edge with source ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "A", next := 166985, root := ~, start := 166972, stop := 166984 ), rec( content := " and ran\ ge ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 167004, root := ~, start := 166996, stop := 167003 ), rec( content := " in ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "A", next := 167022, root := ~, start := 167008, stop := 167021 ), rec( content := "; in\n \ particular, this means that ", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "C", next := 167069, root := ~, start := 167059, stop := 167068 ), rec( content := " may con\ tain duplicates.", count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 11, 6 ], mathmode := "Text", name := "P", next := 167097, root := ~, start := 167093, stop := 167096 ), rec( content := "\n ", count := [ 5, 2, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 3], [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2]\ ]);\n\ngap> OutNeighboursOfV\ ertex(D, 1);\n[ 2, 2, 3 ]\ngap> OutNeighboursOfVertex(D, 3);\n[ 2, 2, 3 ]\n", count := [ 5, 2, 11, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 2, 11, 7 ], mathmode := "Text", name := "Example", next := 167357, root := ~, start := 167102, stop := 167356 ), rec( content := "\n ", count := [ 5, 2, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 11, 5 ], mathmode := "Text", name := "Description", next := 167374, root := ~, start := 166762, stop := 167373 ) ], count := [ 5, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 167388, root := ~, start := 166585, stop := 167387 ), rec( content := "\n\n ", count := [ 5, 2, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "InDegreeOfVerte\ x" ), content := 0, count := [ 5, 2, 12, 2 ], mathmode := "Text", name := "Oper", next := 167462, root := ~, start := 167409, stop := 167461 ), rec( attributes := rec( ), content := [ rec( content := "A non-neg\ ative integer.", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "Returns", next := 167507, root := ~, start := 167465, stop := 167506 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns the in-degree of the vertex ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 167592, root := ~, start := 167579, stop := 167591 ), rec( content := " in the\\ n digraph ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 167626, root := ~, start := 167612, stop := 167625 ), rec( content := ".\n T\ he in-degree of ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 167662, root := ~, start := 167649, stop := 167661 ), rec( content := " is the \ number of edges in ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 167703, root := ~, start := 167689, stop := 167702 ), rec( content := "\n wh\ ose range is ", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "A", next := 167736, root := ~, start := 167723, stop := 167735 ), rec( content := ".\n " , count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 1], [1, 4], [2, 2, 4, 2], [1, 1, 2, 2, 1, 2, 2]\ ]);\n\ngap> InDegreeOfVertex\ (D, 1);\n5\ngap> InDegreeOfVertex(D, 2);\n9\ngap> InDegreeOfVertex(D, 3);\n0\n\ gap> InDegreeOfVertex(D, 4);\n2\n", count := [ 5, 2, 12, 4 ], name := "PCDATA" , root := ~ ) ], count := [ 5, 2, 12, 4 ], mathmode := "Text", name := "Example", next := 168029, root := ~, start := 167742, stop := 168028 ), rec( content := "\n ", count := [ 5, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 12, 3 ], mathmode := "Text", name := "Description", next := 168046, root := ~, start := 167510, stop := 168045 ) ], count := [ 5, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 168060, root := ~, start := 167394, stop := 168059 ), rec( content := "\n\n ", count := [ 5, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "InNeighboursOfV\ ertex" ), content := 0, count := [ 5, 2, 13, 2 ], mathmode := "Text", name := "Oper", next := 168138, root := ~, start := 168081, stop := 168137 ), rec( attributes := rec( Arg := "digraph, vertex" , Name := "InNeighborsOfVe\ rtex" ), content := 0, count := [ 5, 2, 13, 4 ], mathmode := "Text", name := "Oper", next := 168197, root := ~, start := 168141, stop := 168196 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ postitive vertices.", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "Returns", next := 168248, root := ~, start := 168200, stop := 168247 ), rec( attributes := rec( ), content := [ rec( content := "\n\n T\ his operation returns the list ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 168312, root := ~, start := 168302, stop := 168311 ), rec( content := " of vert\ ices of the digraph\n ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "A", next := 168358, root := ~, start := 168344, stop := 168357 ), rec( content := ".\n A\ vertex ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 168381, root := ~, start := 168373, stop := 168380 ), rec( content := " appears\ in the list ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 168412, root := ~, start := 168402, stop := 168411 ), rec( content := " each ti\ me there exists an\n edge with source ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 168469, root := ~, start := 168461, stop := 168468 ), rec( content := " and ran\ ge ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "A", next := 168493, root := ~, start := 168480, stop := 168492 ), rec( content := " in ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "A", next := 168511, root := ~, start := 168497, stop := 168510 ), rec( content := "; in\n \ particular, this means that ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nn", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "C", next := 168558, root := ~, start := 168548, stop := 168557 ), rec( content := " may con\ tain duplicates. ", count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 13, 6 ], mathmode := "Text", name := "P", next := 168587, root := ~, start := 168583, stop := 168586 ), rec( content := "\n\n \ ", count := [ 5, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 2, 3], [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2]\ ]);\n\ngap> InNeighboursOfVe\ rtex(D, 1);\n[ 2, 4, 4, 4 ]\ngap> InNeighboursOfVertex(D, 2);\n[ 1, 1, 3, 3, 4\ , 4, 4, 4 ]\n", count := [ 5, 2, 13, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 7 ], mathmode := "Text", name := "Example", next := 168864, root := ~, start := 168593, stop := 168863 ), rec( content := "\n ", count := [ 5, 2, 13, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 13, 5 ], mathmode := "Text", name := "Description", next := 168881, root := ~, start := 168251, stop := 168880 ) ], count := [ 5, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 168895, root := ~, start := 168066, stop := 168894 ), rec( content := "\n\n ", count := [ 5, 2, 13, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphLoops" ) , content := 0, count := [ 5, 2, 14, 2 ], mathmode := "Text", name := "Attr", next := 168957, root := ~, start := 168916, stop := 168956 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Returns", next := 168998, root := ~, start := 168960, stop := 168997 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "A", next := 169036, root := ~, start := 169022, stop := 169035 ), rec( content := " is a di\ graph, then ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphLoops", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "C", next := 169075, root := ~, start := 169056, stop := 169074 ), rec( content := " returns\ the list\n consisting of the ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Vertices" ), content := 0, count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Ref", next := 169144, root := ~, start := 169115, stop := 169143 ), rec( content := " of ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "A", next := 169162, root := ~, start := 169148, stop := 169161 ), rec( content := "\n at\ which there is a loop. See ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Ref", next := 169226, root := ~, start := 169197, stop := 169225 ), rec( content := ". ", count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 14, 4 ], mathmode := "Text", name := "P", next := 169232, root := ~, start := 169228, stop := 169231 ), rec( content := "\n\n \ ", count := [ 5, 2, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], []]);\n\ngap> DigraphHasLoops(D);\nfalse\ngap> DigraphLoops(D);\n[ ]\ngap> D := D\ igraph([[3, 5], [1], [2, 4, 3], [4], [2, 1]]);\n\ngap> DigraphLoops(D);\n[ 3, 4 ]\ngap> D := Digraph(IsMutableDi\ graph, [[1], [1]]);\n\ngap> DigraphL\ oops(D);\n[ 1 ]\n", count := [ 5, 2, 14, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 5 ], mathmode := "Text", name := "Example", next := 169662, root := ~, start := 169238, stop := 169661 ), rec( content := "\n ", count := [ 5, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 14, 3 ], mathmode := "Text", name := "Description", next := 169679, root := ~, start := 169001, stop := 169678 ) ], count := [ 5, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 169693, root := ~, start := 168901, stop := 169692 ), rec( content := "\n\n ", count := [ 5, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "PartialOrderDig\ raphMeetOfVertices" ), content := 0, count := [ 5, 2, 15, 2 ], mathmode := "Text", name := "Oper", next := 169786, root := ~, start := 169714, stop := 169785 ), rec( attributes := rec( Arg := "digraph, u, v", Name := "PartialOrderDig\ raphJoinOfVertices" ), content := 0, count := [ 5, 2, 15, 4 ], mathmode := "Text", name := "Oper", next := 169861, root := ~, start := 169789, stop := 169860 ), rec( attributes := rec( ), content := [ rec( content := "A positiv\ e integer or ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "K", next := 169906, root := ~, start := 169895, stop := 169905 ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Returns", next := 169916, root := ~, start := 169864, stop := 169915 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ the first argument is a partial order digraph ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsParti\ alOrderDigraph" ), content := 0, count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Ref", next := 170027, root := ~, start := 169988, stop := 170026 ), rec( content := " then th\ ese operations return the meet, or\n the join, of the two input vertices. I\ f the meet (or join) is does not\n exist then ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "K", next := 170171, root := ~, start := 170160, stop := 170170 ), rec( content := " is retu\ rned. The meet (or join) is guaranteed to\n exist when the first argument s\ atisfies\n ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMeetS\ emilatticeDigraph" ), content := 0, count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Ref", next := 170307, root := ~, start := 170269, stop := 170306 ), rec( content := " (or\n \ ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsJoinS\ emilatticeDigraph" ), content := 0, count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Ref", next := 170355, root := ~, start := 170317, stop := 170354 ), rec( content := ") - see \ the documentation for these\n properties for the definition of the meet (or\ the join).\n\n ", count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1], [1, 2], [1, 3], [1, 2, 3, 4]]);\n\ngap> PartialOrderDigraphMeetOfVertices(D, 2, 3);\n4\ \ngap> PartialOrderDigraphJoinOfVertices(D, 2, 3);\n1\ngap> PartialOrderDigrap\ hMeetOfVertices(D, 1, 2);\n2\ngap> PartialOrderDigraphJoinOfVertices(D, 3, 4);\ \n3\ngap> D := Digraph([[1], [2], [1, 2, 3], [1, 2, 4]]);\n\ngap> PartialOrderDigraphMeetOfVertices(D, 3, 4);\nf\ ail\ngap> PartialOrderDigraphJoinOfVertices(D, 3, 4);\nfail\ngap> PartialOrder\ DigraphMeetOfVertices(D, 1, 2);\nfail\ngap> PartialOrderDigraphJoinOfVertices(\ D, 1, 2);\nfail\n", count := [ 5, 2, 15, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 6 ], mathmode := "Text", name := "Example", next := 171108, root := ~, start := 170457, stop := 171107 ), rec( content := "\n ", count := [ 5, 2, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 15, 5 ], mathmode := "Text", name := "Description", next := 171125, root := ~, start := 169921, stop := 171124 ) ], count := [ 5, 2, 15, 1 ], mathmode := "Text", name := "ManSection", next := 171139, root := ~, start := 169699, stop := 171138 ), rec( content := "\n\n ", count := [ 5, 2, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DegreeMatrix" ) , content := 0, count := [ 5, 2, 16, 2 ], mathmode := "Text", name := "Attr", next := 171201, root := ~, start := 171160, stop := 171200 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix of non-negative integers.", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "Returns", next := 171264, root := ~, start := 171204, stop := 171263 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the outdegree matrix ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "C", next := 171338, root := ~, start := 171328, stop := 171337 ), rec( content := " of the \ digraph\n ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "A", next := 171372, root := ~, start := 171358, stop := 171371 ), rec( content := ". The va\ lue of the ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "C", next := 171399, root := ~, start := 171391, stop := 171398 ), rec( content := "th diago\ nal matrix entry is the\n outdegree of the vertex ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "C", next := 171467, root := ~, start := 171459, stop := 171466 ), rec( content := " in ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "A", next := 171485, root := ~, start := 171471, stop := 171484 ), rec( content := ". All of\ f-diagonal entries \n are ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "C", next := 171529, root := ~, start := 171521, stop := 171528 ), rec( content := ".\n I\ f ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "A", next := 171552, root := ~, start := 171538, stop := 171551 ), rec( content := " has no \ vertices, then the empty list is returned. ", count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 16, 4 ], mathmode := "Text", name := "P", next := 171607, root := ~, start := 171603, stop := 171606 ), rec( content := "\n\n \ See ", count := [ 5, 2, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutDegr\ ees" ), content := 0, count := [ 5, 2, 16, 4 ], mathmode := "Text", name := "Ref", next := 171641, root := ~, start := 171617, stop := 171640 ), rec( content := " for mor\ e information.\n ", count := [ 5, 2, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2, 3], [4], [1, 3, 4], []]);\n\ngap> PrintArray(DegreeMatrix(D));\n[ [ 3, 0, 0, 0 ],\ \n [ 0, 1, 0, 0 ],\n [ 0, 0, 3, 0 ],\n [ 0, 0, 0, 0 ] ]\ngap> D\ := CycleDigraph(5);;\ngap> PrintArray(DegreeMatrix(D));\n[ [ 1, 0, 0, 0, \ 0 ],\n [ 0, 1, 0, 0, 0 ],\n [ 0, 0, 1, 0, 0 ],\n [ 0, 0, 0, \ 1, 0 ],\n [ 0, 0, 0, 0, 1 ] ]\ngap> DegreeMatrix(EmptyDigraph(0));\n[ \ ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> Display(DegreeMatrix(D));\n[ [ 1, 0, 0 ],\n [ 0, 1, \ 0 ],\n [ 0, 0, 1 ] ]\n", count := [ 5, 2, 16, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 5 ], mathmode := "Text", name := "Example", next := 172328, root := ~, start := 171668, stop := 172327 ), rec( content := "\n ", count := [ 5, 2, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 16, 3 ], mathmode := "Text", name := "Description", next := 172345, root := ~, start := 171267, stop := 172344 ) ], count := [ 5, 2, 16, 1 ], mathmode := "Text", name := "ManSection", next := 172359, root := ~, start := 171145, stop := 172358 ), rec( content := "\n\n ", count := [ 5, 2, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "LaplacianMatrix\ " ), content := 0, count := [ 5, 2, 17, 2 ], mathmode := "Text", name := "Attr", next := 172424, root := ~, start := 172380, stop := 172423 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix of integers.", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "Returns", next := 172474, root := ~, start := 172427, stop := 172473 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the outdegree Laplacian matrix ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "C", next := 172558, root := ~, start := 172548, stop := 172557 ), rec( content := " of the\\ n digraph ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "A", next := 172592, root := ~, start := 172578, stop := 172591 ), rec( content := ". The ou\ tdegree Laplacian matrix is defined as\n ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ egreeMatrix(digraph) - AdjacencyMatrix(digraph)", count := [ 5, 2, 17, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "C", next := 172698, root := ~, start := 172643, stop := 172697 ), rec( content := ". If\n \ ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "A", next := 172722, root := ~, start := 172708, stop := 172721 ), rec( content := " has no \ vertices, then the empty list is returned. ", count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "P", next := 172777, root := ~, start := 172773, stop := 172776 ), rec( content := "\n\n \ See ", count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DegreeM\ atrix" ), content := 0, count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "Ref", next := 172813, root := ~, start := 172787, stop := 172812 ), rec( content := " and ", count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Adjacen\ cyMatrix" ), content := 0, count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "Ref", next := 172847, root := ~, start := 172818, stop := 172846 ), rec( content := " for mor\ e\n information.\n ", count := [ 5, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[1, 2, 3], [4], [1, 3, 4], []]);\n\ngap> PrintArray(LaplacianMatrix(gr));\n[ [ 2, -1, -\ 1, 0 ],\n [ 0, 1, 0, -1 ],\n [ -1, 0, 2, -1 ],\n [ 0, 0,\ 0, 0 ] ]\ngap> LaplacianMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDi\ graph(IsMutableDigraph, 3);\n\ngap> \ Display(LaplacianMatrix(D));\n[ [ 1, -1, 0 ],\n [ 0, 1, -1 ],\n [ \ -1, 0, 1 ] ]\n", count := [ 5, 2, 17, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 5 ], mathmode := "Text", name := "Example", next := 173381, root := ~, start := 172878, stop := 173380 ), rec( content := "\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 17, 3 ], mathmode := "Text", name := "Description", next := 173398, root := ~, start := 172477, stop := 173397 ) ], count := [ 5, 2, 17, 1 ], mathmode := "Text", name := "ManSection", next := 173412, root := ~, start := 172365, stop := 173411 ), rec( content := "\n\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 173426, root := ~, start := 153194, stop := 173425 ) , rec( content := "\n\n ", count := [ 5, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Reachability and \ connectivity", count := [ 5, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 173487, root := ~, start := 173439, stop := 173486 ), rec( content := "\n ", count := [ 5, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDiameter\ " ), content := 0, count := [ 5, 3, 1, 2 ], mathmode := "Text", name := "Attr", next := 173551, root := ~, start := 173507, stop := 173550 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r or ", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "C", next := 173588, root := ~, start := 173577, stop := 173587 ), rec( content := ".", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 173599, root := ~, start := 173554, stop := 173598 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the diameter of the digraph ", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "A", next := 173684, root := ~, start := 173670, stop := 173683 ), rec( content := ".\n " , count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "P", next := 173694, root := ~, start := 173690, stop := 173693 ), rec( content := "\n\n \ If a digraph ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "A", next := 173727, root := ~, start := 173713, stop := 173726 ), rec( content := " is stro\ ngly connected and has at least 1\n vertex, then the ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ iameter", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "E", next := 173805, root := ~, start := 173790, stop := 173804 ), rec( content := " is the \ maximum shortest distance between\n any pair of distinct vertices. Otherwis\ e then the diameter of\n ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "A", next := 173931, root := ~, start := 173917, stop := 173930 ), rec( content := " is unde\ fined, and this function returns the value\n ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "C", next := 173997, root := ~, start := 173986, stop := 173996 ), rec( content := ". ", count := [ 5, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 1, 5 ], mathmode := "Text", name := "P", next := 174003, root := ~, start := 173999, stop := 174002 ), rec( content := "\n\n \ See ", count := [ 5, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ShortestDistances" ), content := 0, count := [ 5, 3, 1, 5 ], mathmode := "Text", name := "Ref", next := 174051, root := ~, start := 174013, stop := 174050 ), rec( content := ". ", count := [ 5, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 1, 6 ], mathmode := "Text", name := "P", next := 174057, root := ~, start := 174053, stop := 174056 ), rec( content := "\n ", count := [ 5, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [4, 5], [5], [1, 2, 3, 4, 5]]);\n\ngap> DigraphDiameter(D);\n3\ngap> D := ChainDi\ graph(2);\n\ngap> DigraphDiameter(D);\ \nfail\ngap> IsStronglyConnectedDigraph(D);\nfalse\ngap> D := GeneralisedPeter\ senGraph(IsMutableDigraph, 6, 2);\n\ngap> DigraphDiameter(D);\n4\n", count := [ 5, 3, 1, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 7 ], mathmode := "Text", name := "Example", next := 174501, root := ~, start := 174062, stop := 174500 ), rec( content := "\n ", count := [ 5, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 174518, root := ~, start := 173602, stop := 174517 ) ], count := [ 5, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 174532, root := ~, start := 173492, stop := 174531 ), rec( content := "\n\n ", count := [ 5, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Label := "for a digraph \ and two vertices", Name := "DigraphShortestDistance" ), content := 0, count := [ 5, 3, 2, 2 ], mathmode := "Text", name := "Oper", next := 174658, root := ~, start := 174553, stop := 174657 ), rec( attributes := rec( Arg := "digraph, list", Label := "for a digraph \ and a list", Name := "DigraphShortestDistance" ), content := 0, count := [ 5, 3, 2, 4 ], mathmode := "Text", name := "Oper", next := 174760, root := ~, start := 174661, stop := 174759 ), rec( attributes := rec( Arg := "digraph, list1, \ list2", Label := "for a digraph, a list, and a list", Name := "DigraphShortest\ Distance" ), content := 0, count := [ 5, 3, 2, 6 ], mathmode := "Text", name := "Oper", next := 174879, root := ~, start := 174763, stop := 174878 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r or ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "K", next := 174916, root := ~, start := 174905, stop := 174915 ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Returns", next := 174926, root := ~, start := 174882, stop := 174925 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ there is a directed path in the digraph ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "A", next := 175004, root := ~, start := 174990, stop := 175003 ), rec( content := " between\ vertex\n ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "A", next := 175032, root := ~, start := 175024, stop := 175031 ), rec( content := " and ver\ tex ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "A", next := 175052, root := ~, start := 175044, stop := 175051 ), rec( content := ", then t\ his operation returns the length of the\n shortest such directed path. If \ no such directed path exists, then this\n operation returns ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "K", next := 175210, root := ~, start := 175199, stop := 175209 ), rec( content := ". See Se\ ction ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Ref", next := 175274, root := ~, start := 175224, stop := 175273 ), rec( content := " for the\ definition of a directed path. ", count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "P", next := 175318, root := ~, start := 175314, stop := 175317 ), rec( content := "\n\n \ If the second form is used, then ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "A", next := 175368, root := ~, start := 175357, stop := 175367 ), rec( content := " should \ be a list of length two,\n containing two positive integers which correspon\ d to the vertices ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "A", next := 175479, root := ~, start := 175471, stop := 175478 ), rec( content := "\n an\ d ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "A", next := 175496, root := ~, start := 175488, stop := 175495 ), rec( content := ". ", count := [ 5, 3, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 2, 9 ], mathmode := "Text", name := "P", next := 175502, root := ~, start := 175498, stop := 175501 ), rec( content := "\n\n \ Note that as usual, a vertex is considered to be at distance 0 from\n itsel\ f .", count := [ 5, 3, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "P", next := 175592, root := ~, start := 175588, stop := 175591 ), rec( content := "\n\n \ If the third form is used, then ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist1", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175642, root := ~, start := 175630, stop := 175641 ), rec( content := " and ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist2", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175659, root := ~, start := 175647, stop := 175658 ), rec( content := " are bot\ h lists\n of vertices. The shortest directed path between ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist1", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175739, root := ~, start := 175727, stop := 175738 ), rec( content := " and\n \ ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist2", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175760, root := ~, start := 175748, stop := 175759 ), rec( content := " is then\ the length of the shortest directed path which starts\n with a vertex in " , count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist1", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175856, root := ~, start := 175844, stop := 175855 ), rec( content := " and ter\ minates at a vertex in ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist2", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175899, root := ~, start := 175887, stop := 175898 ), rec( content := ", if\n \ such directed path exists. If ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist1", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175950, root := ~, start := 175938, stop := 175949 ), rec( content := " and ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist2", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "A", next := 175967, root := ~, start := 175955, stop := 175966 ), rec( content := " have no\ n-empty\n intersection, the operation returns ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "C", next := 176031, root := ~, start := 176023, stop := 176030 ), rec( content := ".", count := [ 5, 3, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 2, 11 ], mathmode := "Text", name := "P", next := 176036, root := ~, start := 176032, stop := 176035 ), rec( content := "\n ", count := [ 5, 3, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [1, 4], [1, 3], [5]]);\n\ngap> DigraphShortestDistance(D, 1, 3);\n2\ngap> DigraphS\ hortestDistance(D, [3, 3]);\n0\ngap> DigraphShortestDistance(D, 5, 2);\nfail\n\ gap> DigraphShortestDistance(D, [1, 2], [4, 5]);\n2\ngap> DigraphShortestDista\ nce(D, [1, 3], [3, 5]);\n0\n", count := [ 5, 3, 2, 12 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 12 ], mathmode := "Text", name := "Example", next := 176400, root := ~, start := 176041, stop := 176399 ), rec( content := "\n ", count := [ 5, 3, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2, 7 ], mathmode := "Text", name := "Description", next := 176417, root := ~, start := 174929, stop := 176416 ) ], count := [ 5, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 176431, root := ~, start := 174538, stop := 176430 ), rec( content := "\n\n ", count := [ 5, 3, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphShortest\ Distances" ), content := 0, count := [ 5, 3, 3, 2 ], mathmode := "Text", name := "Attr", next := 176505, root := ~, start := 176452, stop := 176504 ), rec( attributes := rec( ), content := [ rec( content := "A square \ matrix.", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "Returns", next := 176543, root := ~, start := 176508, stop := 176542 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "A", next := 176581, root := ~, start := 176567, stop := 176580 ), rec( content := " is a di\ graph with ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "M", next := 176608, root := ~, start := 176600, stop := 176607 ), rec( content := " vertice\ s, then this\n function returns an ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ \\times n", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "M", next := 176670, root := ~, start := 176653, stop := 176669 ), rec( content := " matrix \ ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "C", next := 176688, root := ~, start := 176678, stop := 176687 ), rec( content := ", where \ each entry is\n either a non-negative integer, or ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "K", next := 176759, root := ~, start := 176748, stop := 176758 ), rec( content := ". If ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ = 0", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "M", next := 176777, root := ~, start := 176765, stop := 176776 ), rec( content := ", then a\ n\n empty list is returned. ", count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "P", next := 176819, root := ~, start := 176815, stop := 176818 ), rec( content := "\n\n \ If there is a directed path from vertex ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 176873, root := ~, start := 176865, stop := 176872 ), rec( content := " to vert\ ex ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 176892, root := ~, start := 176884, stop := 176891 ), rec( content := ", then\n\ the value of ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 176932, root := ~, start := 176916, stop := 176931 ), rec( content := " is the \ length of the shortest such directed\n path. If no such directed path exist\ s, then the value of ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 177054, root := ~, start := 177038, stop := 177053 ), rec( content := " is\n \ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 177073, root := ~, start := 177062, stop := 177072 ), rec( content := ". We us\ e the convention that the distance from every vertex to\n itself is ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 177159, root := ~, start := 177151, stop := 177158 ), rec( content := ", i.e. " , count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][i] = 0", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 177186, root := ~, start := 177166, stop := 177185 ), rec( content := " for all\ vertices ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "C", next := 177212, root := ~, start := 177204, stop := 177211 ), rec( content := ".\n " , count := [ 5, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "P", next := 177222, root := ~, start := 177218, stop := 177221 ), rec( content := "\n\n \ The method used in this function is a version of the Floyd-Warshall\n algor\ ithm, and has complexity ", count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (n^3)", count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "M", next := 177343, root := ~, start := 177330, stop := 177342 ), rec( content := ".\n\n \ ", count := [ 5, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [3], [1, 2], [4]]);\n\ngap> mat := DigraphShortestDistances(D);;\ngap> PrintArray(ma\ t);\n[ [ 0, 1, 2, fail ],\n [ 2, 0, 1, fail ],\n [\ 1, 1, 0, fail ],\n [ fail, fail, fail, 0 ] ]\ngap> D := \ CycleDigraph(IsMutableDigraph, 3);\n\ \ngap> DigraphShortestDistances(D);\n[ [ 0, 1, 2 ], [ 2, 0, 1 ], [ 1, 2, 0 ] ]\ \n", count := [ 5, 3, 3, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 6 ], mathmode := "Text", name := "Example", next := 177839, root := ~, start := 177350, stop := 177838 ), rec( content := "\n ", count := [ 5, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 177856, root := ~, start := 176546, stop := 177855 ) ], count := [ 5, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 177870, root := ~, start := 176437, stop := 177869 ), rec( content := "\n\n ", count := [ 5, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, v", Name := "DigraphLongestD\ istanceFromVertex" ), content := 0, count := [ 5, 3, 4, 2 ], mathmode := "Text", name := "Oper", next := 177955, root := ~, start := 177891, stop := 177954 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r, or ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "K", next := 177997, root := ~, start := 177982, stop := 177996 ), rec( content := ".", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 178008, root := ~, start := 177958, stop := 178007 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 178046, root := ~, start := 178032, stop := 178045 ), rec( content := " is a di\ graph and ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 178072, root := ~, start := 178064, stop := 178071 ), rec( content := " is a ve\ rtex in ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 178102, root := ~, start := 178088, stop := 178101 ), rec( content := ",\n t\ hen this operation returns the length of the longest directed walk in\n ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 178197, root := ~, start := 178183, stop := 178196 ), rec( content := " which b\ egins at vertex ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "A", next := 178229, root := ~, start := 178221, stop := 178228 ), rec( content := ". See S\ ection ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 178294, root := ~, start := 178244, stop := 178293 ), rec( content := " for the\ definitions of directed\n walk, directed cycle, and loop.\n ", count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 4, 4 ], mathmode := "Text", name := "P", next := 178371, root := ~, start := 178367, stop := 178370 ), rec( content := "\n\n \ ", count := [ 5, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n If there exists a directed w\ alk starting at vertex ", count := [ 5, 3, 4, 6 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "v", count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ) ], count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "A", next := 1784\ 64, root := ~, start := 178456, stop := 178463 ), rec( content := "\n w\ hich traverses a loop or a directed cycle,\n then we consider there to \ be a walk of infinite length from ", count := [ 5, 3, 4, 6 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "v", count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "A", next\ := 178593, root := ~, start := 178585, stop := 178592 ), rec( content := "\n \ (realised by repeatedly traversing the loop/directed cycle),\n a\ nd so the result is ", count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "infinity", count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 5, 3, 4, 6 ], mathmode := "Text", name := "K", next := \ 178707, root := ~, start := 178692, stop := 178706 ), rec( content := ".\n \ To disallow walks using loops, try using\n ", count := [ 5, 3, 4, 6\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ Oper := "DigraphRemoveLoops" ), content := 0, count := [ 5, 3, 4, 6 ], mathmo\ de := "Text", name := "Ref", next := 178798, root := ~, start := 178766, stop \ := 178797 ), rec( content := ":", count := [ 5, 3, 4, 6 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ \ 5, 3, 4, 7 ], mathmode := "Text", name := "P", next := 178803, root := ~, star\ t := 178799, stop := 178802 ), rec( content := "\n\n ", count := [ 5, 3\ , 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "DigraphLongestDistanceFromVertex(Digr\ aphRemoveLoops(", count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digr\ aph", count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "A", next := 178882\ , root := ~, start := 178868, stop := 178881 ), rec( content := ",", count := \ [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "v", count := [ 5, 3, 4, 7 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 7 ], ma\ thmode := "Text", name := "A", next := 178891, root := ~, start := 178883, sto\ p := 178890 ), rec( content := "))", count := [ 5, 3, 4, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 7 ], mathmode := "Tex\ t", name := "C", next := 178897, root := ~, start := 178813, stop := 178896 ), rec( cont\ ent := ".\n ", count := [ 5, 3, 4, 7 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 5, 3, 4, 6 ], name := "Item", next := 178912, root := ~, start := 178\ 390, stop := 178911 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n Otherwise, if all directed walks\n starting at vertex ", coun\ t := [ 5, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "v", count := [ 5, 3, 4, 9 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 9 ], math\ mode := "Text", name := "A", next := 179003, root := ~, start := 178995, stop \ := 179002 ), rec( content := " have finite length,\n then the length of\ the longest such walk is returned.\n ", count := [ 5, 3, 4, 9 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 9 ], name := "Ite\ m", next := 179099, root := ~, start := 178920, stop := 179098 ) ], count := [ 5, 3, 4, 5 ], mathmode := "Text", name := "List", next := 179111, root := ~, start := 178377, stop := 179110 ), rec( content := "\n\n \ Note that the result is ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "C", next := 179149, root := ~, start := 179141, stop := 179148 ), rec( content := " if and \ only if ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "A", next := 179173, root := ~, start := 179165, stop := 179172 ), rec( content := " is a si\ nk of\n ", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "A", next := 179205, root := ~, start := 179191, stop := 179204 ), rec( content := ". See " , count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Sinks" ), content := 0, count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "Ref", next := 179238, root := ~, start := 179212, stop := 179237 ), rec( content := ".\n\n", count := [ 5, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3, 4], [], [5], [], [6]]);\n\ngap> DigraphLongestDistanceFromVertex(D, 1);\n3\ngap> Di\ graphLongestDistanceFromVertex(D, 3);\n0\ngap> 3 in DigraphSinks(D);\ntrue\nga\ p> DigraphLongestDistanceFromVertex(D, 6);\ninfinity\n", count := [ 5, 3, 4, 12 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 4, 12 ], mathmode := "Text", name := "Example", next := 179550, root := ~, start := 179241, stop := 179549 ), rec( content := "\n ", count := [ 5, 3, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 179567, root := ~, start := 178011, stop := 179566 ) ], count := [ 5, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 179581, root := ~, start := 177876, stop := 179580 ), rec( content := "\n\n ", count := [ 5, 3, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex,\ distance", Label := "for a digraph, a pos int, and an int", Name := "DigraphDistance\ Set" ), content := 0, count := [ 5, 3, 5, 2 ], mathmode := "Text", name := "Oper", next := 179712, root := ~, start := 179602, stop := 179711 ), rec( attributes := rec( Arg := "digraph, vertex,\ distances", Label := "for a digraph, a pos int, and a list", Name := "DigraphDistance\ Set" ), content := 0, count := [ 5, 3, 5, 4 ], mathmode := "Text", name := "Oper", next := 179826, root := ~, start := 179715, stop := 179825 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "Returns", next := 179866, root := ~, start := 179829, stop := 179865 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns the list of all vertices in digraph ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "A", next := 179960, root := ~, start := 179946, stop := 179959 ), rec( content := "\n su\ ch that the shortest distance to a vertex ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "A", next := 180022, root := ~, start := 180009, stop := 180021 ), rec( content := " is\n \ ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ istance", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "A", next := 180045, root := ~, start := 180030, stop := 180044 ), rec( content := " or is i\ n the list ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ istances", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "A", next := 180080, root := ~, start := 180064, stop := 180079 ), rec( content := ". ", count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "P", next := 180086, root := ~, start := 180082, stop := 180085 ), rec( content := "\n\n \ ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "A", next := 180106, root := ~, start := 180092, stop := 180105 ), rec( content := " should \ be a digraph, ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "A", next := 180141, root := ~, start := 180128, stop := 180140 ), rec( content := " should \ be a positive\n integer, ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ istance", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "A", next := 180191, root := ~, start := 180176, stop := 180190 ), rec( content := " should \ be a non-negative integer, and\n ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ istances", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "A", next := 180250, root := ~, start := 180234, stop := 180249 ), rec( content := " should \ be a list of non-negative integers.\n\n ", count := [ 5, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [1, 4], [1, 3]]);\n\ngap> DigraphDistanceSet(D, 2, [1, 2]);\n[ 3, 1, 4 ]\ngap> Dig\ raphDistanceSet(D, 3, 1);\n[ 1, 4 ]\ngap> DigraphDistanceSet(D, 2, 0);\n[ 2 ]\ \n", count := [ 5, 3, 5, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 7 ], mathmode := "Text", name := "Example", next := 180557, root := ~, start := 180299, stop := 180556 ), rec( content := "\n ", count := [ 5, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 5, 5 ], mathmode := "Text", name := "Description", next := 180574, root := ~, start := 179869, stop := 180573 ) ], count := [ 5, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 180588, root := ~, start := 179587, stop := 180587 ), rec( content := "\n\n ", count := [ 5, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphGirth" ) , content := 0, count := [ 5, 3, 6, 2 ], mathmode := "Text", name := "Attr", next := 180650, root := ~, start := 180609, stop := 180649 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r, or ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "K", next := 180692, root := ~, start := 180677, stop := 180691 ), rec( content := ".", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 180703, root := ~, start := 180653, stop := 180702 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s attribute returns the ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ irth", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "E", next := 180763, root := ~, start := 180751, stop := 180762 ), rec( content := " of the \ digraph ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "A", next := 180793, root := ~, start := 180779, stop := 180792 ), rec( content := ".\n T\ he ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ irth", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "E", next := 180815, root := ~, start := 180803, stop := 180814 ), rec( content := " of a di\ graph is the length of its shortest simple circuit.\n See Section ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Ref", next := 180935, root := ~, start := 180891, stop := 180934 ), rec( content := " for the\ definitions\n of simple circuit, directed cycle, and loop.\n ", count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "P", next := 181013, root := ~, start := 181009, stop := 181012 ), rec( content := "\n\n \ If ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "A", next := 181036, root := ~, start := 181022, stop := 181035 ), rec( content := " has no \ directed cycles, then this function will return\n ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "K", next := 181111, root := ~, start := 181096, stop := 181110 ), rec( content := ". If ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "A", next := 181131, root := ~, start := 181117, stop := 181130 ), rec( content := " contain\ s a loop, then this function will\n return ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "C", next := 181192, root := ~, start := 181184, stop := 181191 ), rec( content := ".\n " , count := [ 5, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "P", next := 181202, root := ~, start := 181198, stop := 181201 ), rec( content := "\n\n \ In the worst case, the method used in this function is a version of the\n F\ loyd-Warshall algorithm, and has complexity ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 5, 3, 6, 5 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := \ "Text", name := "A", next := 181342, root := ~, start := 181334, stop := 18134\ 1 ), rec( content := " ^ 3)", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "C", next := 181351, root := ~, start := 181329, stop := 181350 ), rec( content := ", where\\ n ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "A", next := 181371, root := ~, start := 181363, stop := 181370 ), rec( content := " is the \ number of vertices in ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "A", next := 181415, root := ~, start := 181401, stop := 181414 ), rec( content := ". If th\ e digraph has\n known automorphisms [see ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "Ref", next := 181492, root := ~, start := 181466, stop := 181491 ), rec( content := "], then \ the performance\n is likely to be better.\n ", count := [ 5, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 6, 6 ], mathmode := "Text", name := "P", next := 181552, root := ~, start := 181548, stop := 181551 ), rec( content := "\n\n \ For symmetric digraphs, see also ", count := [ 5, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ UndirectedGirth" ), content := 0, count := [ 5, 3, 6, 6 ], mathmode := "Text", name := "Ref", next := 181627, root := ~, start := 181591, stop := 181626 ), rec( content := ". ", count := [ 5, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 6, 7 ], mathmode := "Text", name := "P", next := 181633, root := ~, start := 181629, stop := 181632 ), rec( content := "\n ", count := [ 5, 3, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1], [1]]);\n\ \ngap> DigraphGirth(D);\n1\ngap> D := Digraph([[2, 3], [3], [4], []]);\n\ngap> DigraphGirth(D);\ninfinity\ngap> \ D := Digraph([[2, 3], [3], [4], [1]]);\n\ngap> DigraphGirth(D);\n3\ngap> D := GeneralisedPetersenGraph(IsMutable\ Digraph, 6, 2);\n\ngap> DigraphGir\ th(D);\n2\n", count := [ 5, 3, 6, 8 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 8 ], mathmode := "Text", name := "Example", next := 182131, root := ~, start := 181638, stop := 182130 ), rec( content := "\n ", count := [ 5, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 182148, root := ~, start := 180706, stop := 182147 ) ], count := [ 5, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 182162, root := ~, start := 180594, stop := 182161 ), rec( content := "\n\n ", count := [ 5, 3, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphOddGirth\ " ), content := 0, count := [ 5, 3, 7, 2 ], mathmode := "Text", name := "Attr", next := 182227, root := ~, start := 182183, stop := 182226 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r, or ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "K", next := 182269, root := ~, start := 182254, stop := 182268 ), rec( content := ".", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Returns", next := 182280, root := ~, start := 182230, stop := 182279 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s attribute returns the ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ dd girth", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "E", next := 182344, root := ~, start := 182328, stop := 182343 ), rec( content := " of the \ digraph ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "A", next := 182374, root := ~, start := 182360, stop := 182373 ), rec( content := ".\n T\ he ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ dd girth", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "E", next := 182400, root := ~, start := 182384, stop := 182399 ), rec( content := " of a di\ graph is the length of its shortest simple\n circuit of odd length. See Se\ ction ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Ref", next := 182539, root := ~, start := 182491, stop := 182538 ), rec( content := " for the\ definitions of simple circuit, directed\n cycle, and loop.\n ", count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "P", next := 182617, root := ~, start := 182613, stop := 182616 ), rec( content := "\n\n \ If ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "A", next := 182640, root := ~, start := 182626, stop := 182639 ), rec( content := " has no \ directed cycles of odd length, then this function\n will return ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "K", next := 182729, root := ~, start := 182714, stop := 182728 ), rec( content := ", even i\ f ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "A", next := 182753, root := ~, start := 182739, stop := 182752 ), rec( content := " has a d\ irected cycle\n of even length. If ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "A", next := 182813, root := ~, start := 182799, stop := 182812 ), rec( content := " contain\ s a loop, then this function\n will return ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "C", next := 182874, root := ~, start := 182866, stop := 182873 ), rec( content := ".\n " , count := [ 5, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 7, 5 ], mathmode := "Text", name := "P", next := 182884, root := ~, start := 182880, stop := 182883 ), rec( content := "\n\n \ See also ", count := [ 5, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Girth" ), content := 0, count := [ 5, 3, 7, 5 ], mathmode := "Text", name := "Ref", next := 182925, root := ~, start := 182899, stop := 182924 ), rec( content := ". ", count := [ 5, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 7, 6 ], mathmode := "Text", name := "P", next := 182931, root := ~, start := 182927, stop := 182930 ), rec( content := "\n ", count := [ 5, 3, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3, 1], [1]]);\n\ngap> DigraphOddGirth(D);\n3\ngap> D := CycleDigraph(4);\n\ngap> DigraphOddGirth(D);\ninfinity\ngap> D := D\ igraph([[2], [3], [], [3], [4]]);\n\ngap> DigraphOddGirth(D);\ninfinity\ngap> D := CycleDigraph(IsMutableDigraph\ , 4);\n\ngap> DigraphDisjointUnion(D\ , CycleDigraph(5));\n\ngap> DigraphO\ ddGirth(D);\n5\n", count := [ 5, 3, 7, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 7 ], mathmode := "Text", name := "Example", next := 183511, root := ~, start := 182936, stop := 183510 ), rec( content := "\n ", count := [ 5, 3, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 7, 3 ], mathmode := "Text", name := "Description", next := 183528, root := ~, start := 182283, stop := 183527 ) ], count := [ 5, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 183542, root := ~, start := 182168, stop := 183541 ), rec( content := "\n\n ", count := [ 5, 3, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphUndirect\ edGirth" ), content := 0, count := [ 5, 3, 8, 2 ], mathmode := "Text", name := "Attr", next := 183614, root := ~, start := 183563, stop := 183613 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r or ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "K", next := 183655, root := ~, start := 183640, stop := 183654 ), rec( content := ".", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 183666, root := ~, start := 183617, stop := 183665 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "A", next := 183704, root := ~, start := 183690, stop := 183703 ), rec( content := " is a sy\ mmetric digraph, then this function returns the\n girth of ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "A", next := 183787, root := ~, start := 183773, stop := 183786 ), rec( content := " when tr\ eated as an undirected graph (i.e. each pair\n of edges ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "M", next := 183866, root := ~, start := 183853, stop := 183865 ), rec( content := " and ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j, i]", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "M", next := 183884, root := ~, start := 183871, stop := 183883 ), rec( content := " is trea\ ted as a single edge between\n ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "M", next := 183933, root := ~, start := 183925, stop := 183932 ), rec( content := " and ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "M", next := 183946, root := ~, start := 183938, stop := 183945 ), rec( content := "). ", count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "P", next := 183953, root := ~, start := 183949, stop := 183952 ), rec( content := "\n\n \ The ", count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ irth", count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "E", next := 183975, root := ~, start := 183963, stop := 183974 ), rec( content := " of an u\ ndirected graph is the length of its shortest simple\n cycle, i.e. the shor\ test non-trivial path starting and ending at the same\n vertex and passing \ through no vertex or edge more than once. ", count := [ 5, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "P", next := 184183, root := ~, start := 184179, stop := 184182 ), rec( content := "\n\n \ If ", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "A", next := 184206, root := ~, start := 184192, stop := 184205 ), rec( content := " has no \ cycles, then this function will return\n ", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "K", next := 184272, root := ~, start := 184257, stop := 184271 ), rec( content := ". ", count := [ 5, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 8, 6 ], mathmode := "Text", name := "P", next := 184278, root := ~, start := 184274, stop := 184277 ), rec( content := "\n ", count := [ 5, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 4], [1, 3], [2, 4], [1, 3]]);\n\ngap> DigraphUndirectedGirth(D);\n4\ngap> D := Digraph([\ [2], [1, 3], [2]]);\n\ngap> Digrap\ hUndirectedGirth(D);\ninfinity\ngap> D := Digraph([[1], [], [4], [3]]);\n\ngap> DigraphUndirectedGirth(D);\n1\ng\ ap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> DigraphUndirectedGirth(D);\n5\n", count := [ 5, 3, 8, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 8, 7 ], mathmode := "Text", name := "Example", next := 184830, root := ~, start := 184283, stop := 184829 ), rec( content := "\n ", count := [ 5, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 184847, root := ~, start := 183669, stop := 184846 ) ], count := [ 5, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 184861, root := ~, start := 183548, stop := 184860 ), rec( content := "\n\n ", count := [ 5, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphConnecte\ dComponents" ), content := 0, count := [ 5, 3, 9, 2 ], mathmode := "Text", name := "Attr", next := 184937, root := ~, start := 184882, stop := 184936 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphNrConnec\ tedComponents" ), content := 0, count := [ 5, 3, 9, 4 ], mathmode := "Text", name := "Attr", next := 184997, root := ~, start := 184940, stop := 184996 ), rec( attributes := rec( ), content := [ rec( content := "A record.\ ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 185028, root := ~, start := 185000, stop := 185027 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the record ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "w\ cc", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "C", next := 185092, root := ~, start := 185082, stop := 185091 ), rec( content := " corresp\ onding to the weakly\n connected components of the digraph ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "A", next := 185175, root := ~, start := 185161, stop := 185174 ), rec( content := ". Two v\ ertices of\n ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "A", next := 185212, root := ~, start := 185198, stop := 185211 ), rec( content := " are in \ the same weakly connected component whenever they are\n equal, or there exi\ sts a directed path (ignoring the orientation of edges)\n between them. Mo\ re formally, two vertices are in the same weakly connected\n component of " , count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "A", next := 185464, root := ~, start := 185450, stop := 185463 ), rec( content := " if and \ only if they are in the same strongly\n connected component (see ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ StronglyConnectedComponents" ), content := 0, count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Ref", next := 185587, root := ~, start := 185539, stop := 185586 ), rec( content := ")\n o\ f the ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ SymmetricClosure" ), content := 0, count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Ref", next := 185637, root := ~, start := 185600, stop := 185636 ), rec( content := " of ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "A", next := 185655, root := ~, start := 185641, stop := 185654 ), rec( content := ". ", count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "P", next := 185662, root := ~, start := 185658, stop := 185661 ), rec( content := "\n\n \ The set of weakly connected components is a partition of\n the vertex set o\ f ", count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "A", next := 185761, root := ~, start := 185747, stop := 185760 ), rec( content := ". ", count := [ 5, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "P", next := 185767, root := ~, start := 185763, stop := 185766 ), rec( content := "\n\n \ The record ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "w\ cc", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 185794, root := ~, start := 185784, stop := 185793 ), rec( content := " has 2 c\ omponents: ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 185825, root := ~, start := 185813, stop := 185824 ), rec( content := " and ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ d", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 185839, root := ~, start := 185830, stop := 185838 ), rec( content := ".\n T\ he component ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 185871, root := ~, start := 185859, stop := 185870 ), rec( content := " is a li\ st of the weakly connected components\n of ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "A", next := 185939, root := ~, start := 185925, stop := 185938 ), rec( content := " (each o\ f which is a list of vertices).\n The component ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ d", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 186006, root := ~, start := 185997, stop := 186005 ), rec( content := " is a li\ st such that the vertex ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 186046, root := ~, start := 186038, stop := 186045 ), rec( content := " is an\n\ element of the weakly connected component ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps[id[i]]", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "C", next := 186118, root := ~, start := 186099, stop := 186117 ), rec( content := ". ", count := [ 5, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "P", next := 186124, root := ~, start := 186120, stop := 186123 ), rec( content := "\n\n \ The method used in this function has complexity ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "M", next := 186192, root := ~, start := 186179, stop := 186191 ), rec( content := ", where\\ n ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "M", next := 186212, root := ~, start := 186204, stop := 186211 ), rec( content := " is the \ number of edges and\n ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "M", next := 186252, root := ~, start := 186244, stop := 186251 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 5, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "P", next := 186299, root := ~, start := 186295, stop := 186298 ), rec( content := "\n\n \ ", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrConnectedComponents(", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "A", next := 1863\ 51, root := ~, start := 186337, stop := 186350 ), rec( content := ")", count := [ 5, 3, 9, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "C", next := 186356, root := ~, start := 186305, stop := 186355 ), rec( content := " is simp\ ly a shortcut\n for ", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ength(DigraphConnectedComponents(", count := [ 5, 3, 9, 9 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "digraph", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "A", next := 1864\ 37, root := ~, start := 186423, stop := 186436 ), rec( content := ").comps)", count := [ 5, 3, 9, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "C", next := 186449, root := ~, start := 186386, stop := 186448 ), rec( content := ",\n a\ nd is no more efficient.\n\n ", count := [ 5, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2], [3, 1], []]);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps := [ [ 1, 2, 3 ] ], \ id := [ 1, 1, 1 ] )\ngap> gr := Digraph([[1], [1, 2], []]);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps \ := [ [ 1, 2 ], [ 3 ] ], id := [ 1, 1, 2 ] )\ngap> DigraphNrConnectedComponents\ (gr);\n2\ngap> gr := EmptyDigraph(0);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps := [ ], id := [ ] )\nga\ p> D := CycleDigraph(IsMutableDigraph, 2);\n\ngap> G := CycleDigraph(3);\n\ngap> DigraphDisjointUnion(D, G);\n\ngap> DigraphConnectedComponents(D);\nrec( comps := [ [ 1, 2 ], [ 3, 4, 5 ] \ ], id := [ 1, 1, 2, 2, 2 ] )\n", count := [ 5, 3, 9, 10 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 10 ], mathmode := "Text", name := "Example", next := 187383, root := ~, start := 186486, stop := 187382 ), rec( content := "\n ", count := [ 5, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 187400, root := ~, start := 185031, stop := 187399 ) ], count := [ 5, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 187414, root := ~, start := 184867, stop := 187413 ), rec( content := "\n\n ", count := [ 5, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphConnecte\ dComponent" ), content := 0, count := [ 5, 3, 10, 2 ], mathmode := "Text", name := "Oper", next := 187497, root := ~, start := 187435, stop := 187496 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "Returns", next := 187538, root := ~, start := 187500, stop := 187537 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "A", next := 187575, root := ~, start := 187562, stop := 187574 ), rec( content := " is a ve\ rtex in the digraph ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "A", next := 187617, root := ~, start := 187603, stop := 187616 ), rec( content := ", then\n\ this operation returns the connected component of ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "A", next := 187691, root := ~, start := 187678, stop := 187690 ), rec( content := "\n in\ ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "A", next := 187713, root := ~, start := 187699, stop := 187712 ), rec( content := ".\n S\ ee ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "Ref", next := 187763, root := ~, start := 187723, stop := 187762 ), rec( content := " for mor\ e information.\n ", count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [2], [1, 2], [4]]);\n\ngap> DigraphConnectedComponent(D, 3);\n[ 1, 2, 3 ]\ngap> Digraph\ ConnectedComponent(D, 2);\n[ 1, 2, 3 ]\ngap> DigraphConnectedComponent(D, 4);\ \n[ 4 ]\n", count := [ 5, 3, 10, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 4 ], mathmode := "Text", name := "Example", next := 188055, root := ~, start := 187790, stop := 188054 ), rec( content := "\n ", count := [ 5, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 10, 3 ], mathmode := "Text", name := "Description", next := 188072, root := ~, start := 187541, stop := 188071 ) ], count := [ 5, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 188086, root := ~, start := 187420, stop := 188085 ), rec( content := "\n\n ", count := [ 5, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphStrongly\ ConnectedComponents" ), content := 0, count := [ 5, 3, 11, 2 ], mathmode := "Text", name := "Attr", next := 188170, root := ~, start := 188107, stop := 188169 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphNrStrong\ lyConnectedComponents" ), content := 0, count := [ 5, 3, 11, 4 ], mathmode := "Text", name := "Attr", next := 188238, root := ~, start := 188173, stop := 188237 ), rec( attributes := rec( ), content := [ rec( content := "A record.\ ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "Returns", next := 188269, root := ~, start := 188241, stop := 188268 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the record ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ cc", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "C", next := 188333, root := ~, start := 188323, stop := 188332 ), rec( content := " corresp\ onding to the strongly\n connected components of the digraph ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "A", next := 188418, root := ~, start := 188404, stop := 188417 ), rec( content := ". Two v\ ertices of\n ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "A", next := 188455, root := ~, start := 188441, stop := 188454 ), rec( content := " are in \ the same strongly connected component whenever they\n are equal, or there i\ s a directed path from each vertex to the other. The\n set of strongly con\ nected components is a partition of the vertex set of\n ", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "A", next := 188689, root := ~, start := 188675, stop := 188688 ), rec( content := ".\n " , count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "P", next := 188699, root := ~, start := 188695, stop := 188698 ), rec( content := "\n\n \ The record ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ cc", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 188726, root := ~, start := 188716, stop := 188725 ), rec( content := " has 2 c\ omponents: ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 188757, root := ~, start := 188745, stop := 188756 ), rec( content := " and ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ d", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 188771, root := ~, start := 188762, stop := 188770 ), rec( content := ".\n T\ he component ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 188803, root := ~, start := 188791, stop := 188802 ), rec( content := " is a li\ st of the strongly connected components\n of ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "A", next := 188872, root := ~, start := 188858, stop := 188871 ), rec( content := " (each o\ f which is a list of vertices).\n The component ", count := [ 5, 3, 11, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ d", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 188939, root := ~, start := 188930, stop := 188938 ), rec( content := " is a li\ st such that the vertex ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 188979, root := ~, start := 188971, stop := 188978 ), rec( content := " is an\n\ element of the strongly connected component ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omps[id[i]]", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "C", next := 189053, root := ~, start := 189034, stop := 189052 ), rec( content := ". ", count := [ 5, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "P", next := 189059, root := ~, start := 189055, stop := 189058 ), rec( content := "\n\n \ The method used in this function is a non-recursive version of Gabow's\n Al\ gorithm ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "Gab00" ) , content := 0, count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "Cite", next := 189169, root := ~, start := 189150, stop := 189168 ), rec( content := " and has\ complexity ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "M", next := 189202, root := ~, start := 189189, stop := 189201 ), rec( content := " where\n\ ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "M", next := 189221, root := ~, start := 189213, stop := 189220 ), rec( content := " is the \ number of edges (counting multiple edges as one) and\n ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "M", next := 189294, root := ~, start := 189286, stop := 189293 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 5, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "P", next := 189341, root := ~, start := 189337, stop := 189340 ), rec( content := "\n\n \ ", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrStronglyConnectedComponents(", count := [ 5, 3, 11, 8 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "A", next := 1894\ 01, root := ~, start := 189387, stop := 189400 ), rec( content := ")", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "C", next := 189406, root := ~, start := 189347, stop := 189405 ), rec( content := " is simp\ ly a\n shortcut for\n ", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ength(DigraphStronglyConnectedComponents(", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "A", next := 1894\ 99, root := ~, start := 189485, stop := 189498 ), rec( content := ").comps)", count := [ 5, 3, 11, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "C", next := 189511, root := ~, start := 189440, stop := 189510 ), rec( content := ",\n a\ nd is no more efficient.\n\n ", count := [ 5, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2], [3, 1], []]);\n\ngap> DigraphStronglyConnectedComponents(gr);\nrec( comps := [ [ 3 ], \ [ 1, 2 ] ], id := [ 2, 2, 1 ] )\ngap> DigraphNrStronglyConnectedComponents(gr)\ ;\n2\ngap> D := DigraphDisjointUnion(CycleDigraph(4), CycleDigraph(5));\n\ngap> DigraphStronglyConnectedComponen\ ts(D);\nrec( comps := [ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8, 9 ] ], \n id := [ 1, 1,\ 1, 1, 2, 2, 2, 2, 2 ] )\ngap> DigraphNrStronglyConnectedComponents(D);\n2\nga\ p> D := CycleDigraph(IsMutableDigraph, 2);\n\ngap> G := CycleDigraph(3);\n\ngap> DigraphDisjointUnion(D, G);\n\ngap> DigraphStronglyConnectedComponents(D);\nrec( comps := [ [ 1, 2 ], [ 3,\ 4, 5 ] ], id := [ 1, 1, 2, 2, 2 ] )\n", count := [ 5, 3, 11, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 11, 9 ], mathmode := "Text", name := "Example", next := 190454, root := ~, start := 189548, stop := 190453 ), rec( content := "\n ", count := [ 5, 3, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 190471, root := ~, start := 188272, stop := 190470 ) ], count := [ 5, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 190485, root := ~, start := 188092, stop := 190484 ), rec( content := "\n\n ", count := [ 5, 3, 11, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphStrongly\ ConnectedComponent" ), content := 0, count := [ 5, 3, 12, 2 ], mathmode := "Text", name := "Oper", next := 190576, root := ~, start := 190506, stop := 190575 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "Returns", next := 190617, root := ~, start := 190579, stop := 190616 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "A", next := 190654, root := ~, start := 190641, stop := 190653 ), rec( content := " is a ve\ rtex in the digraph ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "A", next := 190696, root := ~, start := 190682, stop := 190695 ), rec( content := ", then\n\ this operation returns the strongly connected component of ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "A", next := 190779, root := ~, start := 190766, stop := 190778 ), rec( content := "\n in\ ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "A", next := 190801, root := ~, start := 190787, stop := 190800 ), rec( content := ".\n S\ ee ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ StronglyConnectedComponents" ), content := 0, count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "Ref", next := 190859, root := ~, start := 190811, stop := 190858 ), rec( content := " for mor\ e information.\n ", count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [2], [1, 2], [3]]);\n\ngap> DigraphStronglyConnectedComponent(D, 3);\n[ 1, 3 ]\ngap> Di\ graphStronglyConnectedComponent(D, 2);\n[ 2 ]\ngap> DigraphStronglyConnectedCo\ mponent(D, 4);\n[ 4 ]\n", count := [ 5, 3, 12, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 4 ], mathmode := "Text", name := "Example", next := 191166, root := ~, start := 190886, stop := 191165 ), rec( content := "\n ", count := [ 5, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 191183, root := ~, start := 190620, stop := 191182 ) ], count := [ 5, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 191197, root := ~, start := 190491, stop := 191196 ), rec( content := "\n\n ", count := [ 5, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphBicompon\ ents" ), content := 0, count := [ 5, 3, 13, 2 ], mathmode := "Text", name := "Attr", next := 191266, root := ~, start := 191218, stop := 191265 ), rec( attributes := rec( ), content := [ rec( content := "A pair of\ lists of vertices, or ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "K", next := 191321, root := ~, start := 191310, stop := 191320 ), rec( content := ".", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Returns", next := 191332, root := ~, start := 191269, stop := 191331 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "A", next := 191370, root := ~, start := 191356, stop := 191369 ), rec( content := " is a bi\ partite digraph, i.e. if it satisfies ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipar\ titeDigraph" ), content := 0, count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Ref", next := 191454, root := ~, start := 191416, stop := 191453 ), rec( content := ", then " , count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphBicomponents", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "C", next := 191487, root := ~, start := 191461, stop := 191486 ), rec( content := " returns\ a\n pair of bicomponents of ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "A", next := 191540, root := ~, start := 191526, stop := 191539 ), rec( content := ". Otherw\ ise,\n ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphBicomponents", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "C", next := 191583, root := ~, start := 191557, stop := 191582 ), rec( content := " returns\ ", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "K", next := 191603, root := ~, start := 191592, stop := 191602 ), rec( content := ".", count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "P", next := 191608, root := ~, start := 191604, stop := 191607 ), rec( content := "\n\n \ For a bipartite digraph, the vertices can be partitioned into two non-empty\n \ sets such that the source and range of any edge are in distinct sets. The\n\ parts of this partition are called ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ icomponents", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "E", next := 191826, root := ~, start := 191807, stop := 191825 ), rec( content := " of ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "A", next := 191844, root := ~, start := 191830, stop := 191843 ), rec( content := ".\n E\ quivalently, a pair of bicomponents of ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "A", next := 191904, root := ~, start := 191890, stop := 191903 ), rec( content := " consist\ s of the\n color-classes of a 2-coloring of ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "A", next := 191972, root := ~, start := 191958, stop := 191971 ), rec( content := ". ", count := [ 5, 3, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 13, 5 ], mathmode := "Text", name := "P", next := 191978, root := ~, start := 191974, stop := 191977 ), rec( content := "\n\n \ For a bipartite digraph with at least 3 vertices, there is a unique pair of\n \ bicomponents of bipartite if and only if the digraph is connected. See ", count := [ 5, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 5, 3, 13, 5 ], mathmode := "Text", name := "Ref", next := 192173, root := ~, start := 192135, stop := 192172 ), rec( content := " for mor\ e information.\n ", count := [ 5, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(3);\n\ngap> D\ igraphBicomponents(D);\nfail\ngap> D := ChainDigraph(5);\n\ngap> DigraphBicomponents(D);\n[ [ 1, 3, 5 ], [ 2, 4 ] ]\ \ngap> D := Digraph([[5], [1, 4], [5], [5], []]);\n\ngap> DigraphBicomponents(D);\n[ [ 1, 3, 4 ], [ 2, 5 ] ]\nga\ p> D := CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);\n\ngap> DigraphBicomponents(D);\n[ [ 1, 2 ], [ 3, 4, 5 \ ] ]\n", count := [ 5, 3, 13, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 6 ], mathmode := "Text", name := "Example", next := 192766, root := ~, start := 192200, stop := 192765 ), rec( content := "\n ", count := [ 5, 3, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 13, 3 ], mathmode := "Text", name := "Description", next := 192783, root := ~, start := 191335, stop := 192782 ) ], count := [ 5, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 192797, root := ~, start := 191203, stop := 192796 ), rec( content := "\n\n ", count := [ 5, 3, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "ArticulationPoi\ nts" ), content := 0, count := [ 5, 3, 14, 2 ], mathmode := "Text", name := "Attr", next := 192865, root := ~, start := 192818, stop := 192864 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices.", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "Returns", next := 192906, root := ~, start := 192868, stop := 192905 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ onnected digraph is ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ iconnected", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "E", next := 192968, root := ~, start := 192950, stop := 192967 ), rec( content := " if it i\ s still connected (in the\n sense of ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "Ref", next := 193047, root := ~, start := 193015, stop := 193046 ), rec( content := ") when a\ ny vertex is removed. \n If the digraph ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "A", next := 193111, root := ~, start := 193097, stop := 193110 ), rec( content := " is not \ biconnected but is connected, then any\n vertex ", count := [ 5, 3, 14, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "C", next := 193177, root := ~, start := 193169, stop := 193176 ), rec( content := " of ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "A", next := 193195, root := ~, start := 193181, stop := 193194 ), rec( content := " whose r\ emoval makes the resulting digraph\n disconnected is called an ", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ rticulation point", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "E", next := 193293, root := ~, start := 193268, stop := 193292 ), rec( content := ".", count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "P", next := 193298, root := ~, start := 193294, stop := 193297 ), rec( content := "\n\n \ ", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "A\ rticulationPoints", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "C", next := 193329, root := ~, start := 193304, stop := 193328 ), rec( content := " returns\ a list of the articulation points of\n ", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "A", next := 193393, root := ~, start := 193379, stop := 193392 ), rec( content := ", if any\ , and, in particular, returns the empty list if\n ", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "A", next := 193467, root := ~, start := 193453, stop := 193466 ), rec( content := " is not \ connected. ", count := [ 5, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 5 ], mathmode := "Text", name := "P", next := 193490, root := ~, start := 193486, stop := 193489 ), rec( content := "\n\n \ Multiple edges are ignored by this method. ", count := [ 5, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "P", next := 193543, root := ~, start := 193539, stop := 193542 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "M", next := 193611, root := ~, start := 193598, stop := 193610 ), rec( content := " where\n\ ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "M", next := 193630, root := ~, start := 193622, stop := 193629 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "M", next := 193666, root := ~, start := 193658, stop := 193665 ), rec( content := " is the \ number of vertices in\n the digraph.", count := [ 5, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "P", next := 193716, root := ~, start := 193712, stop := 193715 ), rec( content := "\n \ \n If ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "A", next := 193739, root := ~, start := 193731, stop := 193738 ), rec( content := " has a b\ ridge (see ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "Ref", next := 193779, root := ~, start := 193758, stop := 193778 ), rec( content := "), then \ a node incident\n to the bridge is an articulation point if and only if it \ has degree at\n least ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "M", next := 193896, root := ~, start := 193888, stop := 193895 ), rec( content := ". It fo\ llows that if ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "A", next := 193926, root := ~, start := 193918, stop := 193925 ), rec( content := " has a b\ ridge and at least\n ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "M", next := 193965, root := ~, start := 193957, stop := 193964 ), rec( content := " nodes, \ then at least one of the nodes incident to the bridge is\n an articulation \ point. The converse does not hold, there are digraphs\n with articulation \ points, but no bridges.\n ", count := [ 5, 3, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "P", next := 194159, root := ~, start := 194155, stop := 194158 ), rec( content := "\n\n \ See also ", count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "Ref", next := 194208, root := ~, start := 194174, stop := 194207 ), rec( content := " and \n \ ", count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "Ref", next := 194251, root := ~, start := 194218, stop := 194250 ), rec( content := ".\n", count := [ 5, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ArticulationPoints(CycleDigraph(5));\n[ ]\ngap> D := Digraph([[2, 7], [\ 3, 5], [4], [2], [6], [1], []]);;\ngap> ArticulationPoints(D);\n[ 2, 1 ]\ngap>\ ArticulationPoints(ChainDigraph(5));\n[ 4, 3, 2 ]\ngap> ArticulationPoints(Nu\ llDigraph(5));\n[ ]\ngap> D := ChainDigraph(IsMutableDigraph, 4);\n\ngap> ArticulationPoints(D);\n[ 3, 2 ]\n", count := [ 5, 3, 14, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 14, 9 ], mathmode := "Text", name := "Example", next := 194656, root := ~, start := 194253, stop := 194655 ), rec( content := "\n ", count := [ 5, 3, 14, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 194673, root := ~, start := 192909, stop := 194672 ) ], count := [ 5, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 194687, root := ~, start := 192803, stop := 194686 ), rec( content := "\n\n ", count := [ 5, 3, 14, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D", Name := "Bridges" ), content := 0, count := [ 5, 3, 15, 2 ], mathmode := "Text", name := "Attr", next := 194738, root := ~, start := 194708, stop := 194737 ), rec( attributes := rec( ), content := [ rec( content := "A (possib\ ly empty) list of edges.", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "Returns", next := 194793, root := ~, start := 194741, stop := 194792 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ onnected digraph is ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ -edge-connected", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "E", next := 194860, root := ~, start := 194837, stop := 194859 ), rec( content := " if it i\ s still connected (in\n the sense of ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "Ref", next := 194939, root := ~, start := 194907, stop := 194938 ), rec( content := ") when a\ ny edge is\n removed. If the digraph ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "A", next := 194995, root := ~, start := 194987, stop := 194994 ), rec( content := " is not \ 2-edge-connected but is\n connected, then any edge ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, v]", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "C", next := 195069, root := ~, start := 195056, stop := 195068 ), rec( content := " of ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "A", next := 195081, root := ~, start := 195073, stop := 195080 ), rec( content := " whose r\ emoval\n makes the resulting digraph disconnected is called a ", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ ridge", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "E", next := 195166, root := ~, start := 195153, stop := 195165 ), rec( content := ".", count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "P", next := 195171, root := ~, start := 195167, stop := 195170 ), rec( content := "\n\n \ ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "B\ ridges", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "C", next := 195191, root := ~, start := 195177, stop := 195190 ), rec( content := " returns\ a list of the bridges of ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "A", next := 195233, root := ~, start := 195225, stop := 195232 ), rec( content := ", if any\ , and, in\n particular, returns the empty list if ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "A", next := 195301, root := ~, start := 195293, stop := 195300 ), rec( content := " is not \ connected. ", count := [ 5, 3, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "P", next := 195324, root := ~, start := 195320, stop := 195323 ), rec( content := "\n\n \ Multiple edges are ignored by this method. ", count := [ 5, 3, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "P", next := 195377, root := ~, start := 195373, stop := 195376 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "M", next := 195445, root := ~, start := 195432, stop := 195444 ), rec( content := " where\n\ ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "M", next := 195464, root := ~, start := 195456, stop := 195463 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "M", next := 195500, root := ~, start := 195492, stop := 195499 ), rec( content := " is the \ number of vertices in\n the digraph.\n ", count := [ 5, 3, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "P", next := 195555, root := ~, start := 195551, stop := 195554 ), rec( content := "\n\n \ If ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "A", next := 195572, root := ~, start := 195564, stop := 195571 ), rec( content := " has a b\ ridge, then a node incident to the bridge is an\n articulation point (see " , count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "Ref", next := 195688, root := ~, start := 195656, stop := 195687 ), rec( content := ") if and\ only if\n it has degree at least ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "M", next := 195740, root := ~, start := 195732, stop := 195739 ), rec( content := ". It fo\ llows that if ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "A", next := 195770, root := ~, start := 195762, stop := 195769 ), rec( content := " has a\n\ bridge and at least ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "M", next := 195809, root := ~, start := 195801, stop := 195808 ), rec( content := " nodes, \ then at least one of the nodes\n incident to the bridge is an articulation \ point. The converse does not\n hold, there are digraphs with articulation p\ oints, but no bridges.\n ", count := [ 5, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "P", next := 196002, root := ~, start := 195998, stop := 196001 ), rec( content := "\n\n \ See also ", count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "Ref", next := 196051, root := ~, start := 196017, stop := 196050 ), rec( content := " and \n \ ", count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "Ref", next := 196094, root := ~, start := 196061, stop := 196093 ), rec( content := ".\n", count := [ 5, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 5], [1, 3, 4, 5], [2, 4], [2, 3], [1, 2]]);\n\ngap> Bridges(D);\n[ ]\ngap> D := Digrap\ h([[2], [3], [4], [2]]);\n\ngap> B\ ridges(D);\n[ [ 1, 2 ] ]\ngap> Bridges(ChainDigraph(2));\n[ [ 1, 2 ] ]\ngap> A\ rticulationPoints(ChainDigraph(2));\n[ ]\n", count := [ 5, 3, 15, 9 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 15, 9 ], mathmode := "Text", name := "Example", next := 196470, root := ~, start := 196096, stop := 196469 ), rec( content := "\n ", count := [ 5, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 15, 3 ], mathmode := "Text", name := "Description", next := 196487, root := ~, start := 194796, stop := 196486 ) ], count := [ 5, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 196501, root := ~, start := 194693, stop := 196500 ), rec( content := "\n\n ", count := [ 5, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D", Name := "StrongOrientati\ on" ), content := 0, count := [ 5, 3, 16, 2 ], mathmode := "Text", name := "Oper", next := 196562, root := ~, start := 196522, stop := 196561 ), rec( attributes := rec( Arg := "D", Name := "StrongOrientati\ onAttr" ), content := 0, count := [ 5, 3, 16, 4 ], mathmode := "Text", name := "Attr", next := 196609, root := ~, start := 196565, stop := 196608 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ or ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "K", next := 196645, root := ~, start := 196634, stop := 196644 ), rec( content := ".", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "Returns", next := 196656, root := ~, start := 196612, stop := 196655 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ trong orientation", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "E", next := 196704, root := ~, start := 196679, stop := 196703 ), rec( content := " of a co\ nnected symmetric digraph ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 196746, root := ~, start := 196738, stop := 196745 ), rec( content := " (if\n \ it exists) is a strongly connected subdigraph ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 196809, root := ~, start := 196801, stop := 196808 ), rec( content := " of ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 196821, root := ~, start := 196813, stop := 196820 ), rec( content := " such\n \ that for every edge ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ u, v]", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 196864, root := ~, start := 196851, stop := 196863 ), rec( content := " of ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "A", next := 196876, root := ~, start := 196868, stop := 196875 ), rec( content := " either \ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "[\ u, v]", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 196897, root := ~, start := 196884, stop := 196896 ), rec( content := " or\n \ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v, u]", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 196918, root := ~, start := 196905, stop := 196917 ), rec( content := " is an e\ dge of ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "C", next := 196941, root := ~, start := 196933, stop := 196940 ), rec( content := " but not\ both. Robbin's Theorem states\n that a digraph admits a strong orientatio\ n if and only if it is\n bridgeless (see ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "Ref", next := 197102, root := ~, start := 197069, stop := 197101 ), rec( content := "). \n \ ", count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "P", next := 197114, root := ~, start := 197110, stop := 197113 ), rec( content := "\n\n \ This operation returns a strong orientation of the digraph ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 197187, root := ~, start := 197179, stop := 197186 ), rec( content := " if \n \ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 197204, root := ~, start := 197196, stop := 197203 ), rec( content := " is symm\ etric and ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 197230, root := ~, start := 197222, stop := 197229 ), rec( content := " admits \ a strong orientation. If ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 197271, root := ~, start := 197263, stop := 197270 ), rec( content := "\n is\ symmetric but does not admit a strong orientation, then ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "K", next := 197346, root := ~, start := 197335, stop := 197345 ), rec( content := " is\n \ returned. If ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "A", next := 197375, root := ~, start := 197367, stop := 197374 ), rec( content := " is not \ symmetric, then an error is given. \n ", count := [ 5, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "P", next := 197427, root := ~, start := 197423, stop := 197426 ), rec( content := "\n\n \ If ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "A", next := 197444, root := ~, start := 197436, stop := 197443 ), rec( content := " is immu\ table, ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ trongOrientation(", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := \ "Text", name := "A", next := 197488, root := ~, start := 197480, stop := 19748\ 7 ), rec( content := ")", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "C", next := 197493, root := ~, start := 197459, stop := 197492 ), rec( content := " returns\ an\n immutable digraph, and if ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "A", next := 197543, root := ~, start := 197535, stop := 197542 ), rec( content := " is muta\ ble, then\n ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ trongOrientation(", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := \ "Text", name := "A", next := 197594, root := ~, start := 197586, stop := 19759\ 3 ), rec( content := ")", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "C", next := 197599, root := ~, start := 197565, stop := 197598 ), rec( content := " returns\ a mutable digraph. ", count := [ 5, 3, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "P", next := 197631, root := ~, start := 197627, stop := 197630 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "M", next := 197699, root := ~, start := 197686, stop := 197698 ), rec( content := " where\n\ ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "M", next := 197718, root := ~, start := 197710, stop := 197717 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "M", next := 197754, root := ~, start := 197746, stop := 197753 ), rec( content := " is the \ number of vertices in\n the digraph.\n", count := [ 5, 3, 16, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> StrongOrientation(DigraphSymmetricClosure(CycleDigraph(5))) \n> = CycleD\ igraph(5);\ntrue\ngap> D := DigraphSymmetricClosure(Digraph(\n> [[2, 7], [3, 5\ ], [4], [2], [6], [1], []]));;\ngap> IsBridgelessDigraph(D);\nfalse\ngap> Stro\ ngOrientation(D);\nfail\ngap> StrongOrientation(NullDigraph(0));\n\ngap> StrongOrientation(DigraphDisjointUnion(Com\ pleteDigraph(3), \n> CompleteDigraph\ (3)));\nfail\n", count := [ 5, 3, 16, 9 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 9 ], mathmode := "Text", name := "Example", next := 198300, root := ~, start := 197801, stop := 198299 ), rec( content := "\n ", count := [ 5, 3, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 16, 5 ], mathmode := "Text", name := "Description", next := 198317, root := ~, start := 196659, stop := 198316 ) ], count := [ 5, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 198331, root := ~, start := 196507, stop := 198330 ), rec( content := "\n\n ", count := [ 5, 3, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphPeriod" ), content := 0, count := [ 5, 3, 17, 2 ], mathmode := "Text", name := "Attr", next := 198394, root := ~, start := 198352, stop := 198393 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "Returns", next := 198427, root := ~, start := 198397, stop := 198426 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the period of the digraph ", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "A", next := 198510, root := ~, start := 198496, stop := 198509 ), rec( content := ".", count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "P", next := 198515, root := ~, start := 198511, stop := 198514 ), rec( content := "\n\n \ If a digraph ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "A", next := 198548, root := ~, start := 198534, stop := 198547 ), rec( content := " has at \ least one directed cycle, then the period\n is the greatest positive intege\ r which divides the lengths of all directed\n cycles of ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "A", next := 198705, root := ~, start := 198691, stop := 198704 ), rec( content := ". If ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "A", next := 198725, root := ~, start := 198711, stop := 198724 ), rec( content := " has no \ directed cycles, then\n this function returns ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "M", next := 198789, root := ~, start := 198781, stop := 198788 ), rec( content := ". See S\ ection ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "Ref", next := 198854, root := ~, start := 198804, stop := 198853 ), rec( content := " for the\ definition of a directed cycle. ", count := [ 5, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "P", next := 198899, root := ~, start := 198895, stop := 198898 ), rec( content := "\n\n \ A digraph with a period of ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "M", next := 198940, root := ~, start := 198932, stop := 198939 ), rec( content := " is said\ to be ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ periodic", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "E", next := 198971, root := ~, start := 198955, stop := 198970 ), rec( content := ". See\n\ ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsAperi\ odicDigraph" ), content := 0, count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "Ref", next := 199014, root := ~, start := 198982, stop := 199013 ), rec( content := ". ", count := [ 5, 3, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 17, 6 ], mathmode := "Text", name := "P", next := 199020, root := ~, start := 199016, stop := 199019 ), rec( content := "\n ", count := [ 5, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[6], [1], [2], [3], [4, 4], [5]]);\n\ngap> DigraphPeriod(D);\n6\ngap> D := Digraph([[2]\ , [3, 5], [4], [5], [1, 2]]);\n\ng\ ap> DigraphPeriod(D);\n1\ngap> D := ChainDigraph(2);\n\ngap> DigraphPeriod(D);\n0\ngap> IsAcyclicDigraph(D);\ntrue\ \ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 5, 2);\n\ngap> DigraphPeriod(D);\n1\n", count := [ 5, 3, 17, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 17, 7 ], mathmode := "Text", name := "Example", next := 199563, root := ~, start := 199025, stop := 199562 ), rec( content := "\n ", count := [ 5, 3, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 199580, root := ~, start := 198430, stop := 199579 ) ], count := [ 5, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 199594, root := ~, start := 198337, stop := 199593 ), rec( content := "\n\n ", count := [ 5, 3, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, func, n\ opath, edge", Name := "DigraphFloydWarshall" ), content := 0, count := [ 5, 3, 18, 2 ], mathmode := "Text", name := "Oper", next := 199684, root := ~, start := 199615, stop := 199683 ), rec( attributes := rec( ), content := [ rec( content := "A matrix.\ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "Returns", next := 199715, root := ~, start := 199687, stop := 199714 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "A", next := 199753, root := ~, start := 199739, stop := 199752 ), rec( content := " is a di\ graph with ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "M", next := 199780, root := ~, start := 199772, stop := 199779 ), rec( content := " vertice\ s, then\n this operation returns an ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ \\times n", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "M", next := 199843, root := ~, start := 199826, stop := 199842 ), rec( content := " matrix \ ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "C", next := 199861, root := ~, start := 199851, stop := 199860 ), rec( content := " contain\ ing\n the output of a generalised version of the Floyd-Warshall algorithm,\ \n applied to ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "A", next := 199975, root := ~, start := 199961, stop := 199974 ), rec( content := ". ", count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "P", next := 199981, root := ~, start := 199977, stop := 199980 ), rec( content := "\n\n \ The operation ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFloydWarshall", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "C", next := 200028, root := ~, start := 200001, stop := 200027 ), rec( content := " is cust\ omised by the arguments\n ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 200075, root := ~, start := 200064, stop := 200074 ), rec( content := ", ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ opath", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 200090, root := ~, start := 200077, stop := 200089 ), rec( content := ", and ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 200107, root := ~, start := 200096, stop := 200106 ), rec( content := ".\n T\ he arguments ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ opath", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 200140, root := ~, start := 200127, stop := 200139 ), rec( content := " and ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 200156, root := ~, start := 200145, stop := 200155 ), rec( content := " can be \ arbitrary ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " objects\ .\n The argument ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 200217, root := ~, start := 200206, stop := 200216 ), rec( content := " must be\ a function which accepts 4 arguments:\n the matrix ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "C", next := 200289, root := ~, start := 200279, stop := 200288 ), rec( content := ", follow\ ed by 3 postive integers. The function\n ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "A", next := 200351, root := ~, start := 200340, stop := 200350 ), rec( content := " is wher\ e the work to calculate the desired outcome must be\n performed. ", count := [ 5, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "P", next := 200430, root := ~, start := 200426, stop := 200429 ), rec( content := "\n\n \ This method initialises the matrix ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 200481, root := ~, start := 200471, stop := 200480 ), rec( content := " by sett\ ing entry\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 200519, root := ~, start := 200503, stop := 200518 ), rec( content := " to equa\ l ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "A", next := 200540, root := ~, start := 200529, stop := 200539 ), rec( content := " if ther\ e is an edge with source\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 200585, root := ~, start := 200577, stop := 200584 ), rec( content := " and ran\ ge ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 200604, root := ~, start := 200596, stop := 200603 ), rec( content := ", and by\ setting entry ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at[i][j]", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 200643, root := ~, start := 200627, stop := 200642 ), rec( content := " to equa\ l\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ opath", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "A", next := 200670, root := ~, start := 200657, stop := 200669 ), rec( content := " otherwi\ se.\n The final part of ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFloydWarshall", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "C", next := 200731, root := ~, start := 200704, stop := 200730 ), rec( content := " then ca\ lls the function\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "A", next := 200771, root := ~, start := 200760, stop := 200770 ), rec( content := " inside \ three nested for loops, over the vertices of\n ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "A", next := 200842, root := ~, start := 200828, stop := 200841 ), rec( content := ": ", count := [ 5, 3, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 18, 6 ], mathmode := "Text", name := "P", next := 200848, root := ~, start := 200844, stop := 200847 ), rec( content := "\n\n", count := [ 5, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ nfor i in DigraphsVertices(digraph) do\n for j in DigraphsVertices(digraph) d\ o\n for k in DigraphsVertices(digraph) do\n func(mat, i, j, k);\n o\ d;\n od;\nod;\n", count := [ 5, 3, 18, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 7 ], mathmode := "Text", name := "Listing", next := 201046, root := ~, start := 200850, stop := 201045 ), rec( content := "\n\n \ The matrix ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ at", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "C", next := 201073, root := ~, start := 201063, stop := 201072 ), rec( content := " is then\ returned as the result. An example of using\n ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFloydWarshall", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "C", next := 201157, root := ~, start := 201130, stop := 201156 ), rec( content := " to calc\ ulate the shortest (non-zero) distances\n between the vertices of a digraph\ is shown below: ", count := [ 5, 3, 18, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 18, 9 ], mathmode := "Text", name := "P", next := 201263, root := ~, start := 201259, stop := 201262 ), rec( content := "\n\n \ ", count := [ 5, 3, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&EAHQeDB\");\n\ngap> func := function(mat, i, j, k)\n> if mat[i][k] <>\ -1 and mat[k][j] <> -1 then\n> if (mat[i][j] = -1) or (mat[i][j] > mat[i]\ [k] + mat[k][j]) then\n> mat[i][j] := mat[i][k] + mat[k][j];\n> fi;\ \n> fi;\n> end;\nfunction( mat, i, j, k ) ... end\ngap> shortest_distances :\ = DigraphFloydWarshall(D, func, -1, 1);;\ngap> Display(shortest_distances);\n[\ [ 3, -1, -1, 2, 1, 2 ],\n [ 4, 2, 1, 3, 2, 1 ],\n [ \ 3, 1, 2, 2, 1, 2 ],\n [ 1, -1, -1, 1, 1, 2 ],\n [ 2, \ -1, -1, 1, 2, 1 ],\n [ 3, -1, -1, 2, 1, 1 ] ]\n", count := [ 5, 3, 18, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 18, 10 ], mathmode := "Text", name := "Example", next := 201969, root := ~, start := 201269, stop := 201968 ), rec( content := "\n ", count := [ 5, 3, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 201986, root := ~, start := 199718, stop := 201985 ) ], count := [ 5, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 202000, root := ~, start := 199600, stop := 201999 ), rec( content := "\n\n ", count := [ 5, 3, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "IsReachable" ), content := 0, count := [ 5, 3, 19, 2 ], mathmode := "Text", name := "Oper", next := 202067, root := ~, start := 202021, stop := 202066 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 202090, root := ~, start := 202079, stop := 202089 ), rec( content := " or ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 202106, root := ~, start := 202094, stop := 202105 ), rec( content := ".", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Returns", next := 202117, root := ~, start := 202070, stop := 202116 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 202172, root := ~, start := 202161, stop := 202171 ), rec( content := " if ther\ e exists a non-trivial directed\n walk from vertex ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "A", next := 202241, root := ~, start := 202233, stop := 202240 ), rec( content := " to vert\ ex ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "A", next := 202260, root := ~, start := 202252, stop := 202259 ), rec( content := " in the \ digraph ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "A", next := 202290, root := ~, start := 202276, stop := 202289 ), rec( content := ",\n a\ nd ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "K", next := 202312, root := ~, start := 202300, stop := 202311 ), rec( content := " if ther\ e does not exist such a directed walk. See Section\n ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Ref", next := 202420, root := ~, start := 202376, stop := 202419 ), rec( content := " for the\ definition of a\n non-trivial directed walk.\n ", count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "P", next := 202484, root := ~, start := 202480, stop := 202483 ), rec( content := "\n\n \ The method for ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sReachable", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "C", next := 202523, root := ~, start := 202505, stop := 202522 ), rec( content := " has wor\ st case complexity of ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m +\n n)", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "M", next := 202574, root := ~, start := 202553, stop := 202573 ), rec( content := " where " , count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "M", next := 202589, root := ~, start := 202581, stop := 202588 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "M", next := 202625, root := ~, start := 202617, stop := 202624 ), rec( content := " the num\ ber of\n vertices in ", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "A", next := 202670, root := ~, start := 202656, stop := 202669 ), rec( content := ".\n\n", count := [ 5, 3, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [2, 3]]);\n\ngap> IsReachable(D, 1, 3);\ntrue\ngap> IsReachable(D, 2, 1);\nfalse\n\ gap> IsReachable(D, 3, 3);\ntrue\ngap> IsReachable(D, 1, 1);\nfalse\n", count := [ 5, 3, 19, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 19, 5 ], mathmode := "Text", name := "Example", next := 202919, root := ~, start := 202673, stop := 202918 ), rec( content := "\n ", count := [ 5, 3, 19, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 19, 3 ], mathmode := "Text", name := "Description", next := 202936, root := ~, start := 202120, stop := 202935 ) ], count := [ 5, 3, 19, 1 ], mathmode := "Text", name := "ManSection", next := 202950, root := ~, start := 202006, stop := 202949 ), rec( content := "\n\n ", count := [ 5, 3, 19, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, root", Name := "VerticesReachab\ leFrom" ), content := 0, count := [ 5, 3, 20, 2 ], mathmode := "Text", name := "Oper", next := 203027, root := ~, start := 202971, stop := 203026 ), rec( attributes := rec( ), content := [ rec( content := "A list.", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Returns", next := 203056, root := ~, start := 203030, stop := 203055 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns a list of the vertices ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 203132, root := ~, start := 203124, stop := 203131 ), rec( content := ", for wh\ ich there exists\n a non-trivial directed walk from vertex ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ oot", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 203212, root := ~, start := 203201, stop := 203211 ), rec( content := " to vert\ ex ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 203231, root := ~, start := 203223, stop := 203230 ), rec( content := " in the \ digraph\n ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "A", next := 203265, root := ~, start := 203251, stop := 203264 ), rec( content := ". See Se\ ction ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Ref", next := 203323, root := ~, start := 203279, stop := 203322 ), rec( content := "\n fo\ r the definition of a non-trivial directed walk.\n ", count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "P", next := 203387, root := ~, start := 203383, stop := 203386 ), rec( content := "\n\n \ The method for ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ erticesReachableFrom", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "C", next := 203436, root := ~, start := 203408, stop := 203435 ), rec( content := " has wor\ st case complexity of ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m +\n n)", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "M", next := 203487, root := ~, start := 203466, stop := 203486 ), rec( content := " where " , count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "M", next := 203502, root := ~, start := 203494, stop := 203501 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "M", next := 203538, root := ~, start := 203530, stop := 203537 ), rec( content := " the num\ ber of\n vertices in ", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "A", next := 203583, root := ~, start := 203569, stop := 203582 ), rec( content := ".\n\n", count := [ 5, 3, 20, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(5);\n\n\ gap> VerticesReachableFrom(D, 1);\n[ 2, 1, 3, 4, 5 ]\ngap> VerticesReachableFr\ om(D, 3);\n[ 1, 2, 3, 4, 5 ]\ngap> D := EmptyDigraph(5);\n\ngap> VerticesReachableFrom(D, 1);\n[ ]\ngap> VerticesR\ eachableFrom(D, 3);\n[ ]\ngap> D := CycleDigraph(4);\n\ngap> VerticesReachableFrom(D, 1);\n[ 2, 3, 4, 1 ]\ngap> Ve\ rticesReachableFrom(D, 3);\n[ 4, 1, 2, 3 ]\ngap> D := ChainDigraph(5);\n\ngap> VerticesReachableFrom(D, 1);\n[ 2, 3\ , 4, 5 ]\ngap> VerticesReachableFrom(D, 3);\n[ 4, 5 ]\ngap> VerticesReachableF\ rom(D, 5);\n[ ]\n", count := [ 5, 3, 20, 5 ], name := "PCDATA", root := ~ ) ] , count := [ 5, 3, 20, 5 ], mathmode := "Text", name := "Example", next := 204311, root := ~, start := 203586, stop := 204310 ), rec( content := "\n ", count := [ 5, 3, 20, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 20, 3 ], mathmode := "Text", name := "Description", next := 204328, root := ~, start := 203059, stop := 204327 ) ], count := [ 5, 3, 20, 1 ], mathmode := "Text", name := "ManSection", next := 204342, root := ~, start := 202956, stop := 204341 ), rec( content := "\n\n ", count := [ 5, 3, 20, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "DigraphPath" ), content := 0, count := [ 5, 3, 21, 2 ], mathmode := "Text", name := "Oper", next := 204409, root := ~, start := 204363, stop := 204408 ), rec( attributes := rec( ), content := [ rec( content := "A pair of\ lists, or ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "K", next := 204452, root := ~, start := 204441, stop := 204451 ), rec( content := ".", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Returns", next := 204463, root := ~, start := 204412, stop := 204462 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ there exists a non-trivial directed path (or a non-trivial cycle, in the\n \ case that ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "A", next := 204582, root := ~, start := 204574, stop := 204581 ), rec( content := " ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "=\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "C", next := 204591, root := ~, start := 204583, stop := 204590 ), rec( content := " ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "A", next := 204600, root := ~, start := 204592, stop := 204599 ), rec( content := ") from v\ ertex ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "A", next := 204622, root := ~, start := 204614, stop := 204621 ), rec( content := " to vert\ ex\n ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "A", next := 204645, root := ~, start := 204637, stop := 204644 ), rec( content := " in the \ digraph ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "A", next := 204675, root := ~, start := 204661, stop := 204674 ), rec( content := ", then t\ his operation returns such a\n directed path (or directed cycle). Otherwise\ , this operation returns\n ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "K", next := 204800, root := ~, start := 204789, stop := 204799 ), rec( content := ". See S\ ection ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Text", Subsect := "Defi\ nitions" ), content := 0, count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Ref", next := 204856, root := ~, start := 204815, stop := 204855 ), rec( content := " for the\ \n definition of a directed path and a directed cycle.\n ", count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "P", next := 204929, root := ~, start := 204925, stop := 204928 ), rec( content := "\n\n \ A directed path (or directed cycle) of non-zero length ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ -1", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "C", next := 205000, root := ~, start := 204990, stop := 204999 ), rec( content := ",\n " , count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)", count := [ 5, 3, 21, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "M", next := 205052, root := ~, start := 205006, stop := 205051 ), rec( content := ",\n i\ s represented by a pair of lists ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v,a]", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "C", next := 205104, root := ~, start := 205092, stop := 205103 ), rec( content := " as foll\ ows:\n\n ", count := [ 5, 3, 21, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 5, 3, 21, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "v", count := [ 5, 3, 21, 6 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 6 ], mathmode := "Text", name\ := "C", next := 205158, root := ~, start := 205150, stop := 205157 ), rec( cont\ ent := " is the list ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "[v_1, v_2, ..., v_n]", count := [ 5, 3, 21, 6 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 6 ], mathmode := "Text", name \ := "M", next := 205198, root := ~, start := 205171, stop := 205197 ), rec( cont\ ent := ".\n ", count := [ 5, 3, 21, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 21, 6 ], name := "Item", next := 205213, root := ~, start := 205\ 135, stop := 205212 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "a", count :\ = [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 5, 3, 21, 8 ], mathmode := "Text", name := "C", next := 205243, root := ~ , star\ t := 205235, stop := 205242 ), rec( content := " is the list of positive integ\ ers ", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "[a_1, a_2, ...\ , a_{n-1}]", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "M", next \ := 205308, root := ~, start := 205277, stop := 205307 ), rec( content := "\n \ where for each each ", count := [ 5, 3, 21, 8 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "i < n", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "M", next\ := 205352, root := ~, start := 205337, stop := 205351 ), rec( content := ", " , coun\ t := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "a_i", count := [ 5, 3, 21, 8 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 8\ ], mathmode := "Text", name := "M", next := 205364, root := ~, start := 20535\ 4, stop := 205363 ), rec( content := " is the position of\n ", count :=\ [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "v_{i+1}", count := [ 5, 3, 21\ , 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 2\ 1, 8 ], mathmode := "Text", name := "M", next := 205406, root := ~, start := 2\ 05392, stop := 205405 ), rec( content := " in\n ", count := [ 5, 3, 21,\ 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "OutNeighboursOfVertex(", count := [ 5, 3,\ 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3\ , 21, 8 ], mathmode := "Text", name := "C", next := 205447, root := ~, start :\ = 205418, stop := 205446 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "A", next\ := 205461, root := ~, start := 205447, stop := 205460 ), rec( attributes := re\ c( ), content := [ rec( content := ",", count := [ 5, 3, 21, 8 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode :\ = "Text", name := "C", next := 205469, root := ~, start := 205461, stop := 205\ 468 ), rec( attributes := rec( ), content := [ rec( content := "v_i", count :\ = [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 5, 3, 21, 8 ], mathmode := "Text", name := "M", next := 205479, root := ~ , star\ t := 205469, stop := 205478 ), rec( attributes := rec( ), content := [ rec( c\ ontent := ")", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "C", next\ := 205487, root := ~, start := 205479, stop := 205486 ), rec( content := "\n \ corresponding to the edge ", count := [ 5, 3, 21, 8 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "e_i", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "M\ ", next := 205532, root := ~, start := 205522, stop := 205531 ), rec( content \ := ". This is can be useful if the\n position of a vertex in a list of\ out-neighours is significant, for\n example in orbit digraphs.\n \ ", count := [ 5, 3, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 8 ], name := "Ite\ m", next := 205688, root := ~, start := 205220, stop := 205687 ) ], count := [ 5, 3, 21, 5 ], mathmode := "Text", name := "List", next := 205700, root := ~, start := 205122, stop := 205699 ), rec( content := "\n\n \ The method for ", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPath", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "C", next := 205739, root := ~, start := 205721, stop := 205738 ), rec( content := " has wor\ st case complexity of ", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m +\n n)", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "M", next := 205790, root := ~, start := 205769, stop := 205789 ), rec( content := " where " , count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "M", next := 205805, root := ~, start := 205797, stop := 205804 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "M", next := 205841, root := ~, start := 205833, stop := 205840 ), rec( content := " the num\ ber of\n vertices in ", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "A", next := 205886, root := ~, start := 205872, stop := 205885 ), rec( content := ".\n\n", count := [ 5, 3, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], [2, 3]]);\n\ngap> DigraphPath(D, 1, 3);\n[ [ 1, 2, 3 ], [ 1, 1 ] ]\ngap> DigraphPa\ th(D, 2, 1);\nfail\ngap> DigraphPath(D, 3, 3);\n[ [ 3, 3 ], [ 2 ] ]\ngap> Digr\ aphPath(D, 1, 1);\nfail\n", count := [ 5, 3, 21, 11 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 11 ], mathmode := "Text", name := "Example", next := 206169, root := ~, start := 205889, stop := 206168 ), rec( content := "\n ", count := [ 5, 3, 21, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 21, 3 ], mathmode := "Text", name := "Description", next := 206186, root := ~, start := 204466, stop := 206185 ) ], count := [ 5, 3, 21, 1 ], mathmode := "Text", name := "ManSection", next := 206200, root := ~, start := 204348, stop := 206199 ), rec( content := "\n\n ", count := [ 5, 3, 21, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "DigraphShortest\ Path" ), content := 0, count := [ 5, 3, 22, 2 ], mathmode := "Text", name := "Oper", next := 206275, root := ~, start := 206221, stop := 206274 ), rec( attributes := rec( ), content := [ rec( content := "A pair of\ lists, or ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "K", next := 206318, root := ~, start := 206307, stop := 206317 ), rec( content := ".", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Returns", next := 206329, root := ~, start := 206278, stop := 206328 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the shortest directed path in the digraph digraph from the vertex\n ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "A", next := 206436, root := ~, start := 206428, stop := 206435 ), rec( content := " to the \ vertex ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "A", next := 206459, root := ~, start := 206451, stop := 206458 ), rec( content := ", if suc\ h a path exists. If ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "u", count := [ 5, 3, 22, 3 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := \ "Text", name := "A", next := 206498, root := ~, start := 206490, stop := 20649\ 7 ), rec( content := " =\n ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := \ "Text", name := "A", next := 206513, root := ~, start := 206505, stop := 20651\ 2 ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "C", next := 206517, root := ~, start := 206487, stop := 206516 ), rec( content := ", then t\ he shortest non-trivial cycle is returned, again, if it\n exists. Otherwise\ , this operation returns ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "K", next := 206638, root := ~, start := 206627, stop := 206637 ), rec( content := ". See S\ ection\n ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Text", Subsect := "Defi\ nitions" ), content := 0, count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Ref", next := 206698, root := ~, start := 206657, stop := 206697 ), rec( content := " for the\ definition of a directed\n path and a directed cycle.\n ", count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "P", next := 206771, root := ~, start := 206767, stop := 206770 ), rec( content := "\n\n \ See ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Path" ), content := 0, count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "Ref", next := 206806, root := ~, start := 206781, stop := 206805 ), rec( content := " for det\ ails on the output.\n\n The method for ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphShortestPath", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "C", next := 206880, root := ~, start := 206854, stop := 206879 ), rec( content := " has wor\ st case complexity of\n ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n)", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "M", next := 206929, root := ~, start := 206914, stop := 206928 ), rec( content := " where " , count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "M", next := 206944, root := ~, start := 206936, stop := 206943 ), rec( content := " is the \ number of edges and ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "M", next := 206980, root := ~, start := 206972, stop := 206979 ), rec( content := " the\n \ number of vertices in ", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "A", next := 207025, root := ~, start := 207011, stop := 207024 ), rec( content := ".\n\n", count := [ 5, 3, 22, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [3], [2, 4], [1], [2, 4]]);\n\ngap> DigraphShortestPath(D, 5, 1);\n[ [ 5, 4, 1 ], [ \ 2, 1 ] ]\ngap> DigraphShortestPath(D, 3, 3);\n[ [ 3, 2, 3 ], [ 1, 1 ] ]\ngap> \ DigraphShortestPath(D, 5, 5);\nfail\ngap> DigraphShortestPath(D, 1, 1);\n[ [ 1\ , 1 ], [ 1 ] ]\n", count := [ 5, 3, 22, 5 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 5 ], mathmode := "Text", name := "Example", next := 207377, root := ~, start := 207028, stop := 207376 ), rec( content := "\n ", count := [ 5, 3, 22, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 22, 3 ], mathmode := "Text", name := "Description", next := 207394, root := ~, start := 206332, stop := 207393 ) ], count := [ 5, 3, 22, 1 ], mathmode := "Text", name := "ManSection", next := 207408, root := ~, start := 206206, stop := 207407 ), rec( content := "\n\n ", count := [ 5, 3, 22, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, u, v", Name := "IteratorOfPaths\ " ), content := 0, count := [ 5, 3, 23, 2 ], mathmode := "Text", name := "Oper", next := 207479, root := ~, start := 207429, stop := 207478 ), rec( attributes := rec( ), content := [ rec( content := "An iterat\ or.", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "Returns", next := 207513, root := ~, start := 207482, stop := 207512 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 207551, root := ~, start := 207537, stop := 207550 ), rec( content := " is a di\ graph or a list of adjacencies which defines a\n digraph - see ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutNeig\ hbours" ), content := 0, count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "Ref", next := 207651, root := ~, start := 207624, stop := 207650 ), rec( content := " - then \ this operation returns an\n iterator of the non-trivial directed paths (or \ directed cycles, in the case\n that ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 207782, root := ~, start := 207774, stop := 207781 ), rec( content := " ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "=\ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "C", next := 207791, root := ~, start := 207783, stop := 207790 ), rec( content := " ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 207800, root := ~, start := 207792, stop := 207799 ), rec( content := ") in ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 207819, root := ~, start := 207805, stop := 207818 ), rec( content := " from th\ e vertex ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 207844, root := ~, start := 207836, stop := 207843 ), rec( content := "\n to\ the vertex ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "A", next := 207871, root := ~, start := 207863, stop := 207870 ), rec( content := ".\n " , count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "P", next := 207881, root := ~, start := 207877, stop := 207880 ), rec( content := "\n\n \ See ", count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Path" ), content := 0, count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "Ref", next := 207916, root := ~, start := 207891, stop := 207915 ), rec( content := " for mor\ e information about the repesentation\n of a directed path or directed cycl\ e which is used, and see ", count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Sect := "Iterators" ), content := 0, count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "Ref", next := 208070, root := ~, start := 208026, stop := 208069 ), rec( content := " for mor\ e information about iterators.\n See Section ", count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Text", Subsect := "Defi\ nitions" ), content := 0, count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "Ref", next := 208166, root := ~, start := 208125, stop := 208165 ), rec( content := " for the\ definition of\n a directed path and a directed cycle.\n ", count := [ 5, 3, 23, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "P", next := 208239, root := ~, start := 208235, stop := 208238 ), rec( content := "\n\n", count := [ 5, 3, 23, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 4, 4, 2], [3, 5], [2, 3], [1, 2], [4]]);\n\ngap> iter := IteratorOfPaths(D, 1, 4);\ \n\ngap> NextIterator(iter);\n[ [ 1, 4 ], [ 2 ] ]\ngap> NextIterator\ (iter);\n[ [ 1, 4 ], [ 3 ] ]\ngap> NextIterator(iter);\n[ [ 1, 2, 5, 4 ], [ 4,\ 2, 1 ] ]\ngap> IsDoneIterator(iter);\ntrue\ngap> iter := IteratorOfPaths(D, 4\ , 3);\n\ngap> NextIterator(iter);\n[ [ 4, 1, 2, 3 ], [ 1, 4, 1 ] ]\n\ ", count := [ 5, 3, 23, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 6 ], mathmode := "Text", name := "Example", next := 208724, root := ~, start := 208241, stop := 208723 ), rec( content := "\n ", count := [ 5, 3, 23, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 23, 3 ], mathmode := "Text", name := "Description", next := 208741, root := ~, start := 207516, stop := 208740 ) ], count := [ 5, 3, 23, 1 ], mathmode := "Text", name := "ManSection", next := 208755, root := ~, start := 207414, stop := 208754 ), rec( content := "\n\n ", count := [ 5, 3, 23, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphAllSimpl\ eCircuits" ), content := 0, count := [ 5, 3, 24, 2 ], mathmode := "Text", name := "Attr", next := 208829, root := ~, start := 208776, stop := 208828 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ lists of vertices.", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Returns", next := 208879, root := ~, start := 208832, stop := 208878 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "A", next := 208917, root := ~, start := 208903, stop := 208916 ), rec( content := " is a di\ graph, then ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "C", next := 208968, root := ~, start := 208937, stop := 208967 ), rec( content := "\n re\ turns a list of the ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ imple circuits", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "E", next := 209017, root := ~, start := 208995, stop := 209016 ), rec( content := " in ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "A", next := 209035, root := ~, start := 209021, stop := 209034 ), rec( content := ". ", count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "P", next := 209041, root := ~, start := 209037, stop := 209040 ), rec( content := "\n\n \ See Section ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "Ref", next := 209103, root := ~, start := 209059, stop := 209102 ), rec( content := " for the\ definition\n of a simple circuit, and related notions. Note that a loop is\ a simple\n circuit. ", count := [ 5, 3, 24, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "P", next := 209215, root := ~, start := 209211, stop := 209214 ), rec( content := "\n\n \ For a digraph without multiple edges, a simple circuit is uniquely\n determ\ ined by its subsequence of vertices. However this is not the case for\n a m\ ultidigraph. The attribute ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "C", next := 209434, root := ~, start := 209403, stop := 209433 ), rec( content := " ignores\ \n multiple edges, and identifies a simple circuit using only its subsequen\ ce\n of vertices. For example, although the simple circuits ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v, e, v)", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "M", next := 209597, root := ~, start := 209581, stop := 209596 ), rec( content := " and\n \ ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v, e', v)", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "M", next := 209623, root := ~, start := 209606, stop := 209622 ), rec( content := " (for di\ stinct edges ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "M", next := 209652, root := ~, start := 209644, stop := 209651 ), rec( content := " and ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ '", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "M", next := 209666, root := ~, start := 209657, stop := 209665 ), rec( content := ") are\n \ mathematically distinct, ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "C", next := 209732, root := ~, start := 209701, stop := 209731 ), rec( content := " conside\ rs them to\n be the same. ", count := [ 5, 3, 24, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "P", next := 209772, root := ~, start := 209768, stop := 209771 ), rec( content := "\n\n \ With this approach, a directed circuit of length ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "C", next := 209835, root := ~, start := 209827, stop := 209834 ), rec( content := " can be \ determined\n by a list of its first ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "C", next := 209889, root := ~, start := 209881, stop := 209888 ), rec( content := " vertice\ s. Thus a simple circuit ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ v_1,\n e_1, v_2, e_2, ..., e_{n-1}, v_n, e_{n+1}, v_1)", count := [ 5, 3, 24, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "M", next := 209988, root := ~, start := 209922, stop := 209987 ), rec( content := " can be \ represented as\n the list ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ v_1, \\ldots, v_n]", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "M", next := 210049, root := ~, start := 210024, stop := 210048 ), rec( content := ", or any\ cyclic permutation thereof. For\n each simple circuit of ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "A", next := 210132, root := ~, start := 210118, stop := 210131 ), rec( content := ",\n " , count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits(", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "A", next := 2101\ 80, root := ~, start := 210166, stop := 210179 ), rec( content := ")", count := [ 5, 3, 24, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "C", next := 210185, root := ~, start := 210138, stop := 210184 ), rec( content := " include\ s precisely one such\n list to represent the circuit. ", count := [ 5, 3, 24, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 24, 7 ], mathmode := "Text", name := "P", next := 210254, root := ~, start := 210250, stop := 210253 ), rec( content := "\n\n \ ", count := [ 5, 3, 24, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[], [3], [2, 4], [5, 4], [4]]);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 4 ], [ 4, 5 ], [ 2\ , 3 ] ]\ngap> D := ChainDigraph(10);;\ngap> DigraphAllSimpleCircuits(D);\n[ ]\ \ngap> D := Digraph([[3], [1], [1]]);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 1, 3 ] ]\ngap> D := Digraph([[1,\ 1]]);\n\ngap> DigraphAllSimple\ Circuits(D);\n[ [ 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 1,\ 2, 3 ] ]\n", count := [ 5, 3, 24, 8 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 8 ], mathmode := "Text", name := "Example", next := 210907, root := ~, start := 210260, stop := 210906 ), rec( content := "\n ", count := [ 5, 3, 24, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 24, 3 ], mathmode := "Text", name := "Description", next := 210924, root := ~, start := 208882, stop := 210923 ) ], count := [ 5, 3, 24, 1 ], mathmode := "Text", name := "ManSection", next := 210938, root := ~, start := 208761, stop := 210937 ), rec( content := "\n\n ", count := [ 5, 3, 24, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphLongestS\ impleCircuit" ), content := 0, count := [ 5, 3, 25, 2 ], mathmode := "Text", name := "Attr", next := 211015, root := ~, start := 210959, stop := 211014 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ vertices, or ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "K", next := 211061, root := ~, start := 211050, stop := 211060 ), rec( content := ".", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "Returns", next := 211072, root := ~, start := 211018, stop := 211071 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "A", next := 211110, root := ~, start := 211096, stop := 211109 ), rec( content := " is a di\ graph, then ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphLongestSimpleCircuit", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "C", next := 211164, root := ~, start := 211130, stop := 211163 ), rec( content := "\n re\ turns the longest ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ imple circuit", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "E", next := 211210, root := ~, start := 211189, stop := 211209 ), rec( content := " in ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "A", next := 211228, root := ~, start := 211214, stop := 211227 ), rec( content := ". See Se\ ction\n ", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "Ref", next := 211290, root := ~, start := 211246, stop := 211289 ), rec( content := " for the\ definition of simple\n circuit, and the definition of length for a simple \ circuit.", count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "P", next := 211387, root := ~, start := 211383, stop := 211386 ), rec( content := "\n\n \ This attribute computes\n ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphAllSimpleCircuits(", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "C", next := 211453, root := ~, start := 211421, stop := 211452 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "A", next := 211467, root := ~, start := 211453, stop := 211466 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "C", next := 211475, root := ~, start := 211467, stop := 211474 ), rec( content := " to find\ all the\n simple circuits of ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "A", next := 211529, root := ~, start := 211515, stop := 211528 ), rec( content := ", and re\ turns one of maximal length. A\n simple circuit is represented as a list o\ f vertices, in the same way as\n described in ", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ AllSimpleCircuits" ), content := 0, count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "Ref", next := 211700, root := ~, start := 211662, stop := 211699 ), rec( content := ".", count := [ 5, 3, 25, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "P", next := 211705, root := ~, start := 211701, stop := 211704 ), rec( content := "\n\n \ If ", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "A", next := 211728, root := ~, start := 211714, stop := 211727 ), rec( content := " has no \ simple circuits, then this attribute returns\n ", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "K", next := 211796, root := ~, start := 211785, stop := 211795 ), rec( content := ". If ", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "A", next := 211816, root := ~, start := 211802, stop := 211815 ), rec( content := " has mul\ tiple simple circuits of maximal\n length, then this attribute returns one \ of them.", count := [ 5, 3, 25, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 25, 6 ], mathmode := "Text", name := "P", next := 211913, root := ~, start := 211909, stop := 211912 ), rec( content := "\n\n \ ", count := [ 5, 3, 25, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[], [3], [2, 4], [5, 4], [4]]);;\ngap> DigraphLongestSimpl\ eCircuit(D);\n[ 4, 5 ]\ngap> D := ChainDigraph(10);;\ngap> DigraphLongestSimpl\ eCircuit(D);\nfail\ngap> D := Digraph([[3], [1], [1, 4], [1, 1]]);;\ngap> Digr\ aphLongestSimpleCircuit(D);\n[ 1, 3, 4 ]\ngap> D := GeneralisedPetersenGraph(I\ sMutableDigraph, 4, 1);\n\ngap> Dig\ raphLongestSimpleCircuit(D);\n[ 1, 2, 3, 4, 8, 7, 6, 5 ]\n", count := [ 5, 3, 25, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 25, 7 ], mathmode := "Text", name := "Example", next := 212385, root := ~, start := 211919, stop := 212384 ), rec( content := "\n ", count := [ 5, 3, 25, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 25, 3 ], mathmode := "Text", name := "Description", next := 212402, root := ~, start := 211075, stop := 212401 ) ], count := [ 5, 3, 25, 1 ], mathmode := "Text", name := "ManSection", next := 212416, root := ~, start := 210944, stop := 212415 ), rec( content := "\n\n ", count := [ 5, 3, 25, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vertex" , Name := "DigraphLayers" ), content := 0, count := [ 5, 3, 26, 2 ], mathmode := "Text", name := "Oper", next := 212487, root := ~, start := 212437, stop := 212486 ), rec( attributes := rec( ), content := [ rec( content := "A list.", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "Returns", next := 212516, root := ~, start := 212490, stop := 212515 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns a list ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "K", next := 212578, root := ~, start := 212567, stop := 212577 ), rec( content := " such th\ at ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist[i]", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "K", next := 212603, root := ~, start := 212589, stop := 212602 ), rec( content := " is the\\ n list of vertices whose minimum distance from the vertex ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "A", next := 212684, root := ~, start := 212671, stop := 212683 ), rec( content := " in\n \ ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "A", next := 212706, root := ~, start := 212692, stop := 212705 ), rec( content := " is ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ - 1", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "K", next := 212722, root := ~, start := 212710, stop := 212721 ), rec( content := ". Vertex\ ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "A", next := 212744, root := ~, start := 212731, stop := 212743 ), rec( content := " is\n \ assumed to be at distance ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "K", next := 212786, root := ~, start := 212778, stop := 212785 ), rec( content := " from it\ self.\n ", count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(4);;\ngap> DigraphLayers(D, 1);\n[ [ 1 ], [ 2, 3, 4\ ] ]\n", count := [ 5, 3, 26, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 4 ], mathmode := "Text", name := "Example", next := 212916, root := ~, start := 212804, stop := 212915 ), rec( content := "\n ", count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 26, 3 ], mathmode := "Text", name := "Description", next := 212933, root := ~, start := 212519, stop := 212932 ) ], count := [ 5, 3, 26, 1 ], mathmode := "Text", name := "ManSection", next := 212947, root := ~, start := 212422, stop := 212946 ), rec( content := "\n\n ", count := [ 5, 3, 26, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDegenera\ cy" ), content := 0, count := [ 5, 3, 27, 2 ], mathmode := "Text", name := "Attr", next := 213014, root := ~, start := 212968, stop := 213013 ), rec( attributes := rec( ), content := [ rec( content := "A non-neg\ ative integer, or ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "K", next := 213064, root := ~, start := 213053, stop := 213063 ), rec( content := ".", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Returns", next := 213075, root := ~, start := 213017, stop := 213074 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "A", next := 213113, root := ~, start := 213099, stop := 213112 ), rec( content := " is a sy\ mmetric digraph without multiple edges - see\n ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Ref", next := 213202, root := ~, start := 213170, stop := 213201 ), rec( content := " and ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Ref", next := 213235, root := ~, start := 213207, stop := 213234 ), rec( content := " - then\\ n this attribute returns the degeneracy of ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "A", next := 213302, root := ~, start := 213288, stop := 213301 ), rec( content := ". ", count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "P", next := 213308, root := ~, start := 213304, stop := 213307 ), rec( content := "\n\n \ The degeneracy of a digraph is the least integer ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "C", next := 213371, root := ~, start := 213363, stop := 213370 ), rec( content := " such\n \ that every induced of ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "A", next := 213417, root := ~, start := 213403, stop := 213416 ), rec( content := " contain\ s a vertex whose number of\n neighbours (excluding itself) is at most ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "C", next := 213505, root := ~, start := 213497, stop := 213504 ), rec( content := ". Note t\ hat this means\n that loops are ignored.", count := [ 5, 3, 27, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "P", next := 213559, root := ~, start := 213555, stop := 213558 ), rec( content := "\n\n \ If ", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "A", next := 213582, root := ~, start := 213568, stop := 213581 ), rec( content := " is not \ symmetric or has multiple edges then this\n attribute returns ", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "K", next := 213665, root := ~, start := 213654, stop := 213664 ), rec( content := ".\n " , count := [ 5, 3, 27, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphSymmetricClosure(ChainDigraph(5));;\ngap> DigraphDegeneracy(\ D);\n1\ngap> D := CompleteDigraph(5);;\ngap> DigraphDegeneracy(D);\n4\ngap> D \ := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4], [2], []]);\n\ngap> DigraphDegeneracy(D);\n1\ngap> D := Generalis\ edPetersenGraph(IsMutableDigraph, 10, 3);\n\ngap> DigraphDegeneracy(D);\n3\n", count := [ 5, 3, 27, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 27, 6 ], mathmode := "Text", name := "Example", next := 214120, root := ~, start := 213671, stop := 214119 ), rec( content := "\n ", count := [ 5, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 27, 3 ], mathmode := "Text", name := "Description", next := 214137, root := ~, start := 213078, stop := 214136 ) ], count := [ 5, 3, 27, 1 ], mathmode := "Text", name := "ManSection", next := 214151, root := ~, start := 212953, stop := 214150 ), rec( content := "\n\n ", count := [ 5, 3, 27, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphDegenera\ cyOrdering" ), content := 0, count := [ 5, 3, 28, 2 ], mathmode := "Text", name := "Attr", next := 214226, root := ~, start := 214172, stop := 214225 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ integers, or ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "K", next := 214272, root := ~, start := 214261, stop := 214271 ), rec( content := ".", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "Returns", next := 214283, root := ~, start := 214229, stop := 214282 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "A", next := 214321, root := ~, start := 214307, stop := 214320 ), rec( content := " is a di\ graph for which\n ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDegeneracy(", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "C", next := 214374, root := ~, start := 214349, stop := 214373 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "A", next := 214388, root := ~, start := 214374, stop := 214387 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "C", next := 214396, root := ~, start := 214388, stop := 214395 ), rec( content := " is a no\ n-negative integer\n ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "C", next := 214435, root := ~, start := 214427, stop := 214434 ), rec( content := " - see " , count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Degeneracy" ), content := 0, count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "Ref", next := 214473, root := ~, start := 214442, stop := 214472 ), rec( content := " - then \ this attribute\n returns a degeneracy ordering of the vertices of the verti\ ces of\n ", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "A", next := 214583, root := ~, start := 214569, stop := 214582 ), rec( content := ".", count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "P", next := 214588, root := ~, start := 214584, stop := 214587 ), rec( content := "\n\n \ A degeneracy ordering of ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "A", next := 214633, root := ~, start := 214619, stop := 214632 ), rec( content := " is a li\ st ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rdering", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 214659, root := ~, start := 214644, stop := 214658 ), rec( content := " of the\\ n vertices of ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "A", next := 214697, root := ~, start := 214683, stop := 214696 ), rec( content := " ordered\ such that for any\n position ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 214745, root := ~, start := 214737, stop := 214744 ), rec( content := " of the \ list, the vertex ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rdering[i]", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 214788, root := ~, start := 214770, stop := 214787 ), rec( content := " has at \ most\n ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "k\ ", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "C", next := 214813, root := ~, start := 214805, stop := 214812 ), rec( content := " neighbo\ urs in later position of the list.", count := [ 5, 3, 28, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "P", next := 214859, root := ~, start := 214855, stop := 214858 ), rec( content := "\n\n \ If ", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDegeneracy(", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "C", next := 214893, root := ~, start := 214868, stop := 214892 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "A", next := 214907, root := ~, start := 214893, stop := 214906 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "C", next := 214915, root := ~, start := 214907, stop := 214914 ), rec( content := " returns\ ", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "K", next := 214935, root := ~, start := 214924, stop := 214934 ), rec( content := ",\n t\ hen this attribute returns ", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "K", next := 214980, root := ~, start := 214969, stop := 214979 ), rec( content := ".\n " , count := [ 5, 3, 28, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphSymmetricClosure(ChainDigraph(5));;\ngap> DigraphDegeneracyO\ rdering(D);\n[ 5, 4, 3, 2, 1 ]\ngap> D := CompleteDigraph(5);;\ngap> DigraphDe\ generacyOrdering(D);\n[ 5, 4, 3, 2, 1 ]\ngap> D := Digraph([[1], [2, 4, 5], [3\ , 4], [2, 3, 4], [2], []]);\n\nga\ p> DigraphDegeneracyOrdering(D);\n[ 1, 6, 5, 2, 4, 3 ]\ngap> D := GeneralisedP\ etersenGraph(IsMutableDigraph, 3, 1);\n\ngap> DigraphDegeneracyOrdering(D);\n[ 6, 5, 4, 1, 3, 2 ]\n", count := [ 5, 3, 28, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 28, 6 ], mathmode := "Text", name := "Example", next := 215535, root := ~, start := 214986, stop := 215534 ), rec( content := "\n ", count := [ 5, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 28, 3 ], mathmode := "Text", name := "Description", next := 215552, root := ~, start := 214286, stop := 215551 ) ], count := [ 5, 3, 28, 1 ], mathmode := "Text", name := "ManSection", next := 215566, root := ~, start := 214157, stop := 215565 ), rec( content := "\n\n ", count := [ 5, 3, 28, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "HamiltonianPath\ " ), content := 0, count := [ 5, 3, 29, 2 ], mathmode := "Text", name := "Attr", next := 215631, root := ~, start := 215587, stop := 215630 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "K", next := 215664, root := ~, start := 215653, stop := 215663 ), rec( content := ".", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "Returns", next := 215675, root := ~, start := 215634, stop := 215674 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns a Hamiltonian path if one exists, ", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "K", next := 215749, root := ~, start := 215738, stop := 215748 ), rec( content := " if not.\ ", count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "P", next := 215761, root := ~, start := 215757, stop := 215760 ), rec( content := "\n\n \ A ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ amiltonian path", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "E", next := 215792, root := ~, start := 215769, stop := 215791 ), rec( content := " of a di\ graph with ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "C", next := 215819, root := ~, start := 215811, stop := 215818 ), rec( content := " vertice\ s is directed\n cycle of length ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "C", next := 215869, root := ~, start := 215861, stop := 215868 ), rec( content := ". If ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "A", next := 215888, root := ~, start := 215874, stop := 215887 ), rec( content := " is a di\ graph that contains a\n Hamiltonian path, then this function returns one, d\ escribed in the form\n used by ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ AllSimpleCircuits" ), content := 0, count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "Ref", next := 216044, root := ~, start := 216006, stop := 216043 ), rec( content := ". Note i\ f ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "A", next := 216068, root := ~, start := 216054, stop := 216067 ), rec( content := " has\n \ 0 or 1 vertices, then ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ amiltonianPath", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "C", next := 216121, root := ~, start := 216099, stop := 216120 ), rec( content := " returns\ ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ ]", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "C", next := 216139, root := ~, start := 216130, stop := 216138 ), rec( content := " or\n \ ", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1]", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "C", next := 216157, root := ~, start := 216147, stop := 216156 ), rec( content := ", respec\ tively.", count := [ 5, 3, 29, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 29, 5 ], mathmode := "Text", name := "P", next := 216176, root := ~, start := 216172, stop := 216175 ), rec( content := "\n\n \ The method used in this attribute has the same worst case complexity as\n " , count := [ 5, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Monomorphism" ), content := 0, count := [ 5, 3, 29, 5 ], mathmode := "Text", name := "Ref", next := 216291, root := ~, start := 216258, stop := 216290 ), rec( content := ".\n\n \ ", count := [ 5, 3, 29, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[]]);\n\ngap> Hamil\ tonianPath(D);\n[ 1 ]\ngap> D := Digraph([[2], [1]]);\n\ngap> HamiltonianPath(D);\n[ 1, 2 ]\ngap> D := Digraph([\ [3], [], [2]]);\n\ngap> Hamiltonia\ nPath(D);\nfail\ngap> D := Digraph([[2], [3], [1]]);\n\ngap> HamiltonianPath(D);\n[ 1, 2, 3 ]\ngap> D := General\ isedPetersenGraph(IsMutableDigraph, 5, 2);\n\ngap> HamiltonianPath(D);\nfail\n", count := [ 5, 3, 29, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 29, 6 ], mathmode := "Text", name := "Example", next := 216899, root := ~, start := 216298, stop := 216898 ), rec( content := "\n ", count := [ 5, 3, 29, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 29, 3 ], mathmode := "Text", name := "Description", next := 216916, root := ~, start := 215678, stop := 216915 ) ], count := [ 5, 3, 29, 1 ], mathmode := "Text", name := "ManSection", next := 216930, root := ~, start := 215572, stop := 216929 ), rec( content := "\n\n ", count := [ 5, 3, 29, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "NrSpanningTrees\ " ), content := 0, count := [ 5, 3, 30, 2 ], mathmode := "Text", name := "Attr", next := 216995, root := ~, start := 216951, stop := 216994 ), rec( attributes := rec( ), content := [ rec( content := "An intege\ r.", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "Returns", next := 217028, root := ~, start := 216998, stop := 217027 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns the number of spanning trees of the symmetric digraph\n ", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "A", next := 217129, root := ~, start := 217115, stop := 217128 ), rec( content := ".\n " , count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ rSpanningTrees", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "C", next := 217157, root := ~, start := 217135, stop := 217156 ), rec( content := " will re\ turn an error if ", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "A", next := 217196, root := ~, start := 217182, stop := 217195 ), rec( content := " is not \ a\n symmetric digraph. ", count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "P", next := 217233, root := ~, start := 217229, stop := 217232 ), rec( content := " \n\n \ See ", count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "Ref", next := 217276, root := ~, start := 217244, stop := 217275 ), rec( content := " and\n \ ", count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsUndir\ ectedSpanningTree" ), content := 0, count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "Ref", next := 217324, root := ~, start := 217285, stop := 217323 ), rec( content := " for mor\ e information. \n\n ", count := [ 5, 3, 30, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(5);\n\n\ gap> NrSpanningTrees(D);\n125\ngap> D := DigraphSymmetricClosure(CycleDigraph(\ 24));;\ngap> NrSpanningTrees(D);\n24\ngap> NrSpanningTrees(EmptyDigraph(0));\n\ 0\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> NrSpanningTrees(D);\n1134225\n", count := [ 5, 3, 30, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 30, 5 ], mathmode := "Text", name := "Example", next := 217750, root := ~, start := 217353, stop := 217749 ), rec( content := "\n ", count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 30, 3 ], mathmode := "Text", name := "Description", next := 217767, root := ~, start := 217031, stop := 217766 ) ], count := [ 5, 3, 30, 1 ], mathmode := "Text", name := "ManSection", next := 217781, root := ~, start := 216936, stop := 217780 ), rec( content := "\n\n ", count := [ 5, 3, 30, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, source,\ target", Label := "for a source and target", Name := "DigraphDijkstra" ), content := 0, count := [ 5, 3, 31, 2 ], mathmode := "Text", name := "Oper", next := 217894, root := ~, start := 217802, stop := 217893 ), rec( attributes := rec( Arg := "digraph, source" , Label := "for a source", Name := "DigraphDijkstra\ " ), content := 0, count := [ 5, 3, 31, 4 ], mathmode := "Text", name := "Oper", next := 217970, root := ~, start := 217897, stop := 217969 ), rec( attributes := rec( ), content := [ rec( content := "Two lists\ .", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "Returns", next := 218002, root := ~, start := 217973, stop := 218001 ), rec( attributes := rec( ), content := [ rec( content := "\n If ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 218038, root := ~, start := 218024, stop := 218037 ), rec( content := " is a di\ graph and ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 218069, root := ~, start := 218056, stop := 218068 ), rec( content := " and ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ arget", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 218087, root := ~, start := 218074, stop := 218086 ), rec( content := " are\n \ vertices of ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 218121, root := ~, start := 218107, stop := 218120 ), rec( content := ", then " , count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDijkstra", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "C", next := 218150, root := ~, start := 218128, stop := 218149 ), rec( content := " calcula\ tes the\n length of the shortest path from ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 218214, root := ~, start := 218201, stop := 218213 ), rec( content := " to ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ arget", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 218231, root := ~, start := 218218, stop := 218230 ), rec( content := " and ret\ urns\n two lists. Each element of the first list is the distance of the\n co\ rresponding element from ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 218353, root := ~, start := 218340, stop := 218352 ), rec( content := ". If a \ vertex was not visited in\n the process of calculating the shortest distance \ to ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ arget", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 218454, root := ~, start := 218441, stop := 218453 ), rec( content := " or if\n\ there is no path connecting that vertex with ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 218521, root := ~, start := 218508, stop := 218520 ), rec( content := ", then\n\ the corresponding distance is ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "K", next := 218575, root := ~, start := 218560, stop := 218574 ), rec( content := ". Each \ element of the\n second list gives the previous vertex in the shortest path\n\ from ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 218679, root := ~, start := 218666, stop := 218678 ), rec( content := " to the \ corresponding vertex. For\n ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "A", next := 218729, root := ~, start := 218716, stop := 218728 ), rec( content := " and for\ any vertices that remained unvisited this\n will be ", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "-\ 1", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "C", next := 218801, root := ~, start := 218792, stop := 218800 ), rec( content := ".", count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "P", next := 218806, root := ~, start := 218802, stop := 218805 ), rec( content := "\n\n \ If the optional second argument ", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ arget", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "A", next := 218857, root := ~, start := 218844, stop := 218856 ), rec( content := " is not \ present, then\n ", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphDijkstra", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "C", next := 218905, root := ~, start := 218883, stop := 218904 ), rec( content := " returns\ the shortest path from ", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "A", next := 218950, root := ~, start := 218937, stop := 218949 ), rec( content := " to\n \ every vertex that is reachable from ", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ource", count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "A", next := 219007, root := ~, start := 218994, stop := 219006 ), rec( content := ".\n\n " , count := [ 5, 3, 31, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> mat := [[0, 1, 1], [0, 0, 1], [0, 0, 0]];\n[ [ 0, 1, 1 ], [ 0, 0, 1 ], [\ 0, 0, 0 ] ]\ngap> D := DigraphByAdjacencyMatrix(mat);\n\ngap> DigraphDijkstra(D, 2, 3);\n[ [ infinity, 0, 1 ], \ [ -1, -1, 2 ] ]\ngap> DigraphDijkstra(D, 1, 3);\n[ [ 0, 1, 1 ], [ -1, 1, 1 ] ]\ \ngap> DigraphDijkstra(D, 1, 2);\n[ [ 0, 1, 1 ], [ -1, 1, 1 ] ]\n", count := [ 5, 3, 31, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 3, 31, 7 ], mathmode := "Text", name := "Example", next := 219410, root := ~, start := 219012, stop := 219409 ), rec( content := "\n", count := [ 5, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 31, 5 ], mathmode := "Text", name := "Description", next := 219425, root := ~, start := 218005, stop := 219424 ) ], count := [ 5, 3, 31, 1 ], mathmode := "Text", name := "ManSection", next := 219439, root := ~, start := 217787, stop := 219438 ), rec( content := "\n\n ", count := [ 5, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 219453, root := ~, start := 173430, stop := 219452 ) , rec( content := "\n\n ", count := [ 5, 3, 31, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Cayley graphs of \ groups", count := [ 5, 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 219508, root := ~, start := 219466, stop := 219507 ), rec( content := "\n ", count := [ 5, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "GroupOfCayleyDi\ graph" ), content := 0, count := [ 5, 4, 1, 2 ], mathmode := "Text", name := "Attr", next := 219577, root := ~, start := 219528, stop := 219576 ), rec( attributes := rec( Arg := "digraph", Name := "SemigroupOfCayl\ eyDigraph" ), content := 0, count := [ 5, 4, 1, 4 ], mathmode := "Text", name := "Attr", next := 219633, root := ~, start := 219580, stop := 219632 ), rec( attributes := rec( ), content := [ rec( content := "A group o\ r semigroup.", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "Returns", next := 219676, root := ~, start := 219636, stop := 219675 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "A", next := 219714, root := ~, start := 219700, stop := 219713 ), rec( content := " is an i\ mmutable Cayley graph of a group ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "C", next := 219763, root := ~, start := 219755, stop := 219762 ), rec( content := " and\n \ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "A", next := 219786, root := ~, start := 219772, stop := 219785 ), rec( content := " belongs\ to the category ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsCayle\ yDigraph" ), content := 0, count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "Ref", next := 219840, root := ~, start := 219811, stop := 219839 ), rec( content := ", then\n\ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ roupOfCayleyDigraph", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "C", next := 219878, root := ~, start := 219851, stop := 219877 ), rec( content := " returns\ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "C", next := 219895, root := ~, start := 219887, stop := 219894 ), rec( content := ".\n " , count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "P", next := 219905, root := ~, start := 219901, stop := 219904 ), rec( content := "\n\n \ If ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "A", next := 219928, root := ~, start := 219914, stop := 219927 ), rec( content := " is a Ca\ yley graph of a semigroup ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "C", next := 219970, root := ~, start := 219962, stop := 219969 ), rec( content := " and\n \ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "A", next := 219993, root := ~, start := 219979, stop := 219992 ), rec( content := " belongs\ to the category ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsCayle\ yDigraph" ), content := 0, count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "Ref", next := 220047, root := ~, start := 220018, stop := 220046 ), rec( content := ", then\n\ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ emigroupOfCayleyDigraph", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "C", next := 220089, root := ~, start := 220058, stop := 220088 ), rec( content := " returns\ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "S\ ", count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "C", next := 220106, root := ~, start := 220098, stop := 220105 ), rec( content := ".\n " , count := [ 5, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 4, 1, 7 ], mathmode := "Text", name := "P", next := 220116, root := ~, start := 220112, stop := 220115 ), rec( content := "\n\n \ See also ", count := [ 5, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Generat\ orsOfCayleyDigraph" ), content := 0, count := [ 5, 4, 1, 7 ], mathmode := "Text", name := "Ref", next := 220170, root := ~, start := 220131, stop := 220169 ), rec( content := ".\n " , count := [ 5, 4, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := DihedralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> d\ igraph := CayleyDigraph(G);\n\nga\ p> GroupOfCayleyDigraph(digraph) = G;\ntrue\n", count := [ 5, 4, 1, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 1, 8 ], mathmode := "Text", name := "Example", next := 220402, root := ~, start := 220176, stop := 220401 ), rec( content := "\n ", count := [ 5, 4, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 1, 5 ], mathmode := "Text", name := "Description", next := 220419, root := ~, start := 219679, stop := 220418 ) ], count := [ 5, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 220433, root := ~, start := 219513, stop := 220432 ), rec( content := "\n\n ", count := [ 5, 4, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "GeneratorsOfCay\ leyDigraph" ), content := 0, count := [ 5, 4, 2, 2 ], mathmode := "Text", name := "Attr", next := 220508, root := ~, start := 220454, stop := 220507 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ generators.", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "Returns", next := 220551, root := ~, start := 220511, stop := 220550 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "A", next := 220589, root := ~, start := 220575, stop := 220588 ), rec( content := " is an i\ mmutable Cayley graph of a group or semigroup with\n respect to a set of ge\ nerators ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "C", next := 220694, root := ~, start := 220683, stop := 220693 ), rec( content := " and ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "A", next := 220713, root := ~, start := 220699, stop := 220712 ), rec( content := " belongs\ to\n the category ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsCayle\ yDigraph" ), content := 0, count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "Ref", next := 220771, root := ~, start := 220742, stop := 220770 ), rec( content := ", then\n\ ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eneratorsOfCayleyDigraph", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "C", next := 220814, root := ~, start := 220782, stop := 220813 ), rec( content := " return \ the list of generators ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ens", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "C", next := 220856, root := ~, start := 220845, stop := 220855 ), rec( content := "\n ov\ er which ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "A", next := 220886, root := ~, start := 220872, stop := 220885 ), rec( content := " is defi\ ned.\n ", count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "P", next := 220907, root := ~, start := 220903, stop := 220906 ), rec( content := "\n\n \ See also ", count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "GroupOf\ CayleyDigraph" ), content := 0, count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "Ref", next := 220956, root := ~, start := 220922, stop := 220955 ), rec( content := " \n o\ r ", count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Semigro\ upOfCayleyDigraph" ), content := 0, count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "Ref", next := 221003, root := ~, start := 220965, stop := 221002 ), rec( content := ".\n " , count := [ 5, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := DihedralGroup(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> d\ igraph := CayleyDigraph(G);\n\nga\ p> GeneratorsOfCayleyDigraph(digraph) = GeneratorsOfGroup(G);\ntrue\ngap> digr\ aph := CayleyDigraph(G, [()]);\n\n\ gap> GeneratorsOfCayleyDigraph(digraph) = [()];\ntrue", count := [ 5, 4, 2, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 4, 2, 5 ], mathmode := "Text", name := "Example", next := 221396, root := ~, start := 221009, stop := 221395 ), rec( content := "\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 221413, root := ~, start := 220554, stop := 221412 ) ], count := [ 5, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 221427, root := ~, start := 220439, stop := 221426 ), rec( content := "\n\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 221441, root := ~, start := 219457, stop := 221440 ) , rec( content := "\n\n ", count := [ 5, 4, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Associated semigr\ oups", count := [ 5, 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 221494, root := ~, start := 221454, stop := 221493 ), rec( content := "\n ", count := [ 5, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filt, digraph", Label := "for a filter a\ nd a digraph", Name := "AsSemigroup" ), content := 0, count := [ 5, 5, 1, 2 ], mathmode := "Text", name := "Oper", next := 221605, root := ~, start := 221518, stop := 221604 ), rec( attributes := rec( Arg := "filt, digraph", Name := "AsMonoid" ), content := 0, count := [ 5, 5, 1, 4 ], mathmode := "Text", name := "Oper", next := 221653, root := ~, start := 221610, stop := 221652 ), rec( attributes := rec( ), content := [ rec( content := "A semilat\ tice of partial perms.", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Returns", next := 221708, root := ~, start := 221658, stop := 221707 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ he operation ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "C", next := 221765, root := ~, start := 221747, stop := 221764 ), rec( content := " require\ s that ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 221791, root := ~, start := 221780, stop := 221790 ), rec( content := " be equa\ l to\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsPartialPermSemigroup" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 221861, root := ~, start := 221810, stop := 221860 ), rec( content := ".\n \ If ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 221886, root := ~, start := 221872, stop := 221885 ), rec( content := " is a\n \ ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsJoinS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 221936, root := ~, start := 221898, stop := 221935 ), rec( content := " or ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsLatti\ ceDigraph" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 221970, root := ~, start := 221940, stop := 221969 ), rec( content := "\n \ then ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "C", next := 222000, root := ~, start := 221982, stop := 221999 ), rec( content := " returns\ a semigroup of partial perms which is\n isomorphic to the semigroup whos\ e elements are the vertices of\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 222136, root := ~, start := 222122, stop := 222135 ), rec( content := " with th\ e binary operation\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Partial\ OrderDigraphJoinOfVertices" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 222216, root := ~, start := 222169, stop := 222215 ), rec( content := ".\n \ If ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 222241, root := ~, start := 222227, stop := 222240 ), rec( content := "\n \ satisfies ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMeetS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 222297, root := ~, start := 222259, stop := 222296 ), rec( content := " but not\ \n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsJoinS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 222350, root := ~, start := 222312, stop := 222349 ), rec( content := " then\n \ ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "C", next := 222380, root := ~, start := 222362, stop := 222379 ), rec( content := " returns\ a semigroup of partial perms which is\n isomorphic to the semigroup whos\ e elements are the vertices of\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "A", next := 222516, root := ~, start := 222502, stop := 222515 ), rec( content := " with th\ e binary operation\n ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Partial\ OrderDigraphMeetOfVertices" ), content := 0, count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Ref", next := 222596, root := ~, start := 222549, stop := 222595 ), rec( content := ".\n \ ", count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "P", next := 222608, root := ~, start := 222604, stop := 222607 ), rec( content := "\n\n \ The operation ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sMonoid", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "C", next := 222645, root := ~, start := 222630, stop := 222644 ), rec( content := " behaves\ similarly to ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "C", next := 222685, root := ~, start := 222667, stop := 222684 ), rec( content := "\n \ except that ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "A", next := 222715, root := ~, start := 222704, stop := 222714 ), rec( content := " may als\ o be equal to\n ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsPartialPermMonoid" ), content := 0, count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "Ref", next := 222791, root := ~, start := 222743, stop := 222790 ), rec( content := ",\n \ ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "A", next := 222813, root := ~, start := 222799, stop := 222812 ), rec( content := " must sa\ tisfy ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsLatti\ ceDigraph" ), content := 0, count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "Ref", next := 222857, root := ~, start := 222827, stop := 222856 ), rec( content := ", and th\ e\n output satisfies ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsMonoid" ), content := 0, count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "Ref", next := 222927, root := ~, start := 222890, stop := 222926 ), rec( content := ". ", count := [ 5, 5, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "P", next := 222933, root := ~, start := 222929, stop := 222932 ), rec( content := "\n\n \ The output of both of these operations is guaranteed to be of minimal\n \ degree (see ", count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DegreeO\ fPartialPermSemigroup", BookName := "ref" ), content := 0, count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "Ref", next := 223086, root := ~, start := 223029, stop := 223085 ), rec( content := ").\n \ Furthermore the ", count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Generat\ orsOfSemigroup", BookName := "ref" ), content := 0, count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "Ref", next := 223161, root := ~, start := 223111, stop := 223160 ), rec( content := " of\n \ the output is guaranteed to be the unique generating set of minimal size.\n\ ", count := [ 5, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> di := Digraph([[1], [1, 2], [1, 3], [1, 4], [1, 2, 3, 5]]);\n\ngap> S := AsSemigroup(IsPartialPermSemigro\ up, di);\n\ngap> ForAll(El\ ements(S), IsIdempotent);\ntrue\ngap> IsInverseSemigroup(S);\ntrue\ngap> Size(\ S);\n5\ngap> di := Digraph([[1], [1, 2], [1, 2, 3]]);\n\ngap> M := AsMonoid(IsPartialPermMonoid, di);\n\ngap> Size(M);\n3", count := [ 5, 5, 1, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 5, 1, 8 ], mathmode := "Text", name := "Example", next := 223793, root := ~, start := 223251, stop := 223792 ), rec( content := "\n ", count := [ 5, 5, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 1, 5 ], mathmode := "Text", name := "Description", next := 223812, root := ~, start := 221713, stop := 223811 ) ], count := [ 5, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 223828, root := ~, start := 221501, stop := 223827 ), rec( content := "\n\n ", count := [ 5, 5, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filt, Y, gps, ho\ ms", Label := "for a filter, semilattice digraph, and two lists", Name := "AsSemigroup" ), content := 0, count := [ 5, 5, 2, 2 ], mathmode := "Text", name := "Oper", next := 223967, root := ~, start := 223853, stop := 223966 ), rec( attributes := rec( ), content := [ rec( content := " A Cliffo\ rd semigroup of partial perms. ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 224031, root := ~, start := 223972, stop := 224030 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ he operation ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 224088, root := ~, start := 224070, stop := 224087 ), rec( content := " require\ s that ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 224114, root := ~, start := 224103, stop := 224113 ), rec( content := " be equa\ l to\n ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Filt := "IsPartialPermSemigroup" ), content := 0, count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 224184, root := ~, start := 224133, stop := 224183 ), rec( content := ".\n \ If ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Y\ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 224203, root := ~, start := 224195, stop := 224202 ), rec( content := " is a ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsJoinS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 224247, root := ~, start := 224209, stop := 224246 ), rec( content := " or\n \ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMeetS\ emilatticeDigraph" ), content := 0, count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 224295, root := ~, start := 224257, stop := 224294 ), rec( content := ",\n \ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "g\ ps", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 224313, root := ~, start := 224303, stop := 224312 ), rec( content := " is a li\ st of groups corresponding to each vertex, and\n ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ oms", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 224385, root := ~, start := 224374, stop := 224384 ), rec( content := " is a li\ st containing for each edge ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "(\ i, j)", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 224434, root := ~, start := 224421, stop := 224433 ), rec( content := " in\n \ the transitive reduction of ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "A", next := 224486, root := ~, start := 224472, stop := 224485 ), rec( content := " a tripl\ e ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j, hom]", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 224514, root := ~, start := 224496, stop := 224513 ), rec( content := "\n \ where ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ om", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 224537, root := ~, start := 224527, stop := 224536 ), rec( content := " is a gr\ oup homomorphism from ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ps[i]", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 224580, root := ~, start := 224567, stop := 224579 ), rec( content := " to\n \ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ps[j]", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 224603, root := ~, start := 224590, stop := 224602 ), rec( content := ", and th\ e diagram of homomorphisms commutes, then\n ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ sSemigroup", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "C", next := 224677, root := ~, start := 224659, stop := 224676 ), rec( content := " returns\ a semigroup of partial perms which is\n isomorphic to the strong semilat\ tice of groups ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ [Y; gps; homs]", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "M", next := 224799, root := ~, start := 224777, stop := 224798 ), rec( content := ".\n \ ", count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G1 := AlternatingGroup(4);;\ngap> G2 := SymmetricGroup(2);;\ngap> G3 := \ SymmetricGroup(3);;\ngap> gr := Digraph([[1, 3], [2, 3], [3]]);;\ngap> sgn := \ function(x)\n> if SignPerm(x) = 1 then\n> return ();\n> fi;\n> return (1, 2);\ \n> end;;\ngap> hom13 := GroupHomomorphismByFunction(G1, G3, sgn);;\ngap> hom2\ 3 := GroupHomomorphismByFunction(G2, G3, sgn);;\ngap> T := AsSemigroup(IsParti\ alPermSemigroup,\n> gr,\n> [G1, G2, G3], [[1, 3, hom13], [2, 3, hom23]]);;\nga\ p> Size(T);\n20\ngap> D := GreensDClasses(T);;\ngap> List(D, x -> Size(x));\n[\ 6, 12, 2 ]\n", count := [ 5, 5, 2, 4 ], name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 4 ], mathmode := "Text", name := "Example", next := 225376, root := ~, start := 224807, stop := 225375 ), rec( content := "\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 225395, root := ~, start := 224036, stop := 225394 ) ], count := [ 5, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 225411, root := ~, start := 223836, stop := 225410 ), rec( content := "\n\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 225425, root := ~, start := 221445, stop := 225424 ) , rec( content := "\n\n ", count := [ 5, 5, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Planarity", count := [ 5, 6, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 0, 2 ], mathmode := "Text", name := "Heading", next := 225466, root := ~, start := 225438, stop := 225465 ), rec( content := "\n ", count := [ 5, 6, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "KuratowskiPlana\ rSubdigraph" ), content := 0, count := [ 5, 6, 1, 2 ], mathmode := "Text", name := "Attr", next := 225541, root := ~, start := 225486, stop := 225540 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "K", next := 225574, root := ~, start := 225563, stop := 225573 ), rec( content := ".", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "Returns", next := 225585, root := ~, start := 225544, stop := 225584 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "K\ uratowskiPlanarSubdigraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 225639, root := ~, start := 225606, stop := 225638 ), rec( content := " returns\ the immutable list of\n lists of out-neighbours of a (not necessarily indu\ ced) subdigraph of the\n digraph ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "A", next := 225773, root := ~, start := 225759, stop := 225772 ), rec( content := " that wi\ tnesses the fact that ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "A", next := 225817, root := ~, start := 225803, stop := 225816 ), rec( content := " is not\\ n planar, or ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "K", next := 225851, root := ~, start := 225840, stop := 225850 ), rec( content := " if ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "A", next := 225869, root := ~, start := 225855, stop := 225868 ), rec( content := " is plan\ ar. In other words,\n ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "K\ uratowskiPlanarSubdigraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 225935, root := ~, start := 225902, stop := 225934 ), rec( content := " returns\ the out-neighbours of a\n subdigraph of ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "A", next := 226000, root := ~, start := 225986, stop := 225999 ), rec( content := " that is\ homeomorphic to the complete graph\n with ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "5\ ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 226061, root := ~, start := 226053, stop := 226060 ), rec( content := " vertice\ s, or to the complete bipartite graph with vertex sets\n of sizes ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 226145, root := ~, start := 226137, stop := 226144 ), rec( content := " and ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "C", next := 226158, root := ~, start := 226150, stop := 226157 ), rec( content := ". ", count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "P", next := 226164, root := ~, start := 226160, stop := 226163 ), rec( content := "\n \n\ The directions and multiplicities of any edges in ", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "A", next := 226238, root := ~, start := 226224, stop := 226237 ), rec( content := " are\n \ ignored when considering whether or not ", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "A", next := 226301, root := ~, start := 226287, stop := 226300 ), rec( content := " is plan\ ar. ", count := [ 5, 6, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "P", next := 226317, root := ~, start := 226313, stop := 226316 ), rec( content := "\n\n \ See also ", count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPlana\ rDigraph" ), content := 0, count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "Ref", next := 226361, root := ~, start := 226332, stop := 226360 ), rec( content := "\n an\ d ", count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Subdigr\ aphHomeomorphicToK33" ), content := 0, count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "Ref", next := 226411, root := ~, start := 226370, stop := 226410 ), rec( content := ".\n " , count := [ 5, 6, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "P", next := 226421, root := ~, start := 226417, stop := 226420 ), rec( content := "\n \n\ This method uses the reference implementation in\n ", count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 1, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "Cite", next := 226583, root := ~, start := 226565, stop := 226582 ), rec( content := ".\n\n", count := [ 5, 6, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiPlanarSubdigraph(D);\nfail\ngap> D := Digra\ ph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3\ , 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, \ 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> IsPlanarDigraph(D);\nfalse\ngap> KuratowskiPlanarSubdigraph(D);\n[ [\ 2, 9, 7 ], [ 3 ], [ 6 ], [ 5, 9 ], [ 6 ], [ ], [ 4 ], \n [ 7, 9, 3 ], [ ],\ [ ] ]\ngap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\ \n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiPlanarSubdigraph(D)\ ;\nfail\ngap> D := Digraph(IsMutableDigraph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6\ , 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9,\ 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ \ngap> IsPlanarDigraph(D);\nfalse\ \ngap> KuratowskiPlanarSubdigraph(D);\n[ [ 2, 9, 7 ], [ 3 ], [ 6 ], [ 5, 9 ], \ [ 6 ], [ ], [ 4 ], \n [ 7, 9, 3 ], [ ], [ ] ]\n", count := [ 5, 6, 1, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 1, 7 ], mathmode := "Text", name := "Example", next := 227885, root := ~, start := 226586, stop := 227884 ), rec( content := "\n ", count := [ 5, 6, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 1, 3 ], mathmode := "Text", name := "Description", next := 227902, root := ~, start := 225588, stop := 227901 ) ], count := [ 5, 6, 1, 1 ], mathmode := "Text", name := "ManSection", next := 227916, root := ~, start := 225471, stop := 227915 ), rec( content := "\n\n ", count := [ 5, 6, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "KuratowskiOuter\ PlanarSubdigraph" ), content := 0, count := [ 5, 6, 2, 2 ], mathmode := "Text", name := "Attr", next := 227997, root := ~, start := 227937, stop := 227996 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "K", next := 228030, root := ~, start := 228019, stop := 228029 ), rec( content := ".", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "Returns", next := 228041, root := ~, start := 228000, stop := 228040 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "K\ uratowskiOuterPlanarSubdigraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 228100, root := ~, start := 228062, stop := 228099 ), rec( content := " returns\ the immutable list of\n immutable lists of out-neighbours of a (not necess\ arily induced)\n subdigraph of the digraph ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "A", next := 228244, root := ~, start := 228230, stop := 228243 ), rec( content := " that wi\ tnesses the fact\n that ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "A", next := 228292, root := ~, start := 228278, stop := 228291 ), rec( content := " is not \ outer planar, or ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "K", next := 228328, root := ~, start := 228317, stop := 228327 ), rec( content := " if\n \ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "A", next := 228350, root := ~, start := 228336, stop := 228349 ), rec( content := " is oute\ r planar. In other words,\n ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "K\ uratowskiOuterPlanarSubdigraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 228427, root := ~, start := 228389, stop := 228426 ), rec( content := " returns\ the out-neighbours of a\n subdigraph of ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "A", next := 228492, root := ~, start := 228478, stop := 228491 ), rec( content := " that is\ homeomorphic to the complete graph\n with ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "4\ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 228553, root := ~, start := 228545, stop := 228552 ), rec( content := " vertice\ s, or to the complete bipartite graph with vertex sets\n of sizes ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 228637, root := ~, start := 228629, stop := 228636 ), rec( content := " and ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "C", next := 228650, root := ~, start := 228642, stop := 228649 ), rec( content := ". ", count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "P", next := 228656, root := ~, start := 228652, stop := 228655 ), rec( content := "\n \n\ The directions and multiplicities of any edges in ", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "A", next := 228730, root := ~, start := 228716, stop := 228729 ), rec( content := " are\n \ ignored when considering whether or not ", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "A", next := 228793, root := ~, start := 228779, stop := 228792 ), rec( content := " is oute\ r planar. \n ", count := [ 5, 6, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "P", next := 228820, root := ~, start := 228816, stop := 228819 ), rec( content := "\n\n \ See also \n ", count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOuter\ PlanarDigraph" ), content := 0, count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "Ref", next := 228874, root := ~, start := 228840, stop := 228873 ), rec( content := ",\n " , count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Subdigr\ aphHomeomorphicToK4" ), content := 0, count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "Ref", next := 228920, root := ~, start := 228880, stop := 228919 ), rec( content := ", and \n\ ", count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Subdigr\ aphHomeomorphicToK23" ), content := 0, count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "Ref", next := 228972, root := ~, start := 228931, stop := 228971 ), rec( content := ".\n " , count := [ 5, 6, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "P", next := 228982, root := ~, start := 228978, stop := 228981 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 2, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "Cite", next := 229140, root := ~, start := 229122, stop := 229139 ), rec( content := ".\n\n", count := [ 5, 6, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ 3, 5, 10 ], \ [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ]\ ]\ngap> D := Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2,\ 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9,\ 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> IsOuterPlanarDigraph(D);\nfalse\ngap> KuratowskiO\ uterPlanarSubdigraph(D);\n[ [ ], [ ], [ ], [ 8, 9 ], [ ], [ ], [ 9, 4 ], \ [ 7, 9 ], [ ], \n [ ] ]\ngap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [\ 8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1,\ 2], [5, 9]]);\n\ngap> KuratowskiO\ uterPlanarSubdigraph(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11 ], \ [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMutableDigra\ ph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, 2\ , 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7\ , 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> IsOuterPlanarDigraph(D);\nfalse\ngap> KuratowskiOuterPlanarSubdigrap\ h(D);\n[ [ ], [ ], [ ], [ 8, 9 ], [ ], [ ], [ 9, 4 ], [ 7, 9 ], [ ], \n \ [ ] ]", count := [ 5, 6, 2, 7 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 7 ], mathmode := "Text", name := "Example", next := 230633, root := ~, start := 229143, stop := 230632 ), rec( content := "\n ", count := [ 5, 6, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 2, 3 ], mathmode := "Text", name := "Description", next := 230650, root := ~, start := 228044, stop := 230649 ) ], count := [ 5, 6, 2, 1 ], mathmode := "Text", name := "ManSection", next := 230664, root := ~, start := 227922, stop := 230663 ), rec( content := "\n\n ", count := [ 5, 6, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "PlanarEmbedding\ " ), content := 0, count := [ 5, 6, 3, 2 ], mathmode := "Text", name := "Attr", next := 230729, root := ~, start := 230685, stop := 230728 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "K", next := 230762, root := ~, start := 230751, stop := 230761 ), rec( content := ".", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "Returns", next := 230773, root := ~, start := 230732, stop := 230772 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "A", next := 230811, root := ~, start := 230797, stop := 230810 ), rec( content := " is a pl\ anar digraph, then \n ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ lanarEmbedding", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "C", next := 230865, root := ~, start := 230843, stop := 230864 ), rec( content := " returns\ the immutable list of lists of\n out-neighbours of a subdigraph of ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "A", next := 230957, root := ~, start := 230943, stop := 230956 ), rec( content := " such th\ at each vertex's\n neighbours are given in clockwise order. If ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "A", next := 231044, root := ~, start := 231030, stop := 231043 ), rec( content := " is not \ planar,\n then ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "K", next := 231080, root := ~, start := 231069, stop := 231079 ), rec( content := " is retu\ rned. ", count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "P", next := 231098, root := ~, start := 231094, stop := 231097 ), rec( content := "\n \n\ The directions and multiplicities of any edges in ", count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "A", next := 231172, root := ~, start := 231158, stop := 231171 ), rec( content := " are\n \ ignored by ", count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ lanarEmbedding", count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "C", next := 231214, root := ~, start := 231192, stop := 231213 ), rec( content := ".\n " , count := [ 5, 6, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 3, 5 ], mathmode := "Text", name := "P", next := 231224, root := ~, start := 231220, stop := 231223 ), rec( content := "\n\n \ See also \n ", count := [ 5, 6, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPlana\ rDigraph" ), content := 0, count := [ 5, 6, 3, 5 ], mathmode := "Text", name := "Ref", next := 231273, root := ~, start := 231244, stop := 231272 ), rec( content := ".\n " , count := [ 5, 6, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "P", next := 231283, root := ~, start := 231279, stop := 231282 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 3, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 3, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "Cite", next := 231441, root := ~, start := 231423, stop := 231440 ), rec( content := ".\n\n", count := [ 5, 6, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> PlanarEmbedding(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ \ 4 ], [ 6 ], [ 11, 7 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D :=\ Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1\ , 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, \ 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> PlanarEmbedding(D);\nfail\ngap> D := Digraph(IsMutableDigraph,\ [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7\ ], [2, 11], [1, 2], [5, 9]]);\n\ng\ ap> PlanarEmbedding(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11, 7 ]\ , [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMutableDig\ raph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1,\ 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5,\ 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> PlanarEmbedding(D);\nfail\n", count := [ 5, 6, 3, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 3, 7 ], mathmode := "Text", name := "Example", next := 232659, root := ~, start := 231444, stop := 232658 ), rec( content := "\n ", count := [ 5, 6, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 3, 3 ], mathmode := "Text", name := "Description", next := 232676, root := ~, start := 230776, stop := 232675 ) ], count := [ 5, 6, 3, 1 ], mathmode := "Text", name := "ManSection", next := 232690, root := ~, start := 230670, stop := 232689 ), rec( content := "\n\n ", count := [ 5, 6, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "OuterPlanarEmbe\ dding" ), content := 0, count := [ 5, 6, 4, 2 ], mathmode := "Text", name := "Attr", next := 232760, root := ~, start := 232711, stop := 232759 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "K", next := 232793, root := ~, start := 232782, stop := 232792 ), rec( content := ".", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "Returns", next := 232804, root := ~, start := 232763, stop := 232803 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "A", next := 232842, root := ~, start := 232828, stop := 232841 ), rec( content := " is an o\ uter planar digraph, then\n ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ uterPlanarEmbedding", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "C", next := 232907, root := ~, start := 232880, stop := 232906 ), rec( content := " returns\ the immutable list of lists\n of out-neighbours of a subdigraph of ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "A", next := 232999, root := ~, start := 232985, stop := 232998 ), rec( content := " such th\ at each\n vertex's neighbours are given in clockwise order. If ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "A", next := 233086, root := ~, start := 233072, stop := 233085 ), rec( content := " is not\\ n outer planar, then ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "K", next := 233128, root := ~, start := 233117, stop := 233127 ), rec( content := " is retu\ rned. ", count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "P", next := 233146, root := ~, start := 233142, stop := 233145 ), rec( content := "\n \n\ The directions and multiplicities of any edges in ", count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "A", next := 233220, root := ~, start := 233206, stop := 233219 ), rec( content := " are\n \ ignored by ", count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ uterPlanarEmbedding", count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "C", next := 233267, root := ~, start := 233240, stop := 233266 ), rec( content := ".\n " , count := [ 5, 6, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 4, 5 ], mathmode := "Text", name := "P", next := 233277, root := ~, start := 233273, stop := 233276 ), rec( content := "\n\n \ See also ", count := [ 5, 6, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOuter\ PlanarDigraph" ), content := 0, count := [ 5, 6, 4, 5 ], mathmode := "Text", name := "Ref", next := 233326, root := ~, start := 233292, stop := 233325 ), rec( content := ".\n " , count := [ 5, 6, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "P", next := 233336, root := ~, start := 233332, stop := 233335 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 4, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 4, 6 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "Cite", next := 233494, root := ~, start := 233476, stop := 233493 ), rec( content := ".\n\n", count := [ 5, 6, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> D := Digraph([[2\ , 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3, 4, 6\ , 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2\ , 3, 5, 6, 7, 8], [3, 5]]);\n\ng\ ap> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbedding(CompleteBipartit\ eDigraph(2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\ngap> D := Digraph(IsMuta\ bleDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6\ , 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> D := Digraph(IsMutableDigr\ aph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, \ 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, \ 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbedding(CompleteB\ ipartiteDigraph(2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\n", count := [ 5, 6, 4, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 4, 7 ], mathmode := "Text", name := "Example", next := 234732, root := ~, start := 233497, stop := 234731 ), rec( content := "\n ", count := [ 5, 6, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 4, 3 ], mathmode := "Text", name := "Description", next := 234749, root := ~, start := 232807, stop := 234748 ) ], count := [ 5, 6, 4, 1 ], mathmode := "Text", name := "ManSection", next := 234763, root := ~, start := 232696, stop := 234762 ), rec( content := "\n\n ", count := [ 5, 6, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK23" ), content := 0, count := [ 5, 6, 5, 2 ], mathmode := "Text", name := "Attr", next := 234840, root := ~, start := 234784, stop := 234839 ), rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK33" ), content := 0, count := [ 5, 6, 5, 4 ], mathmode := "Text", name := "Attr", next := 234899, root := ~, start := 234843, stop := 234898 ), rec( attributes := rec( Arg := "digraph", Name := "SubdigraphHomeo\ morphicToK4" ), content := 0, count := [ 5, 6, 5, 6 ], mathmode := "Text", name := "Attr", next := 234957, root := ~, start := 234902, stop := 234956 ), rec( attributes := rec( ), content := [ rec( content := "A list or\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "K", next := 234990, root := ~, start := 234979, stop := 234989 ), rec( content := ".", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "Returns", next := 235001, root := ~, start := 234960, stop := 235000 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se attributes return the immutable list of lists of\n out-neighbours of a s\ ubdigraph of the digraph ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "A", next := 235141, root := ~, start := 235127, stop := 235140 ), rec( content := " which i\ s\n homeomorphic to one of the following:\n the complete bipartite graph\ with vertex sets of sizes ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 235260, root := ~, start := 235252, stop := 235259 ), rec( content := " and\n \ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 235277, root := ~, start := 235269, stop := 235276 ), rec( content := "; the co\ mplete bipartite graph with vertex sets of sizes ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 235342, root := ~, start := 235334, stop := 235341 ), rec( content := "\n an\ d ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 235359, root := ~, start := 235351, stop := 235358 ), rec( content := "; or the\ complete graph with ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "4\ ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "C", next := 235396, root := ~, start := 235388, stop := 235395 ), rec( content := " vertice\ s. If\n ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "A", next := 235428, root := ~, start := 235414, stop := 235427 ), rec( content := " has no \ such subdigraphs, then ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "K", next := 235470, root := ~, start := 235459, stop := 235469 ), rec( content := " is retu\ rned. \n ", count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "P", next := 235493, root := ~, start := 235489, stop := 235492 ), rec( content := "\n\n \ See also ", count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPlana\ rDigraph" ), content := 0, count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "Ref", next := 235537, root := ~, start := 235508, stop := 235536 ), rec( content := " and ", count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOuter\ PlanarDigraph" ), content := 0, count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "Ref", next := 235582, root := ~, start := 235542, stop := 235581 ), rec( content := " for mor\ e details.", count := [ 5, 6, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "P", next := 235604, root := ~, start := 235600, stop := 235603 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 5, 6, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 9 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 5, 6, 5, 9 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 5, 6, 5, 9 ], name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 9 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "Cite", next := 235762, root := ~, start := 235744, stop := 235761 ), rec( content := ".\n\n", count := [ 5, 6, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], [1, 7, 11], \n> [4\ , 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\n[ [ 3, 5, 10 ], [ 9, \ 8, 10 ], [ 4 ], [ 6 ], [ 7, 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ \ngap> SubdigraphHomeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [\ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(\ [[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], [1, 11], \n> [4, 7], [6, 8], [2, 7],\ [2, 11], [1, 2], [5, 9]]);\n\ng\ ap> SubdigraphHomeomorphicToK4(D);\nfail\ngap> SubdigraphHomeomorphicToK23(D);\ \n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], \ [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\nfail\ngap> Subdigr\ aphHomeomorphicToK23(NullDigraph(0));\nfail\ngap> SubdigraphHomeomorphicToK33(\ CompleteDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK33(CompleteBipartite\ Digraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ ], [ ], [ ] ]\n\ gap> SubdigraphHomeomorphicToK4(CompleteDigraph(3));\nfail\ngap> D := Digraph(\ IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, \ 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\n[ [ 3, 5, 10 ], [ 9, 8, 1\ 0 ], [ 4 ], [ 6 ], [ 7, 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\nga\ p> SubdigraphHomeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ]\ , [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMu\ tableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 11], [4, 7], [6,\ 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\nfail\ngap> SubdigraphHomeomorphi\ cToK23(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \ \n [ ], [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\nfail\nga\ p> SubdigraphHomeomorphicToK23(NullDigraph(0));\nfail\ngap> SubdigraphHomeomor\ phicToK33(CompleteDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK33(Complet\ eBipartiteDigraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ ], [ ]\ , [ ] ]\ngap> SubdigraphHomeomorphicToK4(CompleteDigraph(3));\nfail\n", count := [ 5, 6, 5, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 5, 6, 5, 10 ], mathmode := "Text", name := "Example", next := 238074, root := ~, start := 235765, stop := 238073 ), rec( content := "\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 5, 7 ], mathmode := "Text", name := "Description", next := 238091, root := ~, start := 235004, stop := 238090 ) ], count := [ 5, 6, 5, 1 ], mathmode := "Text", name := "ManSection", next := 238105, root := ~, start := 234769, stop := 238104 ), rec( content := "\n\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 6, 0, 1 ], mathmode := "Text", name := "Section", next := 238119, root := ~, start := 225429, stop := 238118 ) , rec( content := "\n\n", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 5, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 238131, root := ~, start := 132079, stop := 238130 ), rec( content := "\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " attributes, operations ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 5, 6, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Properties of digraphs" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Properties of digraphs", count := [ 6, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 238248, root := ~, start := 238207, stop := 238247 ) , rec( content := "\n\n ", count := [ 6, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Edge properties", count := [ 6, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 238295, root := ~, start := 238261, stop := 238294 ), rec( content := "\n ", count := [ 6, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphHasLoops\ " ), content := 0, count := [ 6, 1, 1, 2 ], mathmode := "Text", name := "Prop", next := 238359, root := ~, start := 238315, stop := 238358 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "K", next := 238382, root := ~, start := 238371, stop := 238381 ), rec( content := " or ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "K", next := 238398, root := ~, start := 238386, stop := 238397 ), rec( content := ".", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 238409, root := ~, start := 238362, stop := 238408 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "K", next := 238449, root := ~, start := 238438, stop := 238448 ), rec( content := " if the \ digraph ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "A", next := 238479, root := ~, start := 238465, stop := 238478 ), rec( content := " has loo\ ps, and\n ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "K", next := 238511, root := ~, start := 238499, stop := 238510 ), rec( content := " if it d\ oes not. A loop is an edge with equal source and range.\n ", count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "P", next := 238583, root := ~, start := 238579, stop := 238582 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "A", next := 238625, root := ~, start := 238611, stop := 238624 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [2]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 2 ] ]\ngap> DigraphHasLo\ ops(D);\ntrue\ngap> D := Digraph([[2, 3], [1], [2]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], \ [ 3, 2 ] ]\ngap> DigraphHasLoops(D);\nfalse\ngap> D := CompleteDigraph(IsMutab\ leDigraph, 4);\n\ngap> DigraphHasLo\ ops(D);\nfalse", count := [ 6, 1, 1, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 5 ], mathmode := "Text", name := "Example", next := 239220, root := ~, start := 238722, stop := 239219 ), rec( content := "\n ", count := [ 6, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 239237, root := ~, start := 238412, stop := 239236 ) ], count := [ 6, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 239251, root := ~, start := 238300, stop := 239250 ), rec( content := "\n\n ", count := [ 6, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsAntiSymmetric\ Digraph" ), content := 0, count := [ 6, 1, 2, 2 ], mathmode := "Text", name := "Prop", next := 239323, root := ~, start := 239272, stop := 239322 ), rec( attributes := rec( Arg := "digraph", Name := "IsAntisymmetric\ Digraph" ), content := 0, count := [ 6, 1, 2, 4 ], mathmode := "Text", name := "Prop", next := 239377, root := ~, start := 239326, stop := 239376 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "K", next := 239400, root := ~, start := 239389, stop := 239399 ), rec( content := " or ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "K", next := 239416, root := ~, start := 239404, stop := 239415 ), rec( content := ".", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 239427, root := ~, start := 239380, stop := 239426 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "K", next := 239476, root := ~, start := 239465, stop := 239475 ), rec( content := " if the \ digraph ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "A", next := 239506, root := ~, start := 239492, stop := 239505 ), rec( content := "\n is\ antisymmetric, and ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "K", next := 239545, root := ~, start := 239533, stop := 239544 ), rec( content := " if it i\ s not.\n ", count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "P", next := 239568, root := ~, start := 239564, stop := 239567 ), rec( content := "\n\n \ A digraph is ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ ntisymmetric", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "E", next := 239607, root := ~, start := 239587, stop := 239606 ), rec( content := " if when\ ever there is an edge with source\n ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 239661, root := ~, start := 239653, stop := 239660 ), rec( content := " and ran\ ge ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 239680, root := ~, start := 239672, stop := 239679 ), rec( content := ", and an\ edge with source ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 239714, root := ~, start := 239706, stop := 239713 ), rec( content := " and ran\ ge\n ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 239737, root := ~, start := 239729, stop := 239736 ), rec( content := ", then t\ he vertices ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 239765, root := ~, start := 239757, stop := 239764 ), rec( content := " and ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "C", next := 239778, root := ~, start := 239770, stop := 239777 ), rec( content := " are equ\ al.\n ", count := [ 6, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "P", next := 239798, root := ~, start := 239794, stop := 239797 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "A", next := 239840, root := ~, start := 239826, stop := 239839 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := Digraph([[2], [1, 3], [2, 3]]);\n\ngap> IsAntisymmetricDigraph(gr1);\nfalse\ngap> DigraphEdges(gr1)\ {[1, 2]};\n[ [ 1, 2 ], [ 2, 1 ] ]\ngap> gr2 := Digraph([[1, 2], [3, 3], [1]]);\ \n\ngap> IsAntisymmetricDigra\ ph(gr2);\ntrue\ngap> DigraphEdges(gr2);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 3 ], [ 2, \ 3 ], [ 3, 1 ] ]\n", count := [ 6, 1, 2, 8 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 8 ], mathmode := "Text", name := "Example", next := 240367, root := ~, start := 239941, stop := 240366 ), rec( content := "\n ", count := [ 6, 1, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 240384, root := ~, start := 239430, stop := 240383 ) ], count := [ 6, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 240398, root := ~, start := 239257, stop := 240397 ), rec( content := "\n\n ", count := [ 6, 1, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsBipartiteDigr\ aph" ), content := 0, count := [ 6, 1, 3, 2 ], mathmode := "Text", name := "Prop", next := 240466, root := ~, start := 240419, stop := 240465 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "K", next := 240489, root := ~, start := 240478, stop := 240488 ), rec( content := " or ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "K", next := 240505, root := ~, start := 240493, stop := 240504 ), rec( content := ".", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Returns", next := 240516, root := ~, start := 240469, stop := 240515 ), rec( attributes := rec( ), content := [ rec( content := "\n This \ property is ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "K", next := 240563, root := ~, start := 240552, stop := 240562 ), rec( content := " if the \ digraph ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "A", next := 240593, root := ~, start := 240579, stop := 240592 ), rec( content := " is bipa\ rtite, and\n ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "K", next := 240626, root := ~, start := 240614, stop := 240625 ), rec( content := " if it i\ s not. A digraph is bipartite if and only if the\n vertices of ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "A", next := 240712, root := ~, start := 240698, stop := 240711 ), rec( content := " can be \ partitioned into two non-empty sets such\n that the source and range of any e\ dge of ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "A", next := 240818, root := ~, start := 240804, stop := 240817 ), rec( content := " lie in \ distinct sets.\n Equivalently, a digraph is bipartite if and only if it is 2-\ colorable; see\n ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ GreedyColouring", Label := "for a digraph" ), content := 0, count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Ref", next := 240978, root := ~, start := 240920, stop := 240977 ), rec( content := ". ", count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 3, 4 ], mathmode := "Text", name := "P", next := 240984, root := ~, start := 240980, stop := 240983 ), rec( content := "\n\n Se\ e also ", count := [ 6, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Bicomponents" ), content := 0, count := [ 6, 1, 3, 4 ], mathmode := "Text", name := "Ref", next := 241030, root := ~, start := 240997, stop := 241029 ), rec( content := ".\n " , count := [ 6, 1, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "P", next := 241040, root := ~, start := 241036, stop := 241039 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "A", next := 241082, root := ~, start := 241068, stop := 241081 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := ChainDigraph(4);\n\ngap> I\ sBipartiteDigraph(D);\ntrue\ngap> D := CycleDigraph(3);\n\ngap> IsBipartiteDigraph(D);\nfalse\ngap> D := CompleteBi\ partiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBipartiteDigraph(D);\ntrue", count := [ 6, 1, 3, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 3, 6 ], mathmode := "Text", name := "Example", next := 241555, root := ~, start := 241182, stop := 241554 ), rec( content := "\n ", count := [ 6, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 3, 3 ], mathmode := "Text", name := "Description", next := 241572, root := ~, start := 240519, stop := 241571 ) ], count := [ 6, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 241586, root := ~, start := 240404, stop := 241585 ), rec( content := "\n\n ", count := [ 6, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCompleteBipar\ titeDigraph" ), content := 0, count := [ 6, 1, 4, 2 ], mathmode := "Text", name := "Prop", next := 241662, root := ~, start := 241607, stop := 241661 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "K", next := 241685, root := ~, start := 241674, stop := 241684 ), rec( content := " or ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "K", next := 241701, root := ~, start := 241689, stop := 241700 ), rec( content := ".", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "Returns", next := 241712, root := ~, start := 241665, stop := 241711 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "K", next := 241752, root := ~, start := 241741, stop := 241751 ), rec( content := " if the \ digraph ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "A", next := 241782, root := ~, start := 241768, stop := 241781 ), rec( content := " is a co\ mplete bipartite\n digraph, and ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "K", next := 241836, root := ~, start := 241824, stop := 241835 ), rec( content := " if it i\ s not. ", count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "P", next := 241855, root := ~, start := 241851, stop := 241854 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omplete bipartite digraph", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "E", next := 241909, root := ~, start := 241876, stop := 241908 ), rec( content := " if it i\ s bipartite, see\n ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipar\ titeDigraph" ), content := 0, count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "Ref", next := 241970, root := ~, start := 241938, stop := 241969 ), rec( content := ", and th\ ere exists a unique edge with\n source ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "C", next := 242027, root := ~, start := 242019, stop := 242026 ), rec( content := " and ran\ ge ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "C", next := 242046, root := ~, start := 242038, stop := 242045 ), rec( content := " if and \ only if ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "C", next := 242070, root := ~, start := 242062, stop := 242069 ), rec( content := " and ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "C", next := 242083, root := ~, start := 242075, stop := 242082 ), rec( content := " lie\n \ in different bicomponents of ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "A", next := 242135, root := ~, start := 242121, stop := 242134 ), rec( content := ", see ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Bicomponents" ), content := 0, count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "Ref", next := 242180, root := ~, start := 242141, stop := 242179 ), rec( content := ". ", count := [ 6, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "P", next := 242186, root := ~, start := 242182, stop := 242185 ), rec( content := "\n\n \ Equivalently, a bipartite digraph with bicomponents of size ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "M", next := 242260, root := ~, start := 242252, stop := 242259 ), rec( content := " and\n \ ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "M", next := 242277, root := ~, start := 242269, stop := 242276 ), rec( content := " is comp\ lete precisely when it has ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ mn", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "M", next := 242322, root := ~, start := 242312, stop := 242321 ), rec( content := " edges, \ none of which\n are multiple edges. ", count := [ 6, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 4, 6 ], mathmode := "Text", name := "P", next := 242372, root := ~, start := 242368, stop := 242371 ), rec( content := "\n\n \ See also ", count := [ 6, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Complet\ eBipartiteDigraph" ), content := 0, count := [ 6, 1, 4, 6 ], mathmode := "Text", name := "Ref", next := 242425, root := ~, start := 242387, stop := 242424 ), rec( content := ".\n " , count := [ 6, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "P", next := 242435, root := ~, start := 242431, stop := 242434 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "A", next := 242477, root := ~, start := 242463, stop := 242476 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(2);\n\ngap> I\ sCompleteBipartiteDigraph(D);\ntrue\ngap> D := CycleDigraph(4);\n\ngap> IsBipartiteDigraph(D);\ntrue\ngap> IsComple\ teBipartiteDigraph(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutableDig\ raph, 5, 4);\n\ngap> IsCompleteBipa\ rtiteDigraph(D);\ntrue", count := [ 6, 1, 4, 8 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 8 ], mathmode := "Text", name := "Example", next := 243008, root := ~, start := 242578, stop := 243007 ), rec( content := "\n ", count := [ 6, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 243025, root := ~, start := 241715, stop := 243024 ) ], count := [ 6, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 243039, root := ~, start := 241592, stop := 243038 ), rec( content := "\n\n ", count := [ 6, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCompleteDigra\ ph" ), content := 0, count := [ 6, 1, 5, 2 ], mathmode := "Text", name := "Prop", next := 243106, root := ~, start := 243060, stop := 243105 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "K", next := 243129, root := ~, start := 243118, stop := 243128 ), rec( content := " or ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "K", next := 243145, root := ~, start := 243133, stop := 243144 ), rec( content := ".", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "Returns", next := 243156, root := ~, start := 243109, stop := 243155 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "K", next := 243196, root := ~, start := 243185, stop := 243195 ), rec( content := " if the \ digraph ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "A", next := 243226, root := ~, start := 243212, stop := 243225 ), rec( content := " is comp\ lete, and\n ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "K", next := 243260, root := ~, start := 243248, stop := 243259 ), rec( content := " if it i\ s not. ", count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "P", next := 243279, root := ~, start := 243275, stop := 243278 ), rec( content := "\n\n \ A digraph is ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omplete", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "E", next := 243313, root := ~, start := 243298, stop := 243312 ), rec( content := " if it h\ as no loops, and for all\n ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ istinct", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "E", next := 243365, root := ~, start := 243350, stop := 243364 ), rec( content := " vertice\ s ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "C", next := 243383, root := ~, start := 243375, stop := 243382 ), rec( content := " and ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "C", next := 243396, root := ~, start := 243388, stop := 243395 ), rec( content := ",\n t\ here is exactly one edge with source ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "C", next := 243448, root := ~, start := 243440, stop := 243447 ), rec( content := " and ran\ ge ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "C", next := 243467, root := ~, start := 243459, stop := 243466 ), rec( content := ".\n\n \ Equivalently, a digraph with ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "M", next := 243511, root := ~, start := 243503, stop := 243510 ), rec( content := " vertice\ s is complete precisely when\n it has ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ (n - 1)", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "M", next := 243574, root := ~, start := 243559, stop := 243573 ), rec( content := " edges, \ no loops, and no multiple edges.\n ", count := [ 6, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "P", next := 243623, root := ~, start := 243619, stop := 243622 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "A", next := 243665, root := ~, start := 243651, stop := 243664 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3], [1, 3], [1, 2]]);\n\ngap> IsCompleteDigraph(D);\ntrue\ngap> D := Digraph([[2, 2], [1\ ]]);\n\ngap> IsCompleteDigrap\ h(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteDigraph(D);\nfalse", count := [ 6, 1, 5, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 5, 6 ], mathmode := "Text", name := "Example", next := 244173, root := ~, start := 243766, stop := 244172 ), rec( content := "\n ", count := [ 6, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 244190, root := ~, start := 243159, stop := 244189 ) ], count := [ 6, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 244204, root := ~, start := 243045, stop := 244203 ), rec( content := "\n\n ", count := [ 6, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCompleteMulti\ partiteDigraph" ), content := 0, count := [ 6, 1, 6, 2 ], mathmode := "Text", name := "Prop", next := 244283, root := ~, start := 244225, stop := 244282 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "K", next := 244306, root := ~, start := 244295, stop := 244305 ), rec( content := " or ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "K", next := 244322, root := ~, start := 244310, stop := 244321 ), rec( content := ".", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "Returns", next := 244333, root := ~, start := 244286, stop := 244332 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property returns ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "K", next := 244387, root := ~, start := 244376, stop := 244386 ), rec( content := " if ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "A", next := 244405, root := ~, start := 244391, stop := 244404 ), rec( content := " is a co\ mplete\n multipartite digraph, and ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "K", next := 244462, root := ~, start := 244450, stop := 244461 ), rec( content := " if not.\ ", count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "P", next := 244475, root := ~, start := 244471, stop := 244474 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ omplete multipartite digraph", count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "E", next := 244532, root := ~, start := 244496, stop := 244531 ), rec( content := " if and \ only if\n its vertices can be partitioned into at least two maximal indepen\ dent sets,\n where every possible edge between these independent sets occur\ s in the\n digraph exactly once.\n ", count := [ 6, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "P", next := 244737, root := ~, start := 244733, stop := 244736 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "A", next := 244779, root := ~, start := 244765, stop := 244778 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteMultipartiteDigraph([2, 4, 6]);\n\ngap> IsCompleteMultipartiteDigraph\ (D);\ntrue\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteMultipartiteDigraph(D);\ \ntrue", count := [ 6, 1, 6, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 6 ], mathmode := "Text", name := "Example", next := 245222, root := ~, start := 244880, stop := 245221 ), rec( content := "\n ", count := [ 6, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 245239, root := ~, start := 244336, stop := 245238 ) ], count := [ 6, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 245253, root := ~, start := 244210, stop := 245252 ), rec( content := "\n\n ", count := [ 6, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEmptyDigraph"\ ), content := 0, count := [ 6, 1, 7, 2 ], mathmode := "Text", name := "Prop", next := 245317, root := ~, start := 245274, stop := 245316 ), rec( attributes := rec( Arg := "digraph", Name := "IsNullDigraph" ), content := 0, count := [ 6, 1, 7, 4 ], mathmode := "Text", name := "Prop", next := 245362, root := ~, start := 245320, stop := 245361 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "K", next := 245385, root := ~, start := 245374, stop := 245384 ), rec( content := " or ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "K", next := 245401, root := ~, start := 245389, stop := 245400 ), rec( content := ".", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "Returns", next := 245412, root := ~, start := 245365, stop := 245411 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "K", next := 245452, root := ~, start := 245441, stop := 245451 ), rec( content := " if the \ digraph ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "A", next := 245482, root := ~, start := 245468, stop := 245481 ), rec( content := " is empt\ y, and\n ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "K", next := 245513, root := ~, start := 245501, stop := 245512 ), rec( content := " if it i\ s not. A digraph is ", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ mpty", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "E", next := 245553, root := ~, start := 245541, stop := 245552 ), rec( content := " if it h\ as no\n edges.", count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "P", next := 245581, root := ~, start := 245577, stop := 245580 ), rec( content := "\n\n \ ", count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sNullDigraph", count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "C", next := 245607, root := ~, start := 245587, stop := 245606 ), rec( content := " is a sy\ nonym for ", count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sEmptyDigraph", count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "C", next := 245646, root := ~, start := 245625, stop := 245645 ), rec( content := ".\n " , count := [ 6, 1, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "P", next := 245656, root := ~, start := 245652, stop := 245655 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "A", next := 245698, root := ~, start := 245684, stop := 245697 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[], []]);\n\ngap>\ IsEmptyDigraph(D);\ntrue\ngap> IsNullDigraph(D);\ntrue\ngap> D := Digraph([[]\ , [1]]);\n\ngap> IsEmptyDigraph(D);\ \nfalse\ngap> IsNullDigraph(D);\nfalse", count := [ 6, 1, 7, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 7, 8 ], mathmode := "Text", name := "Example", next := 246091, root := ~, start := 245799, stop := 246090 ), rec( content := "\n ", count := [ 6, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 7, 5 ], mathmode := "Text", name := "Description", next := 246108, root := ~, start := 245415, stop := 246107 ) ], count := [ 6, 1, 7, 1 ], mathmode := "Text", name := "ManSection", next := 246122, root := ~, start := 245259, stop := 246121 ), rec( content := "\n\n ", count := [ 6, 1, 7, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEquivalenceDi\ graph" ), content := 0, count := [ 6, 1, 8, 2 ], mathmode := "Text", name := "Prop", next := 246192, root := ~, start := 246143, stop := 246191 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "K", next := 246215, root := ~, start := 246204, stop := 246214 ), rec( content := " or ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "K", next := 246231, root := ~, start := 246219, stop := 246230 ), rec( content := ".", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Returns", next := 246242, root := ~, start := 246195, stop := 246241 ), rec( attributes := rec( ), content := [ rec( content := "\n A d\ igraph is an equivalence digraph if and only if the digraph satisifies\n al\ l of ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRefle\ xiveDigraph" ), content := 0, count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 246380, root := ~, start := 246348, stop := 246379 ), rec( content := ",\n " , count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 246418, root := ~, start := 246386, stop := 246417 ), rec( content := " and ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTrans\ itiveDigraph" ), content := 0, count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Ref", next := 246456, root := ~, start := 246423, stop := 246455 ), rec( content := ".\n A\ partial order ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "A", next := 246492, root := ~, start := 246478, stop := 246491 ), rec( content := " corresp\ onds to an equivalence relation.\n ", count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "P", next := 246541, root := ~, start := 246537, stop := 246540 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "A", next := 246583, root := ~, start := 246569, stop := 246582 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 1, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2], [1, 3]]);\n\ngap> IsEquivalenceDigraph(D);\ntrue\n", count := [ 6, 1, 8, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 8, 5 ], mathmode := "Text", name := "Example", next := 246834, root := ~, start := 246680, stop := 246833 ), rec( content := "\n ", count := [ 6, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 8, 3 ], mathmode := "Text", name := "Description", next := 246851, root := ~, start := 246245, stop := 246850 ) ], count := [ 6, 1, 8, 1 ], mathmode := "Text", name := "ManSection", next := 246865, root := ~, start := 246128, stop := 246864 ), rec( content := "\n\n ", count := [ 6, 1, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsFunctionalDig\ raph" ), content := 0, count := [ 6, 1, 9, 2 ], mathmode := "Text", name := "Prop", next := 246934, root := ~, start := 246886, stop := 246933 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "K", next := 246957, root := ~, start := 246946, stop := 246956 ), rec( content := " or ", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "K", next := 246973, root := ~, start := 246961, stop := 246972 ), rec( content := ".", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "Returns", next := 246984, root := ~, start := 246937, stop := 246983 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "K", next := 247033, root := ~, start := 247022, stop := 247032 ), rec( content := " if the \ digraph ", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "A", next := 247063, root := ~, start := 247049, stop := 247062 ), rec( content := " is\n \ functional. ", count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "P", next := 247087, root := ~, start := 247083, stop := 247086 ), rec( content := "\n\n \ A digraph is ", count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unctional", count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "E", next := 247123, root := ~, start := 247106, stop := 247122 ), rec( content := " if ever\ y vertex is the source of a\n unique edge.\n ", count := [ 6, 1, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "P", next := 247184, root := ~, start := 247180, stop := 247183 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "A", next := 247226, root := ~, start := 247212, stop := 247225 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := Digraph([[3], [2], [2], [1], [6], [5]]);\n\ngap> IsFunctionalDigraph(gr1);\ntrue\ngap> gr2 := Digra\ ph([[1, 2], [1]]);\n\ngap> IsFunct\ ionalDigraph(gr2);\nfalse\ngap> gr3 := Digraph(3, [1, 2, 3], [2, 3, 1]);\n\ngap> IsFunctionalDigraph(gr3);\ntrue\ \n", count := [ 6, 1, 9, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 6 ], mathmode := "Text", name := "Example", next := 247737, root := ~, start := 247326, stop := 247736 ), rec( content := "\n ", count := [ 6, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 9, 3 ], mathmode := "Text", name := "Description", next := 247754, root := ~, start := 246987, stop := 247753 ) ], count := [ 6, 1, 9, 1 ], mathmode := "Text", name := "ManSection", next := 247768, root := ~, start := 246871, stop := 247767 ), rec( content := "\n\n ", count := [ 6, 1, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsMultiDigraph"\ ), content := 0, count := [ 6, 1, 10, 2 ], mathmode := "Text", name := "Prop", next := 247832, root := ~, start := 247789, stop := 247831 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "K", next := 247855, root := ~, start := 247844, stop := 247854 ), rec( content := " or ", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "K", next := 247871, root := ~, start := 247859, stop := 247870 ), rec( content := ".", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "Returns", next := 247882, root := ~, start := 247835, stop := 247881 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ultidigraph", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "E", next := 247924, root := ~, start := 247905, stop := 247923 ), rec( content := " is one \ that has at least two\n edges with equal source and range.", count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "P", next := 247996, root := ~, start := 247992, stop := 247995 ), rec( content := "\n\n \ If the argument ", count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "A", next := 248032, root := ~, start := 248018, stop := 248031 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\"a\", \"b\", \"c\"], [\"a\", \"b\", \"b\"], [\"b\", \"c\"\ , \"a\"]);\n\ngap> IsMultiDigraph(\ D);\nfalse\ngap> D := DigraphFromDigraph6String(\"&Bug\");\n\ngap> IsDuplicateFree(DigraphEdges(D));\ntrue\ngap>\ IsMultiDigraph(D);\nfalse\ngap> D := Digraph([[1, 2, 3, 2], [2, 1], [3]]);\n<\ immutable multidigraph with 3 vertices, 7 edges>\ngap> IsDuplicateFree(Digraph\ Edges(D));\nfalse\ngap> IsMultiDigraph(D);\ntrue\ngap> D := DigraphMutableCopy\ (D); \n\ngap> IsMultiDigraph(D)\ ;\ntrue", count := [ 6, 1, 10, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 5 ], mathmode := "Text", name := "Example", next := 248756, root := ~, start := 248133, stop := 248755 ), rec( content := "\n ", count := [ 6, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 10, 3 ], mathmode := "Text", name := "Description", next := 248773, root := ~, start := 247885, stop := 248772 ) ], count := [ 6, 1, 10, 1 ], mathmode := "Text", name := "ManSection", next := 248787, root := ~, start := 247774, stop := 248786 ), rec( content := "\n\n ", count := [ 6, 1, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsReflexiveDigr\ aph" ), content := 0, count := [ 6, 1, 11, 2 ], mathmode := "Text", name := "Prop", next := 248855, root := ~, start := 248808, stop := 248854 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "K", next := 248878, root := ~, start := 248867, stop := 248877 ), rec( content := " or ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "K", next := 248894, root := ~, start := 248882, stop := 248893 ), rec( content := ".", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "Returns", next := 248905, root := ~, start := 248858, stop := 248904 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "K", next := 248954, root := ~, start := 248943, stop := 248953 ), rec( content := " if the \ digraph ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "A", next := 248984, root := ~, start := 248970, stop := 248983 ), rec( content := " is\n \ reflexive, and ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "K", next := 249019, root := ~, start := 249007, stop := 249018 ), rec( content := " if it i\ s not.\n A digraph is ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ eflexive", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "E", next := 249067, root := ~, start := 249051, stop := 249066 ), rec( content := " if it h\ as a loop at every vertex. ", count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 11, 4 ], mathmode := "Text", name := "P", next := 249106, root := ~, start := 249102, stop := 249105 ), rec( content := "\n ", count := [ 6, 1, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "P", next := 249115, root := ~, start := 249111, stop := 249114 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "A", next := 249157, root := ~, start := 249143, stop := 249156 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [2]]);\n\ngap> IsReflexiveDigraph(D);\ntrue\ngap> D := Digraph([[3, 1], [4, 2], [3],\ [2, 1]]);\n\ngap> IsReflexiveDigr\ aph(D);\nfalse\n", count := [ 6, 1, 11, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 6 ], mathmode := "Text", name := "Example", next := 249530, root := ~, start := 249257, stop := 249529 ), rec( content := "\n ", count := [ 6, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 11, 3 ], mathmode := "Text", name := "Description", next := 249547, root := ~, start := 248908, stop := 249546 ) ], count := [ 6, 1, 11, 1 ], mathmode := "Text", name := "ManSection", next := 249561, root := ~, start := 248793, stop := 249560 ), rec( content := "\n\n ", count := [ 6, 1, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsSymmetricDigr\ aph" ), content := 0, count := [ 6, 1, 12, 2 ], mathmode := "Text", name := "Prop", next := 249629, root := ~, start := 249582, stop := 249628 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "K", next := 249652, root := ~, start := 249641, stop := 249651 ), rec( content := " or ", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "K", next := 249668, root := ~, start := 249656, stop := 249667 ), rec( content := ".", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "Returns", next := 249679, root := ~, start := 249632, stop := 249678 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "K", next := 249728, root := ~, start := 249717, stop := 249727 ), rec( content := " if the \ digraph ", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "A", next := 249758, root := ~, start := 249744, stop := 249757 ), rec( content := "\n is\ symmetric, and ", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "K", next := 249793, root := ~, start := 249781, stop := 249792 ), rec( content := " if it i\ s not.", count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "P", next := 249811, root := ~, start := 249807, stop := 249810 ), rec( content := "\n\n \ A ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ymmetric digraph", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "E", next := 249843, root := ~, start := 249819, stop := 249842 ), rec( content := " is one \ where for each non-loop edge, having\n source ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 249907, root := ~, start := 249899, stop := 249906 ), rec( content := " and ran\ ge ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 249926, root := ~, start := 249918, stop := 249925 ), rec( content := ", there \ is a corresponding edge with\n source ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 249982, root := ~, start := 249974, stop := 249981 ), rec( content := " and ran\ ge ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 250001, root := ~, start := 249993, stop := 250000 ), rec( content := ". If th\ ere are ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 250025, root := ~, start := 250017, stop := 250024 ), rec( content := " edges w\ ith\n source ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 250056, root := ~, start := 250048, stop := 250055 ), rec( content := " and ran\ ge ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 250075, root := ~, start := 250067, stop := 250074 ), rec( content := ", then t\ here must be precisely ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 250114, root := ~, start := 250106, stop := 250113 ), rec( content := "\n ed\ ges with source ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 250145, root := ~, start := 250137, stop := 250144 ), rec( content := " and ran\ ge ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "C", next := 250164, root := ~, start := 250156, stop := 250163 ), rec( content := ". In ot\ her words, a symmetric\n digraph has a symmetric adjacency matrix ", count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Adjacen\ cyMatrix" ), content := 0, count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "Ref", next := 250269, root := ~, start := 250240, stop := 250268 ), rec( content := ".\n " , count := [ 6, 1, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "P", next := 250279, root := ~, start := 250275, stop := 250278 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "A", next := 250321, root := ~, start := 250307, stop := 250320 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := Digraph([[2], [1, 3], [2, 3]]);\n\ngap> IsSymmetricDigraph(gr1);\ntrue\ngap> adj1 := AdjacencyMatri\ x(gr1);;\ngap> Display(adj1);\n[ [ 0, 1, 0 ],\n [ 1, 0, 1 ],\n [ 0, \ 1, 1 ] ]\ngap> adj1 = TransposedMat(adj1);\ntrue\ngap> gr1 = DigraphReverse(g\ r1);\ntrue\ngap> gr2 := Digraph([[2, 3], [1, 3], [2, 3]]);\n\ngap> IsSymmetricDigraph(gr2);\nfalse\ngap> adj2 :=\ AdjacencyMatrix(gr2);;\ngap> Display(adj2);\n[ [ 0, 1, 1 ],\n [ 1, 0, \ 1 ],\n [ 0, 1, 1 ] ]\ngap> adj2 = TransposedMat(adj2);\nfalse\n", count := [ 6, 1, 12, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 12, 6 ], mathmode := "Text", name := "Example", next := 251041, root := ~, start := 250421, stop := 251040 ), rec( content := "\n ", count := [ 6, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 12, 3 ], mathmode := "Text", name := "Description", next := 251058, root := ~, start := 249682, stop := 251057 ) ], count := [ 6, 1, 12, 1 ], mathmode := "Text", name := "ManSection", next := 251072, root := ~, start := 249567, stop := 251071 ), rec( content := "\n\n ", count := [ 6, 1, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsTournament" ) , content := 0, count := [ 6, 1, 13, 2 ], mathmode := "Text", name := "Prop", next := 251134, root := ~, start := 251093, stop := 251133 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "K", next := 251157, root := ~, start := 251146, stop := 251156 ), rec( content := " or ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "K", next := 251173, root := ~, start := 251161, stop := 251172 ), rec( content := ".", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "Returns", next := 251184, root := ~, start := 251137, stop := 251183 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "K", next := 251233, root := ~, start := 251222, stop := 251232 ), rec( content := " if the \ digraph ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "A", next := 251263, root := ~, start := 251249, stop := 251262 ), rec( content := " is a to\ urnament,\n and ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "K", next := 251301, root := ~, start := 251289, stop := 251300 ), rec( content := " if it i\ s not. ", count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 13, 4 ], mathmode := "Text", name := "P", next := 251320, root := ~, start := 251316, stop := 251319 ), rec( content := "\n\n \ A tournament is an orientation of a complete (undirected) graph.\n Specific\ ally, a tournament is a digraph which has a unique directed edge\n (of some\ orientation) between any pair of distinct vertices, and no loops.\n ", count := [ 6, 1, 13, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "P", next := 251555, root := ~, start := 251551, stop := 251554 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "A", next := 251597, root := ~, start := 251583, stop := 251596 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3, 4], [3, 4], [4], []]);\n\ngap> IsTournament(D);\ntrue\ngap> D := Digraph([[2], [1], [\ 3]]);\n\ngap> IsTournament(D);\nfa\ lse\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> IsTournament(D);\ntrue\ngap> DigraphRemoveEdge(D, 1, 2);\ \n\ngap> IsTournament(D);\nfalse\n", count := [ 6, 1, 13, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 13, 6 ], mathmode := "Text", name := "Example", next := 252178, root := ~, start := 251697, stop := 252177 ), rec( content := "\n ", count := [ 6, 1, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 13, 3 ], mathmode := "Text", name := "Description", next := 252195, root := ~, start := 251187, stop := 252194 ) ], count := [ 6, 1, 13, 1 ], mathmode := "Text", name := "ManSection", next := 252209, root := ~, start := 251078, stop := 252208 ), rec( content := "\n\n ", count := [ 6, 1, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsTransitiveDig\ raph" ), content := 0, count := [ 6, 1, 14, 2 ], mathmode := "Text", name := "Prop", next := 252278, root := ~, start := 252230, stop := 252277 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "K", next := 252301, root := ~, start := 252290, stop := 252300 ), rec( content := " or ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "K", next := 252317, root := ~, start := 252305, stop := 252316 ), rec( content := ".", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "Returns", next := 252328, root := ~, start := 252281, stop := 252327 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "K", next := 252377, root := ~, start := 252366, stop := 252376 ), rec( content := " if the \ digraph ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "A", next := 252407, root := ~, start := 252393, stop := 252406 ), rec( content := "\n is\ transitive, and ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "K", next := 252443, root := ~, start := 252431, stop := 252442 ), rec( content := " if it i\ s not.\n\n A digraph is ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ransitive", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "E", next := 252493, root := ~, start := 252476, stop := 252492 ), rec( content := " if when\ ever ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j ]", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "C", next := 252521, root := ~, start := 252506, stop := 252520 ), rec( content := " and\n \ ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ j, k ]", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "C", next := 252545, root := ~, start := 252530, stop := 252544 ), rec( content := " are edg\ es of the digraph, then ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, k ]", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "C", next := 252592, root := ~, start := 252577, stop := 252591 ), rec( content := " is also\ an\n edge of the digraph. ", count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "P", next := 252633, root := ~, start := 252629, stop := 252632 ), rec( content := "\n\n \ Let ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "M", next := 252651, root := ~, start := 252643, stop := 252650 ), rec( content := " be the \ number of vertices of an arbitrary digraph, and let\n ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "M", next := 252727, root := ~, start := 252719, stop := 252726 ), rec( content := " be the \ number of edges.\n For general digraphs, the methods used for this property\ use a version\n of the Floyd-Warshall algorithm, and have complexity ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (n^3)", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "M", next := 252897, root := ~, start := 252884, stop := 252896 ), rec( content := ".\n\n \ However for digraphs which are topologically sortable\n [", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "Ref", next := 252999, root := ~, start := 252963, stop := 252998 ), rec( content := "], then \ methods with\n complexity ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m + n + m \\cdot n)", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "M", next := 253062, root := ~, start := 253035, stop := 253061 ), rec( content := " will be\ used when appropriate.\n ", count := [ 6, 1, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "P", next := 253102, root := ~, start := 253098, stop := 253101 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "A", next := 253144, root := ~, start := 253130, stop := 253143 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 1, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 14, 6 ], mathmode := "Text", name := "P", next := 253248, root := ~, start := 253244, stop := 253247 ), rec( content := "\n ", count := [ 6, 1, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 2], [3], [3]]);\n\ngap> IsTransitiveDigraph(D);\nfalse\ngap> gr2 := Digraph([[1, 2, 3], \ [3], [3]]);\n\ngap> IsTransitiveDi\ graph(gr2);\ntrue\ngap> gr2 = DigraphTransitiveClosure(D);\ntrue\ngap> gr3 := \ Digraph([[1, 2, 2, 3], [3, 3], [3]]);\n\ngap> IsTransitiveDigraph(gr3);\ntrue\n", count := [ 6, 1, 14, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 14, 7 ], mathmode := "Text", name := "Example", next := 253710, root := ~, start := 253253, stop := 253709 ), rec( content := "\n ", count := [ 6, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 14, 3 ], mathmode := "Text", name := "Description", next := 253727, root := ~, start := 252331, stop := 253726 ) ], count := [ 6, 1, 14, 1 ], mathmode := "Text", name := "ManSection", next := 253741, root := ~, start := 252215, stop := 253740 ), rec( content := "\n\n ", count := [ 6, 1, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsPreorderDigra\ ph" ), content := 0, count := [ 6, 1, 15, 2 ], mathmode := "Text", name := "Prop", next := 253808, root := ~, start := 253762, stop := 253807 ), rec( attributes := rec( Arg := "digraph", Name := "IsQuasiorderDig\ raph" ), content := 0, count := [ 6, 1, 15, 4 ], mathmode := "Text", name := "Prop", next := 253859, root := ~, start := 253811, stop := 253858 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "K", next := 253882, root := ~, start := 253871, stop := 253881 ), rec( content := " or ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "K", next := 253898, root := ~, start := 253886, stop := 253897 ), rec( content := ".", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Returns", next := 253909, root := ~, start := 253862, stop := 253908 ), rec( attributes := rec( ), content := [ rec( content := "\n A d\ igraph is a preorder digraph if and only if the digraph satisifies both\n " , count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRefle\ xiveDigraph" ), content := 0, count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Ref", next := 254041, root := ~, start := 254009, stop := 254040 ), rec( content := " and ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTrans\ itiveDigraph" ), content := 0, count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Ref", next := 254079, root := ~, start := 254046, stop := 254078 ), rec( content := ".\n A\ preorder digraph (or quasiorder digraph) ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "A", next := 254142, root := ~, start := 254128, stop := 254141 ), rec( content := " corresp\ onds to\n the preorder relation ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \leq", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "M", next := 254195, root := ~, start := 254184, stop := 254194 ), rec( content := " defined\ by ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ \\leq y", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "M", next := 254222, root := ~, start := 254207, stop := 254221 ), rec( content := " if and \ only\n if ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ x, y]", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "C", next := 254255, root := ~, start := 254242, stop := 254254 ), rec( content := " is an e\ dge of ", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "A", next := 254284, root := ~, start := 254270, stop := 254283 ), rec( content := ".\n " , count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "P", next := 254294, root := ~, start := 254290, stop := 254293 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "A", next := 254336, root := ~, start := 254322, stop := 254335 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 1, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1], [2, 3], [2, 3]]);\n\ngap> IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[1 .. 4], [1 \ .. 4], [1 .. 4], [1 .. 4]]);\n\ng\ ap> IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[2], [3], [4], [5], [1]]);\ \n\ngap> IsPreorderDigraph(D);\nfa\ lse\ngap> D := Digraph([[1], [1, 2], [2, 3]]);\n\ngap> IsQuasiorderDigraph(D);\nfalse\n", count := [ 6, 1, 15, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 15, 7 ], mathmode := "Text", name := "Example", next := 254969, root := ~, start := 254433, stop := 254968 ), rec( content := "\n ", count := [ 6, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 15, 5 ], mathmode := "Text", name := "Description", next := 254986, root := ~, start := 253912, stop := 254985 ) ], count := [ 6, 1, 15, 1 ], mathmode := "Text", name := "ManSection", next := 255000, root := ~, start := 253747, stop := 254999 ), rec( content := "\n\n ", count := [ 6, 1, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsPartialOrderD\ igraph" ), content := 0, count := [ 6, 1, 16, 2 ], mathmode := "Text", name := "Prop", next := 255071, root := ~, start := 255021, stop := 255070 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "K", next := 255094, root := ~, start := 255083, stop := 255093 ), rec( content := " or ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "K", next := 255110, root := ~, start := 255098, stop := 255109 ), rec( content := ".", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Returns", next := 255121, root := ~, start := 255074, stop := 255120 ), rec( attributes := rec( ), content := [ rec( content := "\n A d\ igraph is a partial order digraph if and only if the digraph satisifies\n a\ ll of ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRefle\ xiveDigraph" ), content := 0, count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Ref", next := 255260, root := ~, start := 255228, stop := 255259 ), rec( content := ",\n " , count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsAntis\ ymmetricDigraph" ), content := 0, count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Ref", next := 255302, root := ~, start := 255266, stop := 255301 ), rec( content := " and ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTrans\ itiveDigraph" ), content := 0, count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Ref", next := 255340, root := ~, start := 255307, stop := 255339 ), rec( content := ".\n A\ partial order ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "A", next := 255376, root := ~, start := 255362, stop := 255375 ), rec( content := " corresp\ onds\n to the partial order relation ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \leq", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "M", next := 255434, root := ~, start := 255423, stop := 255433 ), rec( content := " defined\ by ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ \\leq y", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "M", next := 255461, root := ~, start := 255446, stop := 255460 ), rec( content := " if and\\ n only if ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ x, y]", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "C", next := 255494, root := ~, start := 255481, stop := 255493 ), rec( content := " is an e\ dge of ", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "A", next := 255523, root := ~, start := 255509, stop := 255522 ), rec( content := ".\n " , count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "P", next := 255533, root := ~, start := 255529, stop := 255532 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "A", next := 255575, root := ~, start := 255561, stop := 255574 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 1, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsPartialOrderDigraph(D);\ntrue\ngap> D := CycleDigraph(5);\n\ \ngap> IsPartialOrderDigraph(D);\nfal\ se\ngap> D := Digraph([[1, 1], [1, 1, 2], [3], [3, 3, 4, 4]]);\n\ngap> IsPartialOrderDigraph(D);\ntrue\n", count := [ 6, 1, 16, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 16, 5 ], mathmode := "Text", name := "Example", next := 256079, root := ~, start := 255672, stop := 256078 ), rec( content := "\n ", count := [ 6, 1, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 16, 3 ], mathmode := "Text", name := "Description", next := 256096, root := ~, start := 255124, stop := 256095 ) ], count := [ 6, 1, 16, 1 ], mathmode := "Text", name := "ManSection", next := 256110, root := ~, start := 255006, stop := 256109 ), rec( content := "\n\n ", count := [ 6, 1, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsMeetSemilatti\ ceDigraph" ), content := 0, count := [ 6, 1, 17, 2 ], mathmode := "Text", name := "Prop", next := 256184, root := ~, start := 256131, stop := 256183 ), rec( attributes := rec( Arg := "digraph", Name := "IsJoinSemilatti\ ceDigraph" ), content := 0, count := [ 6, 1, 17, 4 ], mathmode := "Text", name := "Prop", next := 256240, root := ~, start := 256187, stop := 256239 ), rec( attributes := rec( Arg := "digraph", Name := "IsLatticeDigrap\ h" ), content := 0, count := [ 6, 1, 17, 6 ], mathmode := "Text", name := "Prop", next := 256288, root := ~, start := 256243, stop := 256287 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 256311, root := ~, start := 256300, stop := 256310 ), rec( content := " or ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 256327, root := ~, start := 256315, stop := 256326 ), rec( content := ".", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "Returns", next := 256338, root := ~, start := 256291, stop := 256337 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMeetSemilatticeDigraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "C", next := 256390, root := ~, start := 256359, stop := 256389 ), rec( content := " returns\ ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 256410, root := ~, start := 256399, stop := 256409 ), rec( content := " if the \ digraph\n ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "A", next := 256444, root := ~, start := 256430, stop := 256443 ), rec( content := " is a me\ et semilattice; ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sJoinSemilatticeDigraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "C", next := 256499, root := ~, start := 256468, stop := 256498 ), rec( content := "\n re\ turns ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 256523, root := ~, start := 256512, stop := 256522 ), rec( content := " if the \ digraph ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "A", next := 256553, root := ~, start := 256539, stop := 256552 ), rec( content := " is a jo\ in semilattice;\n and ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sLatticeDigraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "C", next := 256609, root := ~, start := 256586, stop := 256608 ), rec( content := " returns\ ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "K", next := 256629, root := ~, start := 256618, stop := 256628 ), rec( content := " if the \ digraph\n ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "A", next := 256663, root := ~, start := 256649, stop := 256662 ), rec( content := " is both\ a meet and a join semilattice.\n ", count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "P", next := 256711, root := ~, start := 256707, stop := 256710 ), rec( content := "\n\n \ For a partial order digraph ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsParti\ alOrderDigraph" ), content := 0, count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "Ref", next := 256780, root := ~, start := 256745, stop := 256779 ), rec( content := " the\n \ corresponding partial order is the relation ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \leq", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "M", next := 256844, root := ~, start := 256833, stop := 256843 ), rec( content := ", define\ d by\n ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ \\leq y", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "M", next := 256876, root := ~, start := 256861, stop := 256875 ), rec( content := " if and \ only if ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ x, y]", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "C", next := 256905, root := ~, start := 256892, stop := 256904 ), rec( content := " is an e\ dge.\n A digraph is a ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ eet semilattice", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "E", next := 256960, root := ~, start := 256937, stop := 256959 ), rec( content := " if it i\ s a partial order and every\n pair of vertices has a greatest lower bound (\ meet) with respect to the\n aforementioned relation. A ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ oin semilattice", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "E", next := 257125, root := ~, start := 257102, stop := 257124 ), rec( content := " is a pa\ rtial order where\n every pair of vertices has a least upper bound (join) w\ ith respect to\n the relation.\n ", count := [ 6, 1, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "P", next := 257251, root := ~, start := 257247, stop := 257250 ), rec( content := "\n \ \n If the argument ", count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "A", next := 257293, root := ~, start := 257279, stop := 257292 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 1, 17, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsMeetSemilatticeDigraph(D);\nfalse\ngap> IsJoinSemilatticeDi\ graph(D);\ntrue\ngap> IsLatticeDigraph(D);\nfalse\ngap> D := Digraph([[1], [2]\ , [1 .. 3]]);\n\ngap> IsJoinSemila\ tticeDigraph(D);\nfalse\ngap> IsMeetSemilatticeDigraph(D);\ntrue\ngap> IsLatti\ ceDigraph(D);\nfalse\ngap> D := Digraph([[1 .. 4], [2, 4], [3, 4], [4]]);\n\ngap> IsMeetSemilatticeDigraph(D);\n\ true\ngap> IsJoinSemilatticeDigraph(D);\ntrue\ngap> IsLatticeDigraph(D);\ntrue\ \ngap> D := Digraph([[1, 1, 1], [1, 1, 2, 2],\n> [1, 3, 3], \ [1, 2, 3, 3, 4]]);\n\ngap> I\ sMeetSemilatticeDigraph(D);\ntrue\ngap> IsJoinSemilatticeDigraph(D);\ntrue\nga\ p> IsLatticeDigraph(D);\ntrue\n", count := [ 6, 1, 17, 10 ], name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 10 ], mathmode := "Text", name := "Example", next := 258281, root := ~, start := 257394, stop := 258280 ), rec( content := "\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 17, 7 ], mathmode := "Text", name := "Description", next := 258298, root := ~, start := 256341, stop := 258297 ) ], count := [ 6, 1, 17, 1 ], mathmode := "Text", name := "ManSection", next := 258312, root := ~, start := 256116, stop := 258311 ), rec( content := "\n\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 258326, root := ~, start := 238252, stop := 258325 ) , rec( content := "\n\n ", count := [ 6, 1, 17, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Regularity", count := [ 6, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 258368, root := ~, start := 258339, stop := 258367 ), rec( content := "\n ", count := [ 6, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsInRegularDigr\ aph" ), content := 0, count := [ 6, 2, 1, 2 ], mathmode := "Text", name := "Prop", next := 258435, root := ~, start := 258388, stop := 258434 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "K", next := 258458, root := ~, start := 258447, stop := 258457 ), rec( content := " or ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "K", next := 258474, root := ~, start := 258462, stop := 258473 ), rec( content := ".", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Returns", next := 258485, root := ~, start := 258438, stop := 258484 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "K", next := 258534, root := ~, start := 258523, stop := 258533 ), rec( content := " if ther\ e is an integer ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "C", next := 258566, root := ~, start := 258558, stop := 258565 ), rec( content := " such th\ at\n for every vertex ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "C", next := 258606, root := ~, start := 258598, stop := 258605 ), rec( content := " of digr\ aph ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "A", next := 258632, root := ~, start := 258618, stop := 258631 ), rec( content := " there a\ re exactly\n ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "C", next := 258663, root := ~, start := 258655, stop := 258662 ), rec( content := " edges t\ erminating in ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "C", next := 258693, root := ~, start := 258685, stop := 258692 ), rec( content := ".\n\n \ See also ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOutRe\ gularDigraph" ), content := 0, count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Ref", next := 258742, root := ~, start := 258709, stop := 258741 ), rec( content := " and\n \ ", count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRegul\ arDigraph" ), content := 0, count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Ref", next := 258781, root := ~, start := 258751, stop := 258780 ), rec( content := ".\n " , count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "P", next := 258791, root := ~, start := 258787, stop := 258790 ), rec( content := "\n \ \n If the argument ", count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "A", next := 258833, root := ~, start := 258819, stop := 258832 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 2, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsInRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsInRegularDigraph(C\ hainDigraph(4));\nfalse\n", count := [ 6, 2, 1, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 5 ], mathmode := "Text", name := "Example", next := 259063, root := ~, start := 258933, stop := 259062 ), rec( content := "\n ", count := [ 6, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 1, 3 ], mathmode := "Text", name := "Description", next := 259080, root := ~, start := 258488, stop := 259079 ) ], count := [ 6, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 259094, root := ~, start := 258373, stop := 259093 ), rec( content := "\n\n ", count := [ 6, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsOutRegularDig\ raph" ), content := 0, count := [ 6, 2, 2, 2 ], mathmode := "Text", name := "Prop", next := 259163, root := ~, start := 259115, stop := 259162 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "K", next := 259186, root := ~, start := 259175, stop := 259185 ), rec( content := " or ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "K", next := 259202, root := ~, start := 259190, stop := 259201 ), rec( content := ".", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 259213, root := ~, start := 259166, stop := 259212 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "K", next := 259262, root := ~, start := 259251, stop := 259261 ), rec( content := " if ther\ e is an integer ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "C", next := 259294, root := ~, start := 259286, stop := 259293 ), rec( content := " such th\ at\n for every vertex ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "C", next := 259334, root := ~, start := 259326, stop := 259333 ), rec( content := " of digr\ aph ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "A", next := 259360, root := ~, start := 259346, stop := 259359 ), rec( content := " there a\ re exactly\n ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "C", next := 259391, root := ~, start := 259383, stop := 259390 ), rec( content := " edges s\ tarting at ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "C", next := 259418, root := ~, start := 259410, stop := 259417 ), rec( content := ".\n " , count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "P", next := 259428, root := ~, start := 259424, stop := 259427 ), rec( content := "\n\n \ See also ", count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsInReg\ ularDigraph" ), content := 0, count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "Ref", next := 259475, root := ~, start := 259443, stop := 259474 ), rec( content := " and\n \ ", count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRegul\ arDigraph" ), content := 0, count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "Ref", next := 259514, root := ~, start := 259484, stop := 259513 ), rec( content := ".\n " , count := [ 6, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "P", next := 259524, root := ~, start := 259520, stop := 259523 ), rec( content := "\n \ \n If the argument ", count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "A", next := 259566, root := ~, start := 259552, stop := 259565 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsOutRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsOutRegularDigraph\ (ChainDigraph(4));\nfalse\n", count := [ 6, 2, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 6 ], mathmode := "Text", name := "Example", next := 259798, root := ~, start := 259666, stop := 259797 ), rec( content := "\n ", count := [ 6, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 259815, root := ~, start := 259216, stop := 259814 ) ], count := [ 6, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 259829, root := ~, start := 259100, stop := 259828 ), rec( content := "\n\n ", count := [ 6, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsRegularDigrap\ h" ), content := 0, count := [ 6, 2, 3, 2 ], mathmode := "Text", name := "Prop", next := 259895, root := ~, start := 259850, stop := 259894 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "K", next := 259918, root := ~, start := 259907, stop := 259917 ), rec( content := " or ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "K", next := 259934, root := ~, start := 259922, stop := 259933 ), rec( content := ".", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Returns", next := 259945, root := ~, start := 259898, stop := 259944 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "K", next := 259994, root := ~, start := 259983, stop := 259993 ), rec( content := " if ther\ e is an integer ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "C", next := 260026, root := ~, start := 260018, stop := 260025 ), rec( content := " such th\ at\n for every vertex ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "C", next := 260066, root := ~, start := 260058, stop := 260065 ), rec( content := " of digr\ aph ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "A", next := 260092, root := ~, start := 260078, stop := 260091 ), rec( content := " there a\ re exactly\n ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "C", next := 260123, root := ~, start := 260115, stop := 260122 ), rec( content := " edges s\ tarting and terminating at ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "C", next := 260166, root := ~, start := 260158, stop := 260165 ), rec( content := ". In oth\ er words,\n the property is ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "K", next := 260215, root := ~, start := 260204, stop := 260214 ), rec( content := " if ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "A", next := 260233, root := ~, start := 260219, stop := 260232 ), rec( content := " is both\ in-regular and\n and out-regular.\n\n See also ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsInReg\ ularDigraph" ), content := 0, count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Ref", next := 260324, root := ~, start := 260292, stop := 260323 ), rec( content := " and\n \ ", count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOutRe\ gularDigraph" ), content := 0, count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Ref", next := 260366, root := ~, start := 260333, stop := 260365 ), rec( content := ".\n " , count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "P", next := 260376, root := ~, start := 260372, stop := 260375 ), rec( content := "\n \ \n If the argument ", count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "A", next := 260418, root := ~, start := 260404, stop := 260417 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 2, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsRegularDigraph(Chain\ Digraph(4));\nfalse\n", count := [ 6, 2, 3, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 5 ], mathmode := "Text", name := "Example", next := 260644, root := ~, start := 260518, stop := 260643 ), rec( content := "\n ", count := [ 6, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 3, 3 ], mathmode := "Text", name := "Description", next := 260661, root := ~, start := 259948, stop := 260660 ) ], count := [ 6, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 260675, root := ~, start := 259835, stop := 260674 ), rec( content := "\n\n ", count := [ 6, 2, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsDistanceRegul\ arDigraph" ), content := 0, count := [ 6, 2, 4, 2 ], mathmode := "Text", name := "Prop", next := 260749, root := ~, start := 260696, stop := 260748 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "K", next := 260772, root := ~, start := 260761, stop := 260771 ), rec( content := " or ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "K", next := 260788, root := ~, start := 260776, stop := 260787 ), rec( content := ".", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 260799, root := ~, start := 260752, stop := 260798 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "A", next := 260837, root := ~, start := 260823, stop := 260836 ), rec( content := " is a co\ nnected symmetric graph, this property returns\n ", count := [ 6, 2, 4, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "K", next := 260907, root := ~, start := 260896, stop := 260906 ), rec( content := " if for \ any two vertices ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 260940, root := ~, start := 260932, stop := 260939 ), rec( content := " and ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 260953, root := ~, start := 260945, stop := 260952 ), rec( content := " of ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "A", next := 260971, root := ~, start := 260957, stop := 260970 ), rec( content := "\n an\ d any two integers ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 261005, root := ~, start := 260997, stop := 261004 ), rec( content := " and ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 261018, root := ~, start := 261010, stop := 261017 ), rec( content := " between\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "0\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 261035, root := ~, start := 261027, stop := 261034 ), rec( content := " and the\ \n diameter of ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "A", next := 261074, root := ~, start := 261060, stop := 261073 ), rec( content := ", the nu\ mber of vertices at distance ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 261119, root := ~, start := 261111, stop := 261118 ), rec( content := "\n fr\ om ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 261137, root := ~, start := 261129, stop := 261136 ), rec( content := " and dis\ tance ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 261159, root := ~, start := 261151, stop := 261158 ), rec( content := " from ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 261173, root := ~, start := 261165, stop := 261172 ), rec( content := " depends\ only on\n ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 261202, root := ~, start := 261194, stop := 261201 ), rec( content := ", ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 261212, root := ~, start := 261204, stop := 261211 ), rec( content := ", and th\ e distance between vertices ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 261256, root := ~, start := 261248, stop := 261255 ), rec( content := " and\n \ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "C", next := 261273, root := ~, start := 261265, stop := 261272 ), rec( content := ".", count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "P", next := 261278, root := ~, start := 261274, stop := 261277 ), rec( content := "\n\n \ Alternatively, a distance regular graph is a graph for which there exist\n \ integers ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ _i", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261380, root := ~, start := 261370, stop := 261379 ), rec( content := ", ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ _i", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261392, root := ~, start := 261382, stop := 261391 ), rec( content := ", and ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261406, root := ~, start := 261398, stop := 261405 ), rec( content := " such th\ at for any two\n vertices ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261450, root := ~, start := 261442, stop := 261449 ), rec( content := ", ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261460, root := ~, start := 261452, stop := 261459 ), rec( content := " in ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "A", next := 261478, root := ~, start := 261464, stop := 261477 ), rec( content := " which a\ re distance ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261506, root := ~, start := 261498, stop := 261505 ), rec( content := "\n ap\ art, there are exactly ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ _i", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261546, root := ~, start := 261536, stop := 261545 ), rec( content := " neighbo\ rs of ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261568, root := ~, start := 261560, stop := 261567 ), rec( content := " which a\ re at\n distance ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ - 1", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261607, root := ~, start := 261595, stop := 261606 ), rec( content := " away fr\ om ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261626, root := ~, start := 261618, stop := 261625 ), rec( content := ", and ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ _i", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261642, root := ~, start := 261632, stop := 261641 ), rec( content := " neighbo\ rs of\n ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261668, root := ~, start := 261660, stop := 261667 ), rec( content := " which a\ re at distance ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ + 1", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261703, root := ~, start := 261691, stop := 261702 ), rec( content := " away fr\ om ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "C", next := 261722, root := ~, start := 261714, stop := 261721 ), rec( content := ". This\n\ definition is used to check whether ", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "A", next := 261783, root := ~, start := 261769, stop := 261782 ), rec( content := " is dist\ ance regular.", count := [ 6, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "P", next := 261808, root := ~, start := 261804, stop := 261807 ), rec( content := "\n\n \ In the case where ", count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "A", next := 261846, root := ~, start := 261832, stop := 261845 ), rec( content := " is not \ symmetric or not connected, the\n property is ", count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "K", next := 261914, root := ~, start := 261902, stop := 261913 ), rec( content := ".\n " , count := [ 6, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "P", next := 261924, root := ~, start := 261920, stop := 261923 ), rec( content := "\n \ \n If the argument ", count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "A", next := 261966, root := ~, start := 261952, stop := 261965 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphSymmetricClosure(ChainDigraph(5));;\ngap> IsDistanceRegularD\ igraph(D);\nfalse\ngap> D := Digraph([[2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, \ 3]]);\n\ngap> IsDistanceRegularDi\ graph(D);\ntrue\n", count := [ 6, 2, 4, 7 ], name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 7 ], mathmode := "Text", name := "Example", next := 262342, root := ~, start := 262067, stop := 262341 ), rec( content := "\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 262359, root := ~, start := 260802, stop := 262358 ) ], count := [ 6, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 262373, root := ~, start := 260681, stop := 262372 ), rec( content := "\n\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 262387, root := ~, start := 258330, stop := 262386 ) , rec( content := "\n\n ", count := [ 6, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Connectivity and \ cycles", count := [ 6, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 262442, root := ~, start := 262400, stop := 262441 ), rec( content := "\n ", count := [ 6, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsAcyclicDigrap\ h" ), content := 0, count := [ 6, 3, 1, 2 ], mathmode := "Text", name := "Prop", next := 262507, root := ~, start := 262462, stop := 262506 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "K", next := 262530, root := ~, start := 262519, stop := 262529 ), rec( content := " or ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "K", next := 262546, root := ~, start := 262534, stop := 262545 ), rec( content := ".", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 262557, root := ~, start := 262510, stop := 262556 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "K", next := 262606, root := ~, start := 262595, stop := 262605 ), rec( content := " if the \ digraph ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "A", next := 262636, root := ~, start := 262622, stop := 262635 ), rec( content := " is acyc\ lic, and\n ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "K", next := 262669, root := ~, start := 262657, stop := 262668 ), rec( content := " if it i\ s not. A digraph is ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ cyclic", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "E", next := 262711, root := ~, start := 262697, stop := 262710 ), rec( content := " if ever\ y directed\n cycle on the digraph is trivial. See Section ", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Ref", next := 262829, root := ~, start := 262779, stop := 262828 ), rec( content := " for the\ definition of a directed cycle, and of a trivial\n directed cycle.", count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "P", next := 262910, root := ~, start := 262906, stop := 262909 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "M", next := 262978, root := ~, start := 262965, stop := 262977 ), rec( content := " where\n\ ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "M", next := 262997, root := ~, start := 262989, stop := 262996 ), rec( content := " is the \ number of edges (counting multiple edges as one) and\n ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "M", next := 263070, root := ~, start := 263062, stop := 263069 ), rec( content := " is the \ number of vertices in the digraph. ", count := [ 6, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "P", next := 263117, root := ~, start := 263113, stop := 263116 ), rec( content := "\n\n \ If the argument ", count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "A", next := 263153, root := ~, start := 263139, stop := 263152 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x, y)\ \n> return IsEmpty(Intersection(x, y));\n> end);;\ngap> D := Digraph(Peterse\ n);\n\ngap> IsAcyclicDigraph(D);\ \nfalse\ngap> D := DigraphFromDiSparse6String(\n> \".b_OGCIDBaPGkULEbQHCeRIdrH\ cuZMfRyDAbPhTi|zF\");\n\ngap> Is\ AcyclicDigraph(D);\ntrue\ngap> IsAcyclicDigraph(ChainDigraph(10));\ntrue\ngap>\ D := CompleteDigraph(IsMutableDigraph, 4);\n\ngap> IsAcyclicDigraph(D);\nfalse\ngap> IsAcyclicDigraph(CycleDigra\ ph(10));\nfalse", count := [ 6, 3, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 6 ], mathmode := "Text", name := "Example", next := 263902, root := ~, start := 263254, stop := 263901 ), rec( content := "\n ", count := [ 6, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 263919, root := ~, start := 262560, stop := 263918 ) ], count := [ 6, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 263933, root := ~, start := 262447, stop := 263932 ), rec( content := "\n\n ", count := [ 6, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsChainDigraph"\ ), content := 0, count := [ 6, 3, 2, 2 ], mathmode := "Text", name := "Prop", next := 263997, root := ~, start := 263954, stop := 263996 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "K", next := 264020, root := ~, start := 264009, stop := 264019 ), rec( content := " or ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "K", next := 264036, root := ~, start := 264024, stop := 264035 ), rec( content := ".", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Returns", next := 264047, root := ~, start := 264000, stop := 264046 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sChainDigraph", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "C", next := 264089, root := ~, start := 264068, stop := 264088 ), rec( content := " returns\ ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "K", next := 264109, root := ~, start := 264098, stop := 264108 ), rec( content := " if the \ digraph ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "A", next := 264139, root := ~, start := 264125, stop := 264138 ), rec( content := " is\n \ isomorphic to the chain digraph with the same number of vertices as\n ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "A", next := 264233, root := ~, start := 264219, stop := 264232 ), rec( content := ", and ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "K", next := 264251, root := ~, start := 264239, stop := 264250 ), rec( content := " if it i\ s not; see ", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ChainDi\ graph" ), content := 0, count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Ref", next := 264302, root := ~, start := 264270, stop := 264301 ), rec( content := ".", count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "P", next := 264307, root := ~, start := 264303, stop := 264306 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ hain", count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "E", next := 264340, root := ~, start := 264328, stop := 264339 ), rec( content := " if and \ only if it is a directed tree, in which\n every vertex has out degree at mo\ st one; see ", count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsDirec\ tedTree" ), content := 0, count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "Ref", next := 264465, root := ~, start := 264437, stop := 264464 ), rec( content := "\n an\ d ", count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutDegr\ ees" ), content := 0, count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "Ref", next := 264498, root := ~, start := 264474, stop := 264497 ), rec( content := ".\n " , count := [ 6, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "P", next := 264508, root := ~, start := 264504, stop := 264507 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "A", next := 264550, root := ~, start := 264536, stop := 264549 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 3, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsChainDigraph(D);\nfalse\ngap> D := ChainDigraph(5);\n\ngap> IsChainDigraph(D);\ntrue\ngap> D := \ DigraphReverse(D);\n\ngap> IsChain\ Digraph(D);\ntrue\ngap> D := ChainDigraph(IsMutableDigraph, 5);\n\ngap> IsChainDigraph(D);\ntrue\ngap> DigraphReve\ rse(D);\n\ngap> IsChainDigraph(D);\n\ true", count := [ 6, 3, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 6 ], mathmode := "Text", name := "Example", next := 265209, root := ~, start := 264647, stop := 265208 ), rec( content := "\n ", count := [ 6, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 2, 3 ], mathmode := "Text", name := "Description", next := 265226, root := ~, start := 264050, stop := 265225 ) ], count := [ 6, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 265240, root := ~, start := 263939, stop := 265239 ), rec( content := "\n\n ", count := [ 6, 3, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsConnectedDigr\ aph" ), content := 0, count := [ 6, 3, 3, 2 ], mathmode := "Text", name := "Prop", next := 265308, root := ~, start := 265261, stop := 265307 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "K", next := 265331, root := ~, start := 265320, stop := 265330 ), rec( content := " or ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "K", next := 265347, root := ~, start := 265335, stop := 265346 ), rec( content := ".", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "Returns", next := 265358, root := ~, start := 265311, stop := 265357 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "K", next := 265407, root := ~, start := 265396, stop := 265406 ), rec( content := " if the \ digraph ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "A", next := 265437, root := ~, start := 265423, stop := 265436 ), rec( content := "\n is\ weakly connected and ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "K", next := 265478, root := ~, start := 265466, stop := 265477 ), rec( content := " if it i\ s not. A digraph\n ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "A", next := 265522, root := ~, start := 265508, stop := 265521 ), rec( content := " is ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "w\ eakly connected", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "E", next := 265549, root := ~, start := 265526, stop := 265548 ), rec( content := " if it i\ s possible to travel\n from any vertex to any other vertex by traversing ed\ ges in either\n direction (possibly against the orientation of some of them\ ). ", count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "P", next := 265718, root := ~, start := 265714, stop := 265717 ), rec( content := "\n\n \ The method used in this function has complexity ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m)", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "M", next := 265783, root := ~, start := 265772, stop := 265782 ), rec( content := " if the\\ n digraph's ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Source" ), content := 0, count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "Ref", next := 265832, root := ~, start := 265805, stop := 265831 ), rec( content := " attribu\ te is set, otherwise it has\n complexity ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "M", next := 265896, root := ~, start := 265883, stop := 265895 ), rec( content := " (where\\ n ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "M", next := 265916, root := ~, start := 265908, stop := 265915 ), rec( content := " is the \ number of edges and\n ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "M", next := 265956, root := ~, start := 265948, stop := 265955 ), rec( content := " is the \ number of vertices of the digraph).\n ", count := [ 6, 3, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "P", next := 266008, root := ~, start := 266004, stop := 266007 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "A", next := 266050, root := ~, start := 266036, stop := 266049 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2], [3], []]);;\ngap> IsConnectedDigraph(D);\ntrue\ngap> \ D := Digraph([[1, 3], [4], [3], []]);;\ngap> IsConnectedDigraph(D);\nfalse\nga\ p> D := Digraph(IsMutableDigraph, [[2], [3], []]);;\ngap> IsConnectedDigraph(D\ );\ntrue\ngap> D := Digraph(IsMutableDigraph, [[1, 3], [4], [3], []]);;\ngap> \ IsConnectedDigraph(D);\nfalse", count := [ 6, 3, 3, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 6 ], mathmode := "Text", name := "Example", next := 266512, root := ~, start := 266151, stop := 266511 ), rec( content := "\n ", count := [ 6, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 3, 3 ], mathmode := "Text", name := "Description", next := 266529, root := ~, start := 265361, stop := 266528 ) ], count := [ 6, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 266543, root := ~, start := 265246, stop := 266542 ), rec( content := "\n\n ", count := [ 6, 3, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsBiconnectedDi\ graph" ), content := 0, count := [ 6, 3, 4, 2 ], mathmode := "Text", name := "Prop", next := 266613, root := ~, start := 266564, stop := 266612 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "K", next := 266636, root := ~, start := 266625, stop := 266635 ), rec( content := " or ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "K", next := 266652, root := ~, start := 266640, stop := 266651 ), rec( content := ".", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 266663, root := ~, start := 266616, stop := 266662 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ onnected digraph is ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ iconnected", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "E", next := 266725, root := ~, start := 266707, stop := 266724 ), rec( content := " if it i\ s still connected (in the\n sense of ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 266804, root := ~, start := 266772, stop := 266803 ), rec( content := ") when a\ ny vertex is removed. \n If ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "A", next := 266852, root := ~, start := 266844, stop := 266851 ), rec( content := " has at \ least 3 vertices, then ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBiconnectedDigraph", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "C", next := 266910, root := ~, start := 266883, stop := 266909 ), rec( content := "\n \ implies ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 266958, root := ~, start := 266925, stop := 266957 ), rec( content := ";\n s\ ee ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 267000, root := ~, start := 266968, stop := 266999 ), rec( content := " or ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 267025, root := ~, start := 267004, stop := 267024 ), rec( content := " for a m\ ore\n detailed explanation.\n ", count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "P", next := 267071, root := ~, start := 267067, stop := 267070 ), rec( content := "\n\n \ ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBiconnectedDigraph", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "C", next := 267104, root := ~, start := 267077, stop := 267103 ), rec( content := " returns\ ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "K", next := 267124, root := ~, start := 267113, stop := 267123 ), rec( content := " if the \ digraph\n ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "A", next := 267158, root := ~, start := 267144, stop := 267157 ), rec( content := " is bico\ nnected, and ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "K", next := 267191, root := ~, start := 267179, stop := 267190 ), rec( content := " if it i\ s not. In\n particular, ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBiconnectedDigraph", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "C", next := 267252, root := ~, start := 267225, stop := 267251 ), rec( content := " returns\ ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "K", next := 267273, root := ~, start := 267261, stop := 267272 ), rec( content := " if\n \ ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "A", next := 267295, root := ~, start := 267281, stop := 267294 ), rec( content := " is not \ connected. ", count := [ 6, 3, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "P", next := 267318, root := ~, start := 267314, stop := 267317 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "A", next := 267360, root := ~, start := 267346, stop := 267359 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. ", count := [ 6, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 6 ], mathmode := "Text", name := "P", next := 267459, root := ~, start := 267455, stop := 267458 ), rec( content := "\n\n \ Multiple edges are ignored by this method. ", count := [ 6, 3, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "P", next := 267512, root := ~, start := 267508, stop := 267511 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "M", next := 267580, root := ~, start := 267567, stop := 267579 ), rec( content := " where\n\ ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "M", next := 267599, root := ~, start := 267591, stop := 267598 ), rec( content := " is the \ number of edges and ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "M", next := 267635, root := ~, start := 267627, stop := 267634 ), rec( content := " is the \ number of vertices in\n the digraph.\n ", count := [ 6, 3, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "P", next := 267690, root := ~, start := 267686, stop := 267689 ), rec( content := "\n\n \ See also ", count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "Ref", next := 267726, root := ~, start := 267705, stop := 267725 ), rec( content := ", ", count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "Ref", next := 267760, root := ~, start := 267728, stop := 267759 ), rec( content := ", and \n\ ", count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBridg\ elessDigraph" ), content := 0, count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "Ref", next := 267804, root := ~, start := 267771, stop := 267803 ), rec( content := ".\n " , count := [ 6, 3, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 4, 9 ], mathmode := "Text", name := "P", next := 267814, root := ~, start := 267810, stop := 267813 ), rec( content := "\n \ \n", count := [ 6, 3, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsBiconnectedDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBic\ onnectedDigraph(CycleDigraph(5));\ntrue\ngap> D := Digraph([[1, 1], [1, 1, 2],\ [3], [3, 3, 4, 4]]);;\ngap> IsBiconnectedDigraph(D);\nfalse\ngap> D := Comple\ teBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBiconnectedDigraph(D);\ntrue", count := [ 6, 3, 4, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 4, 10 ], mathmode := "Text", name := "Example", next := 268202, root := ~, start := 267822, stop := 268201 ), rec( content := "\n ", count := [ 6, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 268219, root := ~, start := 266666, stop := 268218 ) ], count := [ 6, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 268233, root := ~, start := 266549, stop := 268232 ), rec( content := "\n\n ", count := [ 6, 3, 4, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D", Name := "IsBridgelessDig\ raph" ), content := 0, count := [ 6, 3, 5, 2 ], mathmode := "Text", name := "Prop", next := 268296, root := ~, start := 268254, stop := 268295 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "K", next := 268319, root := ~, start := 268308, stop := 268318 ), rec( content := " or ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "K", next := 268335, root := ~, start := 268323, stop := 268334 ), rec( content := ".", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Returns", next := 268346, root := ~, start := 268299, stop := 268345 ), rec( attributes := rec( ), content := [ rec( content := "\n A c\ onnected digraph is ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "b\ ridgeless", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "E", next := 268407, root := ~, start := 268390, stop := 268406 ), rec( content := " if it i\ s still connected (in the\n sense of ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsConne\ ctedDigraph" ), content := 0, count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Ref", next := 268486, root := ~, start := 268454, stop := 268485 ), rec( content := ") when a\ ny edge is removed. \n If ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "A", next := 268530, root := ~, start := 268522, stop := 268529 ), rec( content := " has at \ least 3 vertices, then ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Ref", next := 268599, root := ~, start := 268561, stop := 268598 ), rec( content := " implies\ ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "I\ sBridgelessDigraph", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "C", next := 268634, root := ~, start := 268608, stop := 268633 ), rec( content := ";\n s\ ee ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Ref", next := 268676, root := ~, start := 268644, stop := 268675 ), rec( content := " or ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Ref", next := 268701, root := ~, start := 268680, stop := 268700 ), rec( content := " for a m\ ore\n detailed explanation.\n ", count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "P", next := 268747, root := ~, start := 268743, stop := 268746 ), rec( content := "\n\n \ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBridgelessDigraph", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "C", next := 268779, root := ~, start := 268753, stop := 268778 ), rec( content := " returns\ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "K", next := 268799, root := ~, start := 268788, stop := 268798 ), rec( content := " if the \ digraph\n ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "A", next := 268827, root := ~, start := 268819, stop := 268826 ), rec( content := " is brid\ geless, and ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "K", next := 268859, root := ~, start := 268847, stop := 268858 ), rec( content := " if it i\ s not. In\n particular, ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBridgelessDigraph", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "C", next := 268919, root := ~, start := 268893, stop := 268918 ), rec( content := " returns\ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "K", next := 268940, root := ~, start := 268928, stop := 268939 ), rec( content := " if\n \ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "A", next := 268956, root := ~, start := 268948, stop := 268955 ), rec( content := " is not \ connected. ", count := [ 6, 3, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "P", next := 268979, root := ~, start := 268975, stop := 268978 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "A", next := 269015, root := ~, start := 269007, stop := 269014 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. ", count := [ 6, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 6 ], mathmode := "Text", name := "P", next := 269114, root := ~, start := 269110, stop := 269113 ), rec( content := "\n\n \ Multiple edges are ignored by this method. ", count := [ 6, 3, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "P", next := 269167, root := ~, start := 269163, stop := 269166 ), rec( content := "\n\n \ The method used in this operation has complexity ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "M", next := 269235, root := ~, start := 269222, stop := 269234 ), rec( content := " where\n\ ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "M", next := 269254, root := ~, start := 269246, stop := 269253 ), rec( content := " is the \ number of edges and ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "M", next := 269290, root := ~, start := 269282, stop := 269289 ), rec( content := " is the \ number of vertices in\n the digraph.\n ", count := [ 6, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "P", next := 269345, root := ~, start := 269341, stop := 269344 ), rec( content := "\n\n \ See also ", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Bridges\ " ), content := 0, count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "Ref", next := 269381, root := ~, start := 269360, stop := 269380 ), rec( content := ", ", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Articul\ ationPoints" ), content := 0, count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "Ref", next := 269415, root := ~, start := 269383, stop := 269414 ), rec( content := ", and \n\ ", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBicon\ nectedDigraph" ), content := 0, count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "Ref", next := 269460, root := ~, start := 269426, stop := 269459 ), rec( content := ".\n \ \n", count := [ 6, 3, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsBridgelessDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBrid\ gelessDigraph(CycleDigraph(5));\ntrue\ngap> D := Digraph([[1, 1], [1, 1, 2], [\ 3], [3, 3, 4, 4]]);;\ngap> IsBridgelessDigraph(D);\nfalse\ngap> D := CompleteB\ ipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBridgelessDigraph(D);\ntrue\ngap> D := Digraph([[2, 5], [1, 3,\ 4, 5], [2, 4], [2, 3], [1, 2]]);\n\ngap> IsBridgelessDigraph(D);\ntrue\ngap> IsBiconnectedDigraph(D);\nfalse\n\ gap> D := Digraph([[2], [3], [4], [2]]);\n\ngap> IsBridgelessDigraph(D);\nfalse\ngap> IsBiconnectedDigraph(D);\n\ false\ngap> IsBridgelessDigraph(ChainDigraph(2));\nfalse\ngap> IsBiconnectedDi\ graph(ChainDigraph(2));\ntrue\n", count := [ 6, 3, 5, 9 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 9 ], mathmode := "Text", name := "Example", next := 270284, root := ~, start := 269469, stop := 270283 ), rec( content := "\n ", count := [ 6, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 5, 3 ], mathmode := "Text", name := "Description", next := 270301, root := ~, start := 268349, stop := 270300 ) ], count := [ 6, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 270315, root := ~, start := 268239, stop := 270314 ), rec( content := "\n\n ", count := [ 6, 3, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsStronglyConne\ ctedDigraph" ), content := 0, count := [ 6, 3, 6, 2 ], mathmode := "Text", name := "Prop", next := 270391, root := ~, start := 270336, stop := 270390 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "K", next := 270414, root := ~, start := 270403, stop := 270413 ), rec( content := " or ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "K", next := 270430, root := ~, start := 270418, stop := 270429 ), rec( content := ".", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 270441, root := ~, start := 270394, stop := 270440 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "K", next := 270490, root := ~, start := 270479, stop := 270489 ), rec( content := " if the \ digraph ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "A", next := 270520, root := ~, start := 270506, stop := 270519 ), rec( content := " is stro\ ngly\n connected and ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "K", next := 270563, root := ~, start := 270551, stop := 270562 ), rec( content := " if it i\ s not. ", count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "P", next := 270582, root := ~, start := 270578, stop := 270581 ), rec( content := "\n\n \ A digraph ", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "A", next := 270612, root := ~, start := 270598, stop := 270611 ), rec( content := " is ", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ trongly connected", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "E", next := 270641, root := ~, start := 270616, stop := 270640 ), rec( content := " if ther\ e is a directed\n path from every vertex to every other vertex. See Sectio\ n ", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "Ref", next := 270778, root := ~, start := 270728, stop := 270777 ), rec( content := " for the\ definition of a directed\n path. ", count := [ 6, 3, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "P", next := 270829, root := ~, start := 270825, stop := 270828 ), rec( content := "\n\n \ The method used in this operation is based on Gabow's Algorithm ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "Gab00" ) , content := 0, count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "Cite", next := 270924, root := ~, start := 270899, stop := 270923 ), rec( content := " and has\ complexity ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ (m+n)", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "M", next := 270957, root := ~, start := 270944, stop := 270956 ), rec( content := ", where \ ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "M", next := 270973, root := ~, start := 270965, stop := 270972 ), rec( content := " is\n \ the number of edges (counting multiple edges as one) and ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "M", next := 271046, root := ~, start := 271038, stop := 271045 ), rec( content := " is the\\ n number of vertices in the digraph.\n ", count := [ 6, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "P", next := 271101, root := ~, start := 271097, stop := 271100 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "A", next := 271143, root := ~, start := 271129, stop := 271142 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 6, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(250000);\n\ngap> IsStronglyConnectedDigraph(D);\ntrue\ngap> D := DigraphRemoveEdges(D,\ [[250000, 1]]);\n\ngap>\ IsStronglyConnectedDigraph(D);\nfalse\ngap> D := CycleDigraph(IsMutableDigrap\ h, 250000);\n\ngap> IsStro\ nglyConnectedDigraph(D);\ntrue\ngap> DigraphRemoveEdge(D, [250000, 1]);\n\ngap> IsStronglyConnectedDigra\ ph(D);\nfalse\n", count := [ 6, 3, 6, 7 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 7 ], mathmode := "Text", name := "Example", next := 271820, root := ~, start := 271244, stop := 271819 ), rec( content := "\n ", count := [ 6, 3, 6, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 271837, root := ~, start := 270444, stop := 271836 ) ], count := [ 6, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 271851, root := ~, start := 270321, stop := 271850 ), rec( content := "\n\n ", count := [ 6, 3, 6, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsAperiodicDigr\ aph" ), content := 0, count := [ 6, 3, 7, 2 ], mathmode := "Text", name := "Prop", next := 271919, root := ~, start := 271872, stop := 271918 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "K", next := 271942, root := ~, start := 271931, stop := 271941 ), rec( content := " or ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "K", next := 271958, root := ~, start := 271946, stop := 271957 ), rec( content := ".", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Returns", next := 271969, root := ~, start := 271922, stop := 271968 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property is ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "K", next := 272018, root := ~, start := 272007, stop := 272017 ), rec( content := " if the \ digraph ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "A", next := 272048, root := ~, start := 272034, stop := 272047 ), rec( content := "\n is\ aperiodic, i.e. if its ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Period" ), content := 0, count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Ref", next := 272108, root := ~, start := 272079, stop := 272107 ), rec( content := " is equa\ l to 1.\n Otherwise, the property is ", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "K", next := 272167, root := ~, start := 272155, stop := 272166 ), rec( content := ".\n " , count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "P", next := 272177, root := ~, start := 272173, stop := 272176 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "A", next := 272219, root := ~, start := 272205, stop := 272218 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n ", count := [ 6, 3, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[6], [1], [2], [3], [4, 4], [5]]);\n\ngap> IsAperiodicDigraph(D);\nfalse\ngap> D := Dig\ raph([[2], [3, 5], [4], [5], [1, 2]]);\n\ngap> IsAperiodicDigraph(D);\ntrue\ngap> D := Digraph(IsMutableDigraph,\ [[2], [3, 5], [4], [5], [1, 2]]);\n\ \ngap> IsAperiodicDigraph(D);\ntrue", count := [ 6, 3, 7, 5 ], name := "PCDATA\ ", root := ~ ) ], count := [ 6, 3, 7, 5 ], mathmode := "Text", name := "Example", next := 272764, root := ~, start := 272319, stop := 272763 ), rec( content := "\n ", count := [ 6, 3, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 7, 3 ], mathmode := "Text", name := "Description", next := 272781, root := ~, start := 271972, stop := 272780 ) ], count := [ 6, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 272795, root := ~, start := 271857, stop := 272794 ), rec( content := "\n\n ", count := [ 6, 3, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsDirectedTree"\ ), content := 0, count := [ 6, 3, 8, 2 ], mathmode := "Text", name := "Prop", next := 272859, root := ~, start := 272816, stop := 272858 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "K", next := 272882, root := ~, start := 272871, stop := 272881 ), rec( content := " or ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "K", next := 272898, root := ~, start := 272886, stop := 272897 ), rec( content := ".", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 272909, root := ~, start := 272862, stop := 272908 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "K", next := 272949, root := ~, start := 272938, stop := 272948 ), rec( content := " if the \ digraph ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "A", next := 272979, root := ~, start := 272965, stop := 272978 ), rec( content := " is a di\ rected tree, and\n ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "K", next := 273020, root := ~, start := 273008, stop := 273019 ), rec( content := " if it i\ s not. ", count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "P", next := 273039, root := ~, start := 273035, stop := 273038 ), rec( content := "\n\n \ A ", count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ irected tree", count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "E", next := 273067, root := ~, start := 273047, stop := 273066 ), rec( content := " is an a\ cyclic digraph with precisely 1 source,\n such that no two vertices share a\ n out-neighbour.\n Note the empty digraph is not considered a directed\n \ tree as it has no source. ", count := [ 6, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "P", next := 273259, root := ~, start := 273255, stop := 273258 ), rec( content := "\n\n \ See also ", count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Sources" ), content := 0, count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "Ref", next := 273302, root := ~, start := 273274, stop := 273301 ), rec( content := ".\n " , count := [ 6, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "P", next := 273312, root := ~, start := 273308, stop := 273311 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "A", next := 273354, root := ~, start := 273340, stop := 273353 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[], [2]]);\n\ng\ ap> IsDirectedTree(D);\nfalse\ngap> D := Digraph([[3], [3], []]);\n\ngap> IsDirectedTree(D);\nfalse\ngap> D := D\ igraph([[2], [3], []]);\n\ngap> Is\ DirectedTree(D);\ntrue\ngap> D := Digraph([[2, 3], [6], [4, 5], [], [], []]);\ \n\ngap> IsDirectedTree(D);\ntrue\ \n", count := [ 6, 3, 8, 7 ], name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 7 ], mathmode := "Text", name := "Example", next := 273938, root := ~, start := 273455, stop := 273937 ), rec( content := "\n ", count := [ 6, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 273955, root := ~, start := 272912, stop := 273954 ) ], count := [ 6, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 273969, root := ~, start := 272801, stop := 273968 ), rec( content := "\n\n ", count := [ 6, 3, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsUndirectedTre\ e" ), content := 0, count := [ 6, 3, 9, 2 ], mathmode := "Text", name := "Prop", next := 274035, root := ~, start := 273990, stop := 274034 ), rec( attributes := rec( Arg := "digraph", Name := "IsUndirectedFor\ est" ), content := 0, count := [ 6, 3, 9, 4 ], mathmode := "Text", name := "Prop", next := 274085, root := ~, start := 274038, stop := 274084 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 274108, root := ~, start := 274097, stop := 274107 ), rec( content := " or ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 274124, root := ~, start := 274112, stop := 274123 ), rec( content := ".", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 274135, root := ~, start := 274088, stop := 274134 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ property ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedTree", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "C", next := 274192, root := ~, start := 274169, stop := 274191 ), rec( content := " returns\ ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 274212, root := ~, start := 274201, stop := 274211 ), rec( content := " if the \ digraph\n ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "A", next := 274246, root := ~, start := 274232, stop := 274245 ), rec( content := " is an u\ ndirected tree, and the property\n ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sUndirectedForest", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "C", next := 274316, root := ~, start := 274291, stop := 274315 ), rec( content := " returns\ ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 274336, root := ~, start := 274325, stop := 274335 ), rec( content := " if ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "A", next := 274354, root := ~, start := 274340, stop := 274353 ), rec( content := " is an\n\ undirected forest; otherwise, these properties return ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "K", next := 274431, root := ~, start := 274419, stop := 274430 ), rec( content := ". ", count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "P", next := 274437, root := ~, start := 274433, stop := 274436 ), rec( content := "\n\n \ An ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected tree", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "E", next := 274468, root := ~, start := 274446, stop := 274467 ), rec( content := " is a sy\ mmetric digraph without loops, in which for\n any pair of distinct vertices\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "C", next := 274562, root := ~, start := 274554, stop := 274561 ), rec( content := " and ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "C", next := 274575, root := ~, start := 274567, stop := 274574 ), rec( content := ", there \ is exactly one\n directed path from ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "C", next := 274629, root := ~, start := 274621, stop := 274628 ), rec( content := " to ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "C", next := 274641, root := ~, start := 274633, stop := 274640 ), rec( content := ". See ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "Ref", next := 274685, root := ~, start := 274647, stop := 274684 ), rec( content := " and ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "Ref", next := 274719, root := ~, start := 274690, stop := 274718 ), rec( content := ", and se\ e\n Section ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "Ref", next := 274785, root := ~, start := 274741, stop := 274784 ), rec( content := " for the\ definition of\n directed path. This definition implies that an undirected \ tree has\n no multiple edges. ", count := [ 6, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "P", next := 274906, root := ~, start := 274902, stop := 274905 ), rec( content := "\n\n \ An ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ndirected forest", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "E", next := 274939, root := ~, start := 274915, stop := 274938 ), rec( content := " is a di\ graph, each of whose connected components\n is an undirected tree. In other\ words, an undirected forest is isomorphic to\n a disjoint union of undirec\ ted trees. See ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "Ref", next := 275163, root := ~, start := 275116, stop := 275162 ), rec( content := " and ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ DisjointUnion", Label := "for a list of digraphs" ), content := 0, count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "Ref", next := 275240, root := ~, start := 275168, stop := 275239 ), rec( content := ". In par\ ticular, every\n undirected tree is an undirected forest. ", count := [ 6, 3, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 8 ], mathmode := "Text", name := "P", next := 275312, root := ~, start := 275308, stop := 275311 ), rec( content := "\n\n \ Please note that the digraph with zero vertices is considered to be neither\n \ an undirected tree nor an undirected forest.\n ", count := [ 6, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "P", next := 275451, root := ~, start := 275447, stop := 275450 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "A", next := 275493, root := ~, start := 275479, stop := 275492 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [3], [1, 2]]);\n\ngap> IsUndirectedTree(D);\ntrue\ngap> IsSymmetricDigraph(D) and not D\ igraphHasLoops(D);\ntrue\ngap> D := Digraph([[3], [5], [1, 4], [3], [2]]);\n\ngap> IsConnectedDigraph(D);\nfalse\ \ngap> IsUndirectedTree(D);\nfalse\ngap> IsUndirectedForest(D);\ntrue\ngap> D \ := Digraph([[1, 2], [1], [2]]);\n\ \ngap> IsUndirectedTree(D) or IsUndirectedForest(D);\nfalse\ngap> IsSymmetricD\ igraph(D) or not DigraphHasLoops(D);\nfalse", count := [ 6, 3, 9, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 9, 10 ], mathmode := "Text", name := "Example", next := 276194, root := ~, start := 275594, stop := 276193 ), rec( content := "\n ", count := [ 6, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 276211, root := ~, start := 274138, stop := 276210 ) ], count := [ 6, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 276225, root := ~, start := 273975, stop := 276224 ), rec( content := "\n\n ", count := [ 6, 3, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEulerianDigra\ ph" ), content := 0, count := [ 6, 3, 10, 2 ], mathmode := "Text", name := "Prop", next := 276292, root := ~, start := 276246, stop := 276291 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "K", next := 276315, root := ~, start := 276304, stop := 276314 ), rec( content := " or ", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "K", next := 276331, root := ~, start := 276319, stop := 276330 ), rec( content := ".", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "Returns", next := 276342, root := ~, start := 276295, stop := 276341 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property returns true if the digraph ", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "A", next := 276419, root := ~, start := 276405, stop := 276418 ), rec( content := " is Eule\ rian.\n ", count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "P", next := 276441, root := ~, start := 276437, stop := 276440 ), rec( content := "\n\n \ A connected digraph is called ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ ulerian", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "E", next := 276492, root := ~, start := 276477, stop := 276491 ), rec( content := " if ther\ e exists a directed\n circuit on the digraph which includes every edge exac\ tly once. See\n Section ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "Ref", next := 276648, root := ~, start := 276604, stop := 276647 ), rec( content := " for the\ definition of\n a directed circuit. Note that the empty digraph with at mo\ st one vertex is\n considered to be Eulerian.\n ", count := [ 6, 3, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "P", next := 276789, root := ~, start := 276785, stop := 276788 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "A", next := 276831, root := ~, start := 276817, stop := 276830 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[]]);\n\ngap> IsEul\ erianDigraph(D);\ntrue\ngap> D := Digraph([[2], []]);\n\ngap> IsEulerianDigraph(D);\nfalse\ngap> D := Digraph([[3\ ], [], [2]]);\n\ngap> IsEulerianDi\ graph(D);\nfalse\ngap> D := Digraph([[2], [3], [1]]);\n\ngap> IsEulerianDigraph(D);\ntrue\n", count := [ 6, 3, 10, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 10, 6 ], mathmode := "Text", name := "Example", next := 277394, root := ~, start := 276932, stop := 277393 ), rec( content := "\n ", count := [ 6, 3, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 10, 3 ], mathmode := "Text", name := "Description", next := 277411, root := ~, start := 276345, stop := 277410 ) ], count := [ 6, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 277425, root := ~, start := 276231, stop := 277424 ), rec( content := "\n\n ", count := [ 6, 3, 10, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsHamiltonianDi\ graph" ), content := 0, count := [ 6, 3, 11, 2 ], mathmode := "Text", name := "Prop", next := 277495, root := ~, start := 277446, stop := 277494 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "K", next := 277518, root := ~, start := 277507, stop := 277517 ), rec( content := " or ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "K", next := 277534, root := ~, start := 277522, stop := 277533 ), rec( content := ".", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "Returns", next := 277545, root := ~, start := 277498, stop := 277544 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "A", next := 277583, root := ~, start := 277569, stop := 277582 ), rec( content := " is Hami\ ltonian, then this property returns\n ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "K", next := 277642, root := ~, start := 277631, stop := 277641 ), rec( content := ", and ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "K", next := 277660, root := ~, start := 277648, stop := 277659 ), rec( content := " if it i\ s not. ", count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "P", next := 277679, root := ~, start := 277675, stop := 277678 ), rec( content := "\n\n \ A digraph with ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "C", next := 277708, root := ~, start := 277700, stop := 277707 ), rec( content := " vertice\ s is ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ amiltonian", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "E", next := 277739, root := ~, start := 277721, stop := 277738 ), rec( content := " if it h\ as a\n directed cycle of length ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "C", next := 277789, root := ~, start := 277781, stop := 277788 ), rec( content := ". See Se\ ction ", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Style := "Number\ ", Subsect := "Definitions" ), content := 0, count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "Ref", next := 277852, root := ~, start := 277803, stop := 277851 ), rec( content := " for the\ definition of a directed cycle.\n Note the empty digraphs on 0 and 1 verti\ ces are considered to be\n Hamiltonian.", count := [ 6, 3, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "P", next := 277982, root := ~, start := 277978, stop := 277981 ), rec( content := "\n\n \ The method used in this operation has the worst case complexity as\n ", count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Monomorphism" ), content := 0, count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "Ref", next := 278092, root := ~, start := 278059, stop := 278091 ), rec( content := ".\n " , count := [ 6, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "P", next := 278102, root := ~, start := 278098, stop := 278101 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "A", next := 278144, root := ~, start := 278130, stop := 278143 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> g := Digraph([[]]);\n\ngap> IsHam\ iltonianDigraph(g);\ntrue\ngap> g := Digraph([[2], [1]]);\n\ngap> IsHamiltonianDigraph(g);\ntrue\ngap> g := Digr\ aph([[3], [], [2]]);\n\ngap> IsHam\ iltonianDigraph(g);\nfalse\ngap> g := Digraph([[2], [3], [1]]);\n\ngap> IsHamiltonianDigraph(g);\ntrue\n", count := [ 6, 3, 11, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 11, 7 ], mathmode := "Text", name := "Example", next := 278720, root := ~, start := 278245, stop := 278719 ), rec( content := "\n ", count := [ 6, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 11, 3 ], mathmode := "Text", name := "Description", next := 278737, root := ~, start := 277548, stop := 278736 ) ], count := [ 6, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 278751, root := ~, start := 277431, stop := 278750 ), rec( content := "\n\n ", count := [ 6, 3, 11, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsCycleDigraph"\ ), content := 0, count := [ 6, 3, 12, 2 ], mathmode := "Text", name := "Prop", next := 278815, root := ~, start := 278772, stop := 278814 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "K", next := 278838, root := ~, start := 278827, stop := 278837 ), rec( content := " or ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "K", next := 278854, root := ~, start := 278842, stop := 278853 ), rec( content := ".", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "Returns", next := 278865, root := ~, start := 278818, stop := 278864 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sCycleDigraph", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "C", next := 278907, root := ~, start := 278886, stop := 278906 ), rec( content := " returns\ ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "K", next := 278927, root := ~, start := 278916, stop := 278926 ), rec( content := " if the \ digraph ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "A", next := 278957, root := ~, start := 278943, stop := 278956 ), rec( content := " is\n \ isomorphic to the cycle digraph with the same number of vertices as\n ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "A", next := 279051, root := ~, start := 279037, stop := 279050 ), rec( content := ", and ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "K", next := 279069, root := ~, start := 279057, stop := 279068 ), rec( content := " if it i\ s not; see ", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CycleDi\ graph" ), content := 0, count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "Ref", next := 279120, root := ~, start := 279088, stop := 279119 ), rec( content := ".", count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "P", next := 279125, root := ~, start := 279121, stop := 279124 ), rec( content := "\n\n \ A digraph is a ", count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ycle", count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "E", next := 279158, root := ~, start := 279146, stop := 279157 ), rec( content := " if and \ only if it is strongly connected and has\n the same number of edges as vert\ ices. \n ", count := [ 6, 3, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "P", next := 279258, root := ~, start := 279254, stop := 279257 ), rec( content := "\n \ \n If the argument ", count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "A", next := 279300, root := ~, start := 279286, stop := 279299 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n", count := [ 6, 3, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsCycleDigraph(D);\nfalse\ngap> D := CycleDigraph(5);\n\ngap> IsCycleDigraph(D);\ntrue\ngap> D := \ OnDigraphs(D, (1, 2, 3));\n\ngap> \ D = CycleDigraph(5);\nfalse\ngap> IsCycleDigraph(D);\ntrue", count := [ 6, 3, 12, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 3, 12, 6 ], mathmode := "Text", name := "Example", next := 279785, root := ~, start := 279397, stop := 279784 ), rec( content := "\n ", count := [ 6, 3, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 279802, root := ~, start := 278868, stop := 279801 ) ], count := [ 6, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 279816, root := ~, start := 278757, stop := 279815 ), rec( content := "\n\n ", count := [ 6, 3, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 279830, root := ~, start := 262391, stop := 279829 ) , rec( content := "\n \n ", count := [ 6, 3, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Planarity", count := [ 6, 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 279873, root := ~, start := 279845, stop := 279872 ), rec( content := "\n ", count := [ 6, 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsPlanarDigraph\ " ), content := 0, count := [ 6, 4, 1, 2 ], mathmode := "Text", name := "Prop", next := 279937, root := ~, start := 279893, stop := 279936 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "K", next := 279960, root := ~, start := 279949, stop := 279959 ), rec( content := " or ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "K", next := 279976, root := ~, start := 279964, stop := 279975 ), rec( content := ".", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "Returns", next := 279987, root := ~, start := 279940, stop := 279986 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ lanar", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "E", next := 280023, root := ~, start := 280010, stop := 280022 ), rec( content := " digraph\ is a digraph that can be embedded in the plane in\n such a way that its ed\ ges do not intersect. A digraph is planar if and only\n if it does not have\ a subdigraph that is homeomorphic to either the\n complete graph on ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "5\ ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "C", next := 280264, root := ~, start := 280256, stop := 280263 ), rec( content := " vertice\ s or the complete bipartite graph with\n vertex sets of sizes ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "C", next := 280344, root := ~, start := 280336, stop := 280343 ), rec( content := " and ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "C", next := 280357, root := ~, start := 280349, stop := 280356 ), rec( content := ".\n " , count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "P", next := 280367, root := ~, start := 280363, stop := 280366 ), rec( content := "\n \n\ ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPlanarDigraph", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "C", next := 280399, root := ~, start := 280377, stop := 280398 ), rec( content := " returns\ ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "K", next := 280419, root := ~, start := 280408, stop := 280418 ), rec( content := " if the \ digraph ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "A", next := 280449, root := ~, start := 280435, stop := 280448 ), rec( content := " is\n \ planar and ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "K", next := 280480, root := ~, start := 280468, stop := 280479 ), rec( content := " if it i\ s not. The directions and multiplicities of\n any edges in ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "A", next := 280563, root := ~, start := 280549, stop := 280562 ), rec( content := " are ign\ ored by ", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPlanarDigraph", count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "C", next := 280601, root := ~, start := 280579, stop := 280600 ), rec( content := ".\n " , count := [ 6, 4, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 1, 5 ], mathmode := "Text", name := "P", next := 280611, root := ~, start := 280607, stop := 280610 ), rec( content := " \n\n \ See also ", count := [ 6, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOuter\ PlanarDigraph" ), content := 0, count := [ 6, 4, 1, 5 ], mathmode := "Text", name := "Ref", next := 280661, root := ~, start := 280627, stop := 280660 ), rec( content := ".\n " , count := [ 6, 4, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "P", next := 280671, root := ~, start := 280667, stop := 280670 ), rec( content := "\n\n \ This method uses the reference implementation in\n ", count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 6, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 6 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 6, 4, 1, 6 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 6, 4, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 6 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "Cite", next := 280829, root := ~, start := 280811, stop := 280828 ), rec( content := ".\n\n \ ", count := [ 6, 4, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsPlanarDigraph(CompleteDigraph(4));\ntrue\ngap> IsPlanarDigraph(Complet\ eDigraph(5));\nfalse\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(2, 3));\nt\ rue\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(3, 3));\nfalse\ngap> IsPlan\ arDigraph(CompleteDigraph(IsMutableDigraph, 4));\ntrue\ngap> IsPlanarDigraph(C\ ompleteDigraph(IsMutableDigraph, 5));\nfalse\ngap> IsPlanarDigraph(CompleteBip\ artiteDigraph(IsMutableDigraph, 2, 3));\ntrue\ngap> IsPlanarDigraph(CompleteBi\ partiteDigraph(IsMutableDigraph, 3, 3));\nfalse\n", count := [ 6, 4, 1, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 4, 1, 7 ], mathmode := "Text", name := "Example", next := 281368, root := ~, start := 280836, stop := 281367 ), rec( content := "\n ", count := [ 6, 4, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 1, 3 ], mathmode := "Text", name := "Description", next := 281385, root := ~, start := 279990, stop := 281384 ) ], count := [ 6, 4, 1, 1 ], mathmode := "Text", name := "ManSection", next := 281399, root := ~, start := 279878, stop := 281398 ), rec( content := "\n\n ", count := [ 6, 4, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsOuterPlanarDi\ graph" ), content := 0, count := [ 6, 4, 2, 2 ], mathmode := "Text", name := "Prop", next := 281469, root := ~, start := 281420, stop := 281468 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "K", next := 281492, root := ~, start := 281481, stop := 281491 ), rec( content := " or ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "K", next := 281508, root := ~, start := 281496, stop := 281507 ), rec( content := ".", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "Returns", next := 281519, root := ~, start := 281472, stop := 281518 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ uter planar", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "E", next := 281562, root := ~, start := 281543, stop := 281561 ), rec( content := " digraph\ is a digraph that can be embedded in the\n plane in such a way that its ed\ ges do not intersect, and all vertices\n belong to the unbounded face of th\ e embedding. A digraph is outer planar\n if and only if it does not have a\ subdigraph that is homeomorphic to either\n the complete graph on ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "4\ ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "C", next := 281878, root := ~, start := 281870, stop := 281877 ), rec( content := " vertice\ s or the complete bipartite graph\n with vertex sets of sizes ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "C", next := 281958, root := ~, start := 281950, stop := 281957 ), rec( content := " and ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "3\ ", count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "C", next := 281971, root := ~, start := 281963, stop := 281970 ), rec( content := ".\n " , count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "P", next := 281981, root := ~, start := 281977, stop := 281980 ), rec( content := "\n \n\ ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sOuterPlanarDigraph", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "C", next := 282018, root := ~, start := 281991, stop := 282017 ), rec( content := " returns\ ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "K", next := 282038, root := ~, start := 282027, stop := 282037 ), rec( content := " if the \ digraph\n ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "A", next := 282072, root := ~, start := 282058, stop := 282071 ), rec( content := " is oute\ r planar and ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "K", next := 282105, root := ~, start := 282093, stop := 282104 ), rec( content := " if it i\ s not. The\n directions and multiplicities of any edges in ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "A", next := 282188, root := ~, start := 282174, stop := 282187 ), rec( content := " are ign\ ored by\n ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sPlanarDigraph", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "C", next := 282230, root := ~, start := 282208, stop := 282229 ), rec( content := ". ", count := [ 6, 4, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "P", next := 282237, root := ~, start := 282233, stop := 282236 ), rec( content := "\n\n \ See also ", count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPlana\ rDigraph" ), content := 0, count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "Ref", next := 282281, root := ~, start := 282252, stop := 282280 ), rec( content := ".\n\n \ This method uses the reference implementation in\n ", count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "edge-addition-planarity-suite" ), content := [ rec( content := \ "https://github.com/graph-algorithms/edge-addition-planarity-suite", count := [ \ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 5 ], mathmode := \ "Text", name := "URL", next := 140, root := ~, start := 17, stop := 139 ) ], count := [ 6, 4, 2, 5 ], name := "Alt", next := 146, root := ~, start := 1, stop := 145 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "edge-addition-planarity-suite", count \ := [ 6, 4, 2, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 5 ], name := "Pac\ kage", next := 218, root := ~, start := 170, stop := 217 ) ], count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "Alt", next := 224, root := ~, start := 146, stop := 223 ), rec( content := " by John\ Boyer of the algorithms described\n in ", count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "BM06" ), content := 0, count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "Cite", next := 282440, root := ~, start := 282422, stop := 282439 ), rec( content := ".\n\n", count := [ 6, 4, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsOuterPlanarDigraph(CompleteDigraph(4));\nfalse\ngap> IsOuterPlanarDigr\ aph(CompleteDigraph(5));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipartiteDi\ graph(2, 3));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipartiteDigraph(3, 3)\ );\nfalse\ngap> IsOuterPlanarDigraph(CycleDigraph(10));\ntrue\ngap> IsOuterPla\ narDigraph(CompleteDigraph(IsMutableDigraph, 4));\nfalse\ngap> IsOuterPlanarDi\ graph(CompleteDigraph(IsMutableDigraph, 5));\nfalse\ngap> IsOuterPlanarDigraph\ (CompleteBipartiteDigraph(IsMutableDigraph,\n> \ 2, 3));\nfalse\ngap> IsOuterPlanarDigraph(CompleteBipartiteD\ igraph(IsMutableDigraph,\n> 3\ , 3));\nfalse\ngap> IsOuterPlanarDigraph(CycleDigraph(IsMutableDigraph, 10));\ \ntrue\n\n", count := [ 6, 4, 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 6 ], mathmode := "Text", name := "Example", next := 283240, root := ~, start := 282443, stop := 283239 ), rec( content := "\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 2, 3 ], mathmode := "Text", name := "Description", next := 283257, root := ~, start := 281522, stop := 283256 ) ], count := [ 6, 4, 2, 1 ], mathmode := "Text", name := "ManSection", next := 283271, root := ~, start := 281405, stop := 283270 ), rec( content := "\n\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 4, 0, 1 ], mathmode := "Text", name := "Section", next := 283285, root := ~, start := 279836, stop := 283284 ) , rec( content := "\n\n ", count := [ 6, 4, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Homomorphisms and\ transformations", count := [ 6, 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 283350, root := ~, start := 283298, stop := 283349 ), rec( content := "\n ", count := [ 6, 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsDigraphCore" ), content := 0, count := [ 6, 5, 1, 2 ], mathmode := "Text", name := "Prop", next := 283412, root := ~, start := 283370, stop := 283411 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "K", next := 283435, root := ~, start := 283424, stop := 283434 ), rec( content := " or ", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "K", next := 283451, root := ~, start := 283439, stop := 283450 ), rec( content := ".", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "Returns", next := 283462, root := ~, start := 283415, stop := 283461 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s property returns ", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "K", next := 283516, root := ~, start := 283505, stop := 283515 ), rec( content := " if ", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "A", next := 283534, root := ~, start := 283520, stop := 283533 ), rec( content := " is a co\ re,\n and ", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "K", next := 283566, root := ~, start := 283554, stop := 283565 ), rec( content := " if it i\ s not.", count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "P", next := 283584, root := ~, start := 283580, stop := 283583 ), rec( content := "\n\n \ A digraph ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 283608, root := ~, start := 283600, stop := 283607 ), rec( content := " is a ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ore", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "E", next := 283625, root := ~, start := 283614, stop := 283624 ), rec( content := " if and \ only if it has no proper\n subdigraphs ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 283682, root := ~, start := 283674, stop := 283681 ), rec( content := " such th\ at there exists a homomorphism from ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 283734, root := ~, start := 283726, stop := 283733 ), rec( content := "\n to\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "A\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 283750, root := ~, start := 283742, stop := 283749 ), rec( content := ". In oth\ er words, a digraph ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 283786, root := ~, start := 283778, stop := 283785 ), rec( content := " is a co\ re if and only if\n every endomorphism on ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 283846, root := ~, start := 283838, stop := 283845 ), rec( content := " is an a\ utomorphism on ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "C", next := 283877, root := ~, start := 283869, stop := 283876 ), rec( content := ".\n " , count := [ 6, 5, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "P", next := 283887, root := ~, start := 283883, stop := 283886 ), rec( content := "\n \ \n If the argument ", count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "A", next := 283929, root := ~, start := 283915, stop := 283928 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 5, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(6);\n\n\ gap> IsDigraphCore(D);\ntrue\ngap> D := DigraphSymmetricClosure(CycleDigraph(6\ ));\n\ngap> DigraphHomo\ morphism(D, CompleteDigraph(2));\nTransformation( [ 1, 2, 1, 2, 1, 2 ] )\ngap>\ IsDigraphCore(D);\nfalse\n", count := [ 6, 5, 1, 6 ], name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 6 ], mathmode := "Text", name := "Example", next := 284390, root := ~, start := 284030, stop := 284389 ), rec( content := "\n ", count := [ 6, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 1, 3 ], mathmode := "Text", name := "Description", next := 284407, root := ~, start := 283465, stop := 284406 ) ], count := [ 6, 5, 1, 1 ], mathmode := "Text", name := "ManSection", next := 284421, root := ~, start := 283355, stop := 284420 ), rec( content := "\n\n ", count := [ 6, 5, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsEdgeTransitiv\ e" ), content := 0, count := [ 6, 5, 2, 2 ], mathmode := "Text", name := "Prop", next := 284487, root := ~, start := 284442, stop := 284486 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "K", next := 284510, root := ~, start := 284499, stop := 284509 ), rec( content := " or ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "K", next := 284526, root := ~, start := 284514, stop := 284525 ), rec( content := ".", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Returns", next := 284537, root := ~, start := 284490, stop := 284536 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "A", next := 284575, root := ~, start := 284561, stop := 284574 ), rec( content := " is a di\ graph without multiple edges, then\n ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sEdgeTransitive", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "C", next := 284645, root := ~, start := 284622, stop := 284644 ), rec( content := " returns\ ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "K", next := 284665, root := ~, start := 284654, stop := 284664 ), rec( content := " if ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "A", next := 284683, root := ~, start := 284669, stop := 284682 ), rec( content := "\n is\ edge transitive, and ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "K", next := 284724, root := ~, start := 284712, stop := 284723 ), rec( content := " otherwi\ se. A digraph is\n ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge transitive", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "E", next := 284775, root := ~, start := 284753, stop := 284774 ), rec( content := " if its \ automorphism group acts\n transitively on its edges (via the action\n ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "OnPairs" ), content := 0, count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Ref", next := 284893, root := ~, start := 284857, stop := 284892 ), rec( content := ").\n \ ", count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "P", next := 284904, root := ~, start := 284900, stop := 284903 ), rec( content := "\n \ \n If the argument ", count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "A", next := 284946, root := ~, start := 284932, stop := 284945 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 5, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsEdgeTransitive(CompleteDigraph(2));\ntrue\ngap> IsEdgeTransitive(Chain\ Digraph(3));\nfalse\ngap> IsEdgeTransitive(Digraph([[2], [3, 3, 3], []]));\nEr\ ror, the argument must be a digraph with no multiple edges,\n", count := [ 6, 5, 2, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 6, 5, 2, 5 ], mathmode := "Text", name := "Example", next := 285293, root := ~, start := 285047, stop := 285292 ), rec( content := "\n ", count := [ 6, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 2, 3 ], mathmode := "Text", name := "Description", next := 285310, root := ~, start := 284540, stop := 285309 ) ], count := [ 6, 5, 2, 1 ], mathmode := "Text", name := "ManSection", next := 285324, root := ~, start := 284427, stop := 285323 ), rec( content := "\n\n ", count := [ 6, 5, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "IsVertexTransit\ ive" ), content := 0, count := [ 6, 5, 3, 2 ], mathmode := "Text", name := "Prop", next := 285392, root := ~, start := 285345, stop := 285391 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "K", next := 285415, root := ~, start := 285404, stop := 285414 ), rec( content := " or ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "K", next := 285431, root := ~, start := 285419, stop := 285430 ), rec( content := ".", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "Returns", next := 285442, root := ~, start := 285395, stop := 285441 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "A", next := 285480, root := ~, start := 285466, stop := 285479 ), rec( content := " is a di\ graph, then ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sVertexTransitive", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "C", next := 285525, root := ~, start := 285500, stop := 285524 ), rec( content := " returns\ \n ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "K", next := 285549, root := ~, start := 285538, stop := 285548 ), rec( content := " if ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "A", next := 285567, root := ~, start := 285553, stop := 285566 ), rec( content := " is vert\ ex transitive, and ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "K", next := 285606, root := ~, start := 285594, stop := 285605 ), rec( content := "\n ot\ herwise. A digraph is ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex transitive", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "E", next := 285659, root := ~, start := 285635, stop := 285658 ), rec( content := " if its \ automorphism group\n acts transitively on its vertices.\n ", count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "P", next := 285733, root := ~, start := 285729, stop := 285732 ), rec( content := "\n \ \n If the argument ", count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "A", next := 285775, root := ~, start := 285761, stop := 285774 ), rec( content := " is muta\ ble, then the return value of\n this property is recomputed every time it i\ s called. \n\n ", count := [ 6, 5, 3, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> IsVertexTransitive(CompleteDigraph(2));\ntrue\ngap> IsVertexTransitive(C\ hainDigraph(3));\nfalse\n", count := [ 6, 5, 3, 5 ], name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 5 ], mathmode := "Text", name := "Example", next := 286006, root := ~, start := 285876, stop := 286005 ), rec( content := "\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 3, 3 ], mathmode := "Text", name := "Description", next := 286023, root := ~, start := 285445, stop := 286022 ) ], count := [ 6, 5, 3, 1 ], mathmode := "Text", name := "ManSection", next := 286037, root := ~, start := 285330, stop := 286036 ), rec( content := "\n\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 5, 0, 1 ], mathmode := "Text", name := "Section", next := 286051, root := ~, start := 283289, stop := 286050 ) , rec( content := "\n\n \n", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 6, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 286066, root := ~, start := 238167, stop := 286065 ), rec( content := "\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " properties ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 6, 5, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Homomorphisms" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Homomorphisms", count := [ 7, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 286153, root := ~, start := 286121, stop := 286152 ) , rec( content := "\n ", count := [ 7, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := "Let $E_a=(E_a^0,E_a^1,\\ra_a,\\s\ o_a)$ and $E_b=(E_b^0,E_b^1,\\ra_b,\\so_b)$ be two\ngraphs, and let $\\phi_0 :\ E_a^0 \\to E_b^0$ and $\\phi_1 : E_a^1 \\to E_b^1$ be\nfunctions. Then the pa\ ir $\\phi = (\\phi_0, \\phi_1)$ is a \\emph{graph homomorphism\nfrom} $E_a$ \\\ emph{to} $E_b$ if $\\phi_0(\\so_a(e)) = \\so_b(\\phi_1(e))$ and\n$\\phi_0(\\ra\ _a(e)) = \\ra_b(\\phi_1(e))$ for every $e \\in E_a^1$. If $\\phi_0$ and\n$\\ph\ i_1$ are in addition bijective, then $\\phi$ is a \\emph{graph isomorphism\nfr\ om} $E_a$ \\emph{to} $E_b$. In this case we say that $E_a$ and $E_b$ are\n\\em\ ph{isomorphic} and write $E_a \\cong E_b$.", count := [ 7, 0, 0, 3 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n\n ", count := [ 7, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Acting on digraph\ s", count := [ 7, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 286797, root := ~, start := 286760, stop := 286796 ), rec( content := "\n ", count := [ 7, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, perm", Label := "for a digraph \ and a perm", Name := "OnDigraphs" ), content := 0, count := [ 7, 1, 1, 2 ], mathmode := "Text", name := "Oper", next := 286899, root := ~, start := 286817, stop := 286898 ), rec( attributes := rec( Arg := "digraph, trans", Label := "for a digraph \ and a transformation", Name := "OnDigraphs" ), content := 0, count := [ 7, 1, 1, 4 ], mathmode := "Text", name := "Oper", next := 286995, root := ~, start := 286902, stop := 286994 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "Returns", next := 287027, root := ~, start := 286998, stop := 287026 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 287065, root := ~, start := 287051, stop := 287064 ), rec( content := " is a di\ graph, and the second argument ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 287115, root := ~, start := 287104, stop := 287114 ), rec( content := " is a\n \ ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ermutation", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "E", next := 287143, root := ~, start := 287125, stop := 287142 ), rec( content := " of the \ vertices of ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 287177, root := ~, start := 287163, stop := 287176 ), rec( content := ", then t\ his operation\n returns a digraph constructed by relabelling the vertices o\ f\n ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 287282, root := ~, start := 287268, stop := 287281 ), rec( content := " accordi\ ng to ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 287307, root := ~, start := 287296, stop := 287306 ), rec( content := ". Note \ that for an automorphism\n ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "C", next := 287352, root := ~, start := 287344, stop := 287351 ), rec( content := " of a di\ graph, we have ", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs(", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 7, 1, 1, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := \ "Text", name := "A", next := 287403, root := ~, start := 287389, stop := 28740\ 2 ), rec( content := ", f) =\n ", count := [ 7, 1, 1, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "C", next := 287418, root := ~, start := 287375, stop := 287417 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "A", next := 287432, root := ~, start := 287418, stop := 287431 ), rec( content := ".\n " , count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "P", next := 287442, root := ~, start := 287438, stop := 287441 ), rec( content := "\n\n \ If the second argument is a ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ransformation", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "E", next := 287497, root := ~, start := 287476, stop := 287496 ), rec( content := " ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 287510, root := ~, start := 287498, stop := 287509 ), rec( content := " of the\\ n vertices of ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 287548, root := ~, start := 287534, stop := 287547 ), rec( content := ", then t\ his operation returns a digraph\n constructed by transforming the source an\ d range of each edge according to\n ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 287683, root := ~, start := 287671, stop := 287682 ), rec( content := ". Thus a\ vertex which does not appear in the image of\n ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 287753, root := ~, start := 287741, stop := 287752 ), rec( content := " will be\ isolated in the returned digraph, and the returned\n digraph may contain m\ ultiple edges, even if ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 287875, root := ~, start := 287861, stop := 287874 ), rec( content := " does no\ t.\n If ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 287905, root := ~, start := 287893, stop := 287904 ), rec( content := " is math\ ematically a permutation, then the result coincides\n with ", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs(", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 7, 1, 1, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := \ "Text", name := "A", next := 288002, root := ~, start := 287988, stop := 28800\ 1 ), rec( content := ", AsPermutation(", count := [ 7, 1, 1, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "trans", count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "A", next := 2880\ 30, root := ~, start := 288018, stop := 288029 ), rec( content := "))", count := [ 7, 1, 1, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "C", next := 288036, root := ~, start := 287974, stop := 288035 ), rec( content := ".\n " , count := [ 7, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "P", next := 288046, root := ~, start := 288042, stop := 288045 ), rec( content := "\n\n \ The ", count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "Ref", next := 288089, root := ~, start := 288056, stop := 288088 ), rec( content := " of ", count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "A", next := 288107, root := ~, start := 288093, stop := 288106 ), rec( content := " will no\ t be retained\n in the returned digraph. ", count := [ 7, 1, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "P", next := 288162, root := ~, start := 288158, stop := 288161 ), rec( content := "\n\n \ If ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "A", next := 288185, root := ~, start := 288171, stop := 288184 ), rec( content := " belongs\ to ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "Ref", next := 288227, root := ~, start := 288197, stop := 288226 ), rec( content := ", then\n\ relabelling of the vertices is performed directly on ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "A", next := 288305, root := ~, start := 288291, stop := 288304 ), rec( content := ".\n I\ f ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "A", next := 288328, root := ~, start := 288314, stop := 288327 ), rec( content := " belongs\ to ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "Ref", next := 288372, root := ~, start := 288340, stop := 288371 ), rec( content := ", an imm\ utable\n copy of ", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "A", next := 288413, root := ~, start := 288399, stop := 288412 ), rec( content := " with th\ e vertices relabelled is returned.", count := [ 7, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 1, 9 ], mathmode := "Text", name := "P", next := 288459, root := ~, start := 288455, stop := 288458 ), rec( content := "\n\n \ ", count := [ 7, 1, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> new := OnDigraphs(D, (1, 2));\n\ngap> OutNeighbours(new);\n[ [ 2, 3, 5 \ ], [ 3 ], [ 2 ], [ 2, 1, 4 ], [ 1, 3, 5 ] ]\ngap> D := Digraph([[2], [], [2]])\ ;\n\ngap> t := Transformation([1, \ 2, 1]);;\ngap> new := OnDigraphs(D, t);\n\ngap> OutNeighbours(new);\n[ [ 2, 2 ], [ ], [ ] ]\ngap> ForAll(\ DigraphEdges(D),\n> e -> IsDigraphEdge(new, [e[1] ^ t, e[2] ^ t]));\ntrue\n", count := [ 7, 1, 1, 10 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 1, 1, 10 ], mathmode := "Text", name := "Example", next := 289102, root := ~, start := 288465, stop := 289101 ), rec( content := "\n ", count := [ 7, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 1, 5 ], mathmode := "Text", name := "Description", next := 289119, root := ~, start := 287030, stop := 289118 ) ], count := [ 7, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 289133, root := ~, start := 286802, stop := 289132 ), rec( content := "\n\n ", count := [ 7, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, pair", Name := "OnMultiDigraphs\ " ), content := 0, count := [ 7, 1, 2, 2 ], mathmode := "Text", name := "Oper", next := 289204, root := ~, start := 289154, stop := 289203 ), rec( attributes := rec( Arg := "digraph, perm1, \ perm2", Label := "for a digraph, perm, and perm", Name := "OnMultiDigraphs" ), content := 0, count := [ 7, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 289307, root := ~, start := 289207, stop := 289306 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "Returns", next := 289339, root := ~, start := 289310, stop := 289338 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 289377, root := ~, start := 289363, stop := 289376 ), rec( content := " is a di\ graph, and ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ air", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 289407, root := ~, start := 289396, stop := 289406 ), rec( content := " is a pa\ ir consisting of a\n permutation of the vertices and a permutation of the e\ dges of\n ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 289518, root := ~, start := 289504, stop := 289517 ), rec( content := ", then t\ his operation returns a digraph\n constructed by relabelling the vertices a\ nd edges of ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 289629, root := ~, start := 289615, stop := 289628 ), rec( content := "\n ac\ cording to ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm[1]", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 289661, root := ~, start := 289647, stop := 289660 ), rec( content := " and ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm[2]", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "A", next := 289680, root := ~, start := 289666, stop := 289679 ), rec( content := ", respec\ tively. ", count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "P", next := 289700, root := ~, start := 289696, stop := 289699 ), rec( content := "\n\n \ In its second form, ", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nMultiDigraphs", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "C", next := 289748, root := ~, start := 289726, stop := 289747 ), rec( content := " returns\ a digraph with vertices\n and edges permuted by ", count := [ 7, 1, 2, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm1", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "A", next := 289819, root := ~, start := 289807, stop := 289818 ), rec( content := " and ", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm2", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "A", next := 289836, root := ~, start := 289824, stop := 289835 ), rec( content := ", respec\ tively. ", count := [ 7, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "P", next := 289856, root := ~, start := 289852, stop := 289855 ), rec( content := "\n\n \ Note that ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs(", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 7, 1, 2, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := \ "Text", name := "A", next := 289900, root := ~, start := 289886, stop := 28989\ 9 ), rec( content := ",\n perm)=OnMultiDigraphs(", count := [ 7, 1, 2, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 2899\ 44, root := ~, start := 289930, stop := 289943 ), rec( content := ", [perm, ()\ ])", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 289961, root := ~, start := 289872, stop := 289960 ), rec( content := " where " , count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ erm", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 289979, root := ~, start := 289968, stop := 289978 ), rec( content := " is\n \ a permutation of the vertices of ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 290034, root := ~, start := 290020, stop := 290033 ), rec( content := ". If you\ are only interested\n in the action of a permutation on the vertices of a \ digraph, then you can\n use ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 290166, root := ~, start := 290149, stop := 290165 ), rec( content := " instead\ of ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ nMultiDigraphs", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "C", next := 290200, root := ~, start := 290178, stop := 290199 ), rec( content := ".\n\n \ If ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 290224, root := ~, start := 290210, stop := 290223 ), rec( content := " belongs\ to ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 290266, root := ~, start := 290236, stop := 290265 ), rec( content := ", then\n\ relabelling of the vertices is performed directly on ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 290344, root := ~, start := 290330, stop := 290343 ), rec( content := ".\n I\ f ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 290367, root := ~, start := 290353, stop := 290366 ), rec( content := " belongs\ to ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "Ref", next := 290411, root := ~, start := 290379, stop := 290410 ), rec( content := ", an imm\ utable\n copy of ", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "A", next := 290452, root := ~, start := 290438, stop := 290451 ), rec( content := " with th\ e vertices relabelled is returned.", count := [ 7, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 1, 2, 8 ], mathmode := "Text", name := "P", next := 290498, root := ~, start := 290494, stop := 290497 ), rec( content := "\n\n \ ", count := [ 7, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := Digraph([\n> [3, 6, 3], [], [3], [9, 10], [9], [], [], [10, 4, 10\ ], [], []]);\n\ngap> p := B\ lissCanonicalLabelling(D1);\n[ (1,7)(3,6)(4,10)(5,9), () ]\ngap> D2 := OnMulti\ Digraphs(D1, p);\n\ngap> Ou\ tNeighbours(D2);\n[ [ ], [ ], [ ], [ ], [ ], [ 6 ], [ 6, 3, 6 ], [ 4, 10,\ 4 ], \n [ 5 ], [ 5, 4 ] ]", count := [ 7, 1, 2, 9 ], name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 9 ], mathmode := "Text", name := "Example", next := 290943, root := ~, start := 290504, stop := 290942 ), rec( content := "\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 2, 5 ], mathmode := "Text", name := "Description", next := 290960, root := ~, start := 289342, stop := 290959 ) ], count := [ 7, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 290974, root := ~, start := 289139, stop := 290973 ), rec( content := "\n\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 290988, root := ~, start := 286751, stop := 290987 ) , rec( content := "\n\n ", count := [ 7, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Isomorphisms and canonical lab\ ellings" ), content := [ rec( content := "\n ", count := [ 7, 2, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Isomorphisms and \ canonical labellings", count := [ 7, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 291108, root := ~, start := 291052, stop := 291107 ), rec( content := "\n From version 0.11.\ 0 of ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " it is possible to use e\ ither ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), content := [ rec( content := "http://ww\ w.tcs.tkk.fi/Software/bliss/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " or\n ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "nauty" ), content := [ rec( content := "http://pa\ llini.di.uniroma1.it/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 124, root := ~, start := 100, stop := 123 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " (via ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "NautyTracesInte\ rface" ), content := [ rec( content := "https://github.com/gap-packages/NautyT\ racesInterface", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ) , rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "NautyTrac\ esInterface", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := ") to calculate canonical\ labellings and\n automorphism groups of digraphs; see ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "JK07" ), content := 0, count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Cite", next := 291325, root := ~, start := 291307, stop := 291324 ), rec( content := " and ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Key := "MP14" ), content := 0, count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Cite", next := 291354, root := ~, start := 291330, stop := 291353 ), rec( content := " for more details about \ ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "bliss" ), content := [ rec( content := "http://ww\ w.tcs.tkk.fi/Software/bliss/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 131, root := ~, start := 107, stop := 130 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " and ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "nauty" ), content := [ rec( content := "http://pa\ llini.di.uniroma1.it/", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 0, 3 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes := rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 3 ], name := "Package", next := 124, root := ~, start := 100, stop := 123 ) ], count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := ",\n respectively. \n\\ n ", count := [ 7, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "", Name := "DigraphsUseNaut\ y" ), content := 0, count := [ 7, 2, 1, 2 ], mathmode := "Text", name := "Func", next := 291476, root := ~, start := 291438, stop := 291475 ), rec( attributes := rec( Arg := "", Name := "DigraphsUseBlis\ s" ), content := 0, count := [ 7, 2, 1, 4 ], mathmode := "Text", name := "Func", next := 291517, root := ~, start := 291479, stop := 291516 ), rec( attributes := rec( ), content := [ rec( content := "Nothing." , count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 291547, root := ~, start := 291520, stop := 291546 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se functions can be used to specify whether ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 1, 5 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " or ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 1, 5 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " should \ be\n used by default by ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ". If ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 1, 5 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 1, 5 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is not \ available,\n then these functions do nothing. Otherwise, by calling\n ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphsUseNauty", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "C", next := 291809, root := ~, start := 291786, stop := 291808 ), rec( content := " subsequ\ ent computations will default to using\n ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 1, 5 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " rather \ than ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 1, 5 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 1, 5 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := ", where \ possible.\n ", count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "P", next := 291913, root := ~, start := 291909, stop := 291912 ), rec( content := "\n\n \ You can call these functions at any point in a ", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ AP", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := " session\ , as many\n times as you like, it is guaranteed that existing digraphs rema\ in\n valid, and that comparison of existing digraphs and newly created digr\ aphs\n via\n ", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsIsomorphicDigraph" ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Ref", next := 292204, root := ~, start := 292150, stop := 292203 ), rec( content := ",\n " , count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsIsomorphicDigraph" ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Ref", next := 292292, root := ~, start := 292210, stop := 292291 ), rec( content := ",\n " , count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Ref", next := 292352, root := ~, start := 292298, stop := 292351 ), rec( content := ", and\n \ ", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "Ref", next := 292444, root := ~, start := 292362, stop := 292443 ), rec( content := " are als\ o valid.", count := [ 7, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "P", next := 292464, root := ~, start := 292460, stop := 292463 ), rec( content := "\n\n \ It is also possible to compute the automorphism group of a specific digraph\n \ using both ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 1, 7 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 1, 7 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 7 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " and ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 1, 7 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 1, 7 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 7 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " using " , count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Ref", next := 292623, root := ~, start := 292587, stop := 292622 ), rec( content := "\n an\ d ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "Ref", next := 292690, root := ~, start := 292632, stop := 292689 ), rec( content := ", respec\ tively.\n ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 292722, root := ~, start := 291550, stop := 292721 ) ], count := [ 7, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 292736, root := ~, start := 291423, stop := 292735 ), rec( content := "\n\n ", count := [ 7, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "AutomorphismGro\ up" ), content := 0, count := [ 7, 2, 2, 2 ], mathmode := "Text", name := "Attr", next := 292825, root := ~, start := 292757, stop := 292824 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "Returns", next := 292867, root := ~, start := 292828, stop := 292866 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "A", next := 292905, root := ~, start := 292891, stop := 292904 ), rec( content := " is a di\ graph, then this attribute contains the group of\n automorphisms of ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "A", next := 292997, root := ~, start := 292983, stop := 292996 ), rec( content := ". An ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "E", next := 293022, root := ~, start := 293003, stop := 293021 ), rec( content := " of ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "A", next := 293040, root := ~, start := 293026, stop := 293039 ), rec( content := "\n is\ an isomorphism from ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "A", next := 293082, root := ~, start := 293068, stop := 293081 ), rec( content := " to itse\ lf. See ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "Ref", next := 293159, root := ~, start := 293098, stop := 293158 ), rec( content := " for mor\ e information\n about isomorphisms of digraphs. ", count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "P", next := 293221, root := ~, start := 293217, stop := 293220 ), rec( content := "\n\n \ If ", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "A", next := 293244, root := ~, start := 293230, stop := 293243 ), rec( content := " is not \ a multidigraph then the automorphism group is\n returned as a group of perm\ utations on the set of vertices of\n ", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "A", next := 293382, root := ~, start := 293368, stop := 293381 ), rec( content := ". ", count := [ 7, 2, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "P", next := 293388, root := ~, start := 293384, stop := 293387 ), rec( content := "\n\n \ If ", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "A", next := 293411, root := ~, start := 293397, stop := 293410 ), rec( content := " is a mu\ ltidigraph then the automorphism group is returned\n as the direct product \ of a group of permutations on the set of vertices of\n ", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "A", next := 293567, root := ~, start := 293553, stop := 293566 ), rec( content := " with a \ group of permutations on the set of edges of\n ", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "A", next := 293638, root := ~, start := 293624, stop := 293637 ), rec( content := ". These \ groups can be accessed using ", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Label := "for a domain and a positive integer", Oper := "Projection" ), content := 0, count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "Ref", next := 293764, root := ~, start := 293675, stop := 293763 ), rec( content := " on the\\ n returned group.", count := [ 7, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "P", next := 293795, root := ~, start := 293791, stop := 293794 ), rec( content := "\n\n \ By default, the automorphism group is found using ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 2, 6 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 2, 6 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 2, 6 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 2, 6 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 2, 6 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 2, 6 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 2, 6 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 6 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Ref", next := 294080, root := ~, start := 294022, stop := 294079 ), rec( content := ", \n \ ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Ref", next := 294123, root := ~, start := 294087, stop := 294122 ), rec( content := ", ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Ref", next := 294155, root := ~, start := 294125, stop := 294154 ), rec( content := ",\n a\ nd ", count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "Ref", next := 294195, root := ~, start := 294165, stop := 294194 ), rec( content := ".\n " , count := [ 7, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "P", next := 294205, root := ~, start := 294201, stop := 294204 ), rec( content := "\n\n \ If the argument ", count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "A", next := 294241, root := ~, start := 294227, stop := 294240 ), rec( content := " is muta\ ble, then the return value of\n this attribute is recomputed every time it \ is called. \n\n ", count := [ 7, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> johnson := DigraphFromGraph6String(\"E}lw\");\n\ngap> G := AutomorphismGroup(johnson);\nGroup([ (3,4), (2\ ,3)(4,5), (1,2)(5,6) ])\ngap> cycle := CycleDigraph(9);\n\ngap> G := AutomorphismGroup(cycle);\nGroup([ (1,2,3,4,5,\ 6,7,8,9) ])\ngap> IsCyclic(G) and Size(G) = 9;\ntrue", count := [ 7, 2, 2, 8 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 2, 8 ], mathmode := "Text", name := "Example", next := 294727, root := ~, start := 294343, stop := 294726 ), rec( content := "\n ", count := [ 7, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 2, 3 ], mathmode := "Text", name := "Description", next := 294746, root := ~, start := 292870, stop := 294745 ) ], count := [ 7, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 294762, root := ~, start := 292742, stop := 294761 ), rec( content := "\n\n ", count := [ 7, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "BlissAutomorphi\ smGroup" ), content := 0, count := [ 7, 2, 3, 2 ], mathmode := "Text", name := "Attr", next := 294856, root := ~, start := 294783, stop := 294855 ), rec( attributes := rec( Arg := "digraph, vertex_\ colours", Label := "for a digraph and homogenous list", Name := "BlissAutomorphi\ smGroup" ), content := 0, count := [ 7, 2, 3, 4 ], mathmode := "Text", name := "Oper", next := 294973, root := ~, start := 294859, stop := 294972 ), rec( attributes := rec( Arg := "digraph, vertex_\ colours, edge_colours", Label := "for a digraph, homogenous list, and list", Name := "BlissAutomorphi\ smGroup" ), content := 0, count := [ 7, 2, 3, 6 ], mathmode := "Text", name := "Oper", next := 295111, root := ~, start := 294976, stop := 295110 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "Returns", next := 295153, root := ~, start := 295114, stop := 295152 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "A", next := 295191, root := ~, start := 295177, stop := 295190 ), rec( content := " is a di\ graph, then this attribute contains the group of\n automorphisms of ", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "A", next := 295283, root := ~, start := 295269, stop := 295282 ), rec( content := " as calc\ ulated using ", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 3, 7 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 3, 7 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. ", count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "P", next := 295357, root := ~, start := 295353, stop := 295356 ), rec( content := "\n\n \ The attribute ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 3, 8 ] , mathmode := "Text", name := "Ref", next := 295430, root := ~, start := 295377, stop := 295429 ), rec( content := " and\n \ operation ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a homogeneous list", Oper := "AutomorphismGroup" ), content := 0, count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "Ref", next := 295532, root := ~, start := 295449, stop := 295531 ), rec( content := " returns\ \n the value of either ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ lissAutomorphismGroup", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "C", next := 295594, root := ~, start := 295565, stop := 295593 ), rec( content := " or ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "Ref", next := 295640, root := ~, start := 295598, stop := 295639 ), rec( content := ". These \ groups are, of course, equal but\n their generating sets may differ. ", count := [ 7, 2, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "P", next := 295723, root := ~, start := 295719, stop := 295722 ), rec( content := "\n Th\ e attribute ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph, homogeneous list, and list" ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Ref", next := 295830, root := ~, start := 295742, stop := 295829 ), rec( content := " returns\ the value of\n ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ lissAutomorphismGroup", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "C", next := 295885, root := ~, start := 295856, stop := 295884 ), rec( content := " as it i\ s not implemented for ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 3, 9 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 3, 9 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 9 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " The \n \ requirements for the optional arguments ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ertex_colours", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "A", next := 295993, root := ~, start := 295972, stop := 295992 ), rec( content := " and \n \ ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge_colours", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "A", next := 296022, root := ~, start := 296003, stop := 296021 ), rec( content := " are doc\ umented in ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph, homogeneous list, and list" ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Ref", next := 296129, root := ~, start := 296041, stop := 296128 ), rec( content := ".\n\n \ See also ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Ref", next := 296175, root := ~, start := 296145, stop := 296174 ), rec( content := ", and ", count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "Ref", next := 296211, root := ~, start := 296181, stop := 296210 ), rec( content := ".\n " , count := [ 7, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "P", next := 296221, root := ~, start := 296217, stop := 296220 ), rec( content := "\n\n \ If the argument ", count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "A", next := 296257, root := ~, start := 296243, stop := 296256 ), rec( content := " is muta\ ble, then the return value of\n this attribute is recomputed every time it \ is called. \n\n ", count := [ 7, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := BlissAutomorphismGroup(JohnsonDigraph(5, 2));;\ngap> IsSymmetricGro\ up(G);\ntrue\ngap> Size(G);\n120", count := [ 7, 2, 3, 11 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 11 ], mathmode := "Text", name := "Example", next := 296496, root := ~, start := 296359, stop := 296495 ), rec( content := "\n ", count := [ 7, 2, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 3, 7 ], mathmode := "Text", name := "Description", next := 296515, root := ~, start := 295156, stop := 296514 ) ], count := [ 7, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 296531, root := ~, start := 294768, stop := 296530 ), rec( content := "\n\n ", count := [ 7, 2, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, vert_c\ olours]", Name := "NautyAutomorphismGroup" ), content := 0, count := [ 7, 2, 4, 2 ], mathmode := "Text", name := "Attr", next := 296619, root := ~, start := 296552, stop := 296618 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 296661, root := ~, start := 296622, stop := 296660 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "A", next := 296699, root := ~, start := 296685, stop := 296698 ), rec( content := " is a di\ graph, then this attribute contains the group of\n automorphisms of ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "A", next := 296791, root := ~, start := 296777, stop := 296790 ), rec( content := " as calc\ ulated using ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 4, 3 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 4, 3 ], name \ := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno via ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 4, 3 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 4, 3 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 4, 3 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := ".\n\n \ ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( content := " FIXME b\ e more explicit about what the second arg is ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "XMLCOMMENT" , root := ~ ), rec( content := "\n\n \ The attribute ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 4, 3 ] , mathmode := "Text", name := "Ref", next := 297026, root := ~, start := 296973, stop := 297025 ), rec( content := " and\n \ operation ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a homogeneous list", Oper := "AutomorphismGroup" ), content := 0, count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Ref", next := 297128, root := ~, start := 297045, stop := 297127 ), rec( content := " returns\ \n the value of either ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ autyAutomorphismGroup", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "C", next := 297190, root := ~, start := 297161, stop := 297189 ), rec( content := " or ", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Ref", next := 297259, root := ~, start := 297195, stop := 297258 ), rec( content := ".\n T\ hese groups are, of course, equal but their generating sets may differ.", count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "P", next := 297341, root := ~, start := 297337, stop := 297340 ), rec( content := "\n\n \ See also ", count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "Ref", next := 297386, root := ~, start := 297356, stop := 297385 ), rec( content := ", and ", count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "Ref", next := 297422, root := ~, start := 297392, stop := 297421 ), rec( content := ".\n " , count := [ 7, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "P", next := 297432, root := ~, start := 297428, stop := 297431 ), rec( content := "\n\n \ If the argument ", count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "A", next := 297468, root := ~, start := 297454, stop := 297467 ), rec( content := " is muta\ ble, then the return value of\n this attribute is recomputed every time it \ is called. \n\n ", count := [ 7, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> NautyAutomorphismGroup(JohnsonDigraph(5, 2));\nGroup([ (3,4)(6,7)(8,9), \ (2,3)(5,6)(9,10), (2,5)(3,6)(4,7),\n (1,2)(6,8)(7,9) ])", count := [ 7, 2, 4, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 4, 6 ], mathmode := "Text", name := "Log", next := 297724, root := ~, start := 297570, stop := 297723 ), rec( content := "\n ", count := [ 7, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 297743, root := ~, start := 296664, stop := 297742 ) ], count := [ 7, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 297759, root := ~, start := 296537, stop := 297758 ), rec( content := "\n\n ", count := [ 7, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vert_co\ lours", Label := "for a digraph and a homogeneous list", Name := "AutomorphismGro\ up" ), content := 0, count := [ 7, 2, 5, 2 ], mathmode := "Text", name := "Oper", next := 297889, root := ~, start := 297780, stop := 297888 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "Returns", next := 297931, root := ~, start := 297892, stop := 297930 ), rec( attributes := rec( ), content := [ rec( content := "\n\n T\ his operation computes the automorphism group of a vertex-coloured digraph.\n \ A vertex-coloured digraph can be specified by its underlying digraph\n " , count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "A", next := 298121, root := ~, start := 298107, stop := 298120 ), rec( content := " and its\ colouring ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "A", next := 298159, root := ~, start := 298140, stop := 298158 ), rec( content := ". Let ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "C", next := 298173, root := ~, start := 298165, stop := 298172 ), rec( content := " be the\\ n number of vertices of ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "A", next := 298221, root := ~, start := 298207, stop := 298220 ), rec( content := ". The co\ louring ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "A", next := 298256, root := ~, start := 298237, stop := 298255 ), rec( content := " may\n \ have one of the following two forms:\n\n ", count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n a list of ", count := [ 7, 2\ , 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 7, 2, 5, 5 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ], mathmode\ := "Text", name := "C", next := 298353, root := ~, start := 298345, stop := 2\ 98352 ), rec( content := " integers, where ", count := [ 7, 2, 5, 5 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "vert_colours", count := [ 7, 2, 5, 5 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ], mathmode := "T\ ext", name := "A", next := 298389, root := ~, start := 298370, stop := 298388 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 5, 5 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ] , math\ mode := "Text", name := "C", next := 298399, root := ~, start := 298389, stop \ := 298398 ), rec( content := " is the\n colour of vertex ", count := [ \ 7, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "i", count := [ 7, 2, 5, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ], math\ mode := "Text", name := "C", next := 298440, root := ~, start := 298432, stop \ := 298439 ), rec( content := ", using the colours ", count := [ 7, 2, 5, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 .. m]", count := [ 7, 2, 5, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 5 ], mathmode := "Tex\ t", name := "C", next := 298475, root := ~, start := 298460, stop := 298474 ), rec( cont\ ent := " for some\n ", count := [ 7, 2, 5, 5 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "m <= n", count := [ 7, 2, 5, 5 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 7, 2, 5, 5 ], mathmode := "Text", name := "C", next\ := 298509, root := ~, start := 298493, stop := 298508 ), rec( content := "; o\ r\n ", count := [ 7, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 5, 5 ], name := "Item", next := 298527, root := ~, start := 298\ 320, stop := 298526 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n a list of non-empty disjoint lists whose union is\n ", coun\ t := [ 7, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphVertices(", count := [\ 7, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 5, 7 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 \ ], mathmode := "Text", name := "A", next := 298640, root := ~, start := 298626\ , stop := 298639 ), rec( content := ")", count := [ 7, 2, 5, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], mathmode := \ "Text", name := "C", next := 298645, root := ~, start := 298607, stop := 29864\ 4 ), rec( content := ", such that\n ", count := [ 7, 2, 5, 7 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "vert_colours", count := [ 7, 2, 5, 7 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], mathmode := "T\ ext", name := "A", next := 298684, root := ~, start := 298665, stop := 298683 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 5, 7 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ] , math\ mode := "Text", name := "C", next := 298694, root := ~, start := 298684, stop \ := 298693 ), rec( content := " is the list of all vertices with colour\n \ ", count := [ 7, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 5, 7 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], math\ mode := "Text", name := "C", next := 298751, root := ~, start := 298743, stop \ := 298750 ), rec( content := ".\n ", count := [ 7, 2, 5, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 7 ], name := "Ite\ m", next := 298766, root := ~, start := 298534, stop := 298765 ) ], count := [ 7, 2, 5, 4 ], mathmode := "Text", name := "List", next := 298778, root := ~, start := 298307, stop := 298777 ), rec( content := "\n\n \ The ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism group", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "E", next := 298813, root := ~, start := 298788, stop := 298812 ), rec( content := " of a co\ loured digraph ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 298850, root := ~, start := 298836, stop := 298849 ), rec( content := " with\n \ colouring ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 298889, root := ~, start := 298870, stop := 298888 ), rec( content := " is the \ group consisting of its automorphisms; an\n ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "E", next := 298962, root := ~, start := 298943, stop := 298961 ), rec( content := " of ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 298980, root := ~, start := 298966, stop := 298979 ), rec( content := " is an i\ somorphism of coloured\n digraphs from ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 299043, root := ~, start := 299029, stop := 299042 ), rec( content := " to itse\ lf. This group is equal to the\n subgroup of ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ utomorphismGroup(", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "di\ graph", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 2991\ 34, root := ~, start := 299120, stop := 299133 ), rec( content := ")", count := [ 7, 2, 5, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "C", next := 299139, root := ~, start := 299099, stop := 299138 ), rec( content := " consist\ ing of those\n automorphisms that preserve the colouring specified by ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "A", next := 299238, root := ~, start := 299219, stop := 299237 ), rec( content := ". See\n \ ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 5, 9 ] , mathmode := "Text", name := "Ref", next := 299302, root := ~, start := 299248, stop := 299301 ), rec( content := ", and se\ e ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "Ref", next := 299395, root := ~, start := 299312, stop := 299394 ), rec( content := "\n fo\ r more information about isomorphisms of coloured digraphs. ", count := [ 7, 2, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "P", next := 299466, root := ~, start := 299462, stop := 299465 ), rec( content := "\n\n \ If ", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "A", next := 299489, root := ~, start := 299475, stop := 299488 ), rec( content := " is not \ a multidigraph then the automorphism group is\n returned as a group of perm\ utations on the set of vertices of\n ", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "A", next := 299627, root := ~, start := 299613, stop := 299626 ), rec( content := ". ", count := [ 7, 2, 5, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "P", next := 299633, root := ~, start := 299629, stop := 299632 ), rec( content := "\n\n \ If ", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "A", next := 299656, root := ~, start := 299642, stop := 299655 ), rec( content := " is a mu\ ltidigraph then the automorphism group is returned\n as the direct product \ of a group of permutations on the set of vertices of\n ", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "A", next := 299812, root := ~, start := 299798, stop := 299811 ), rec( content := " with a \ group of permutations on the set of edges of\n ", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "A", next := 299883, root := ~, start := 299869, stop := 299882 ), rec( content := ". These \ groups can be accessed using ", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Label := "for a domain and a positive integer", Oper := "Projection" ), content := 0, count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "Ref", next := 300009, root := ~, start := 299920, stop := 300008 ), rec( content := " on the\\ n returned group.", count := [ 7, 2, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "P", next := 300040, root := ~, start := 300036, stop := 300039 ), rec( content := "\n\n \ By default, the automorphism group is found using ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 5, 12 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 5, 12 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 5, 12 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 5, 12 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 5, 12 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 5, 12 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 5, 12 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 12 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph and homogenous list" ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Ref", next := 300351, root := ~, start := 300267, stop := 300350 ), rec( content := ",\n " , count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Ref", next := 300393, root := ~, start := 300357, stop := 300392 ), rec( content := ", ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Ref", next := 300425, root := ~, start := 300395, stop := 300424 ), rec( content := ",\n a\ nd ", count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "Ref", next := 300465, root := ~, start := 300435, stop := 300464 ), rec( content := ".\n\n " , count := [ 7, 2, 5, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> cycle := CycleDigraph(9);\n\nga\ p> G := AutomorphismGroup(cycle);;\ngap> IsCyclic(G) and Size(G) = 9;\ntrue\ng\ ap> colours := [[1, 4, 7], [2, 5, 8], [3, 6, 9]];;\ngap> H := AutomorphismGrou\ p(cycle, colours);;\ngap> Size(H);\n3\ngap> H = AutomorphismGroup(cycle, [1, 2\ , 3, 1, 2, 3, 1, 2, 3]);\ntrue\ngap> H = SubgroupByProperty(G, p -> OnTuplesSe\ ts(colours, p) = colours);\ntrue\ngap> IsTrivial(AutomorphismGroup(cycle, [1, \ 1, 2, 2, 2, 2, 2, 2, 2]));\ntrue", count := [ 7, 2, 5, 13 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 13 ], mathmode := "Text", name := "Example", next := 300987, root := ~, start := 300470, stop := 300986 ), rec( content := "\n ", count := [ 7, 2, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 301004, root := ~, start := 297934, stop := 301003 ) ], count := [ 7, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 301018, root := ~, start := 297765, stop := 301017 ), rec( content := "\n\n ", count := [ 7, 2, 5, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, vert_co\ lours, edge_colours", Label := "for a digraph, homogeneous list, and list", Name := "AutomorphismGro\ up" ), content := 0, count := [ 7, 2, 6, 2 ], mathmode := "Text", name := "Oper", next := 301171, root := ~, start := 301039, stop := 301170 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "Returns", next := 301213, root := ~, start := 301174, stop := 301212 ), rec( attributes := rec( ), content := [ rec( content := "\n\n T\ his operation computes the automorphism group of a vertex- and/or\n edge-co\ loured digraph.\n A coloured digraph can be specified by its underlying dig\ raph\n ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "A", next := 301413, root := ~, start := 301399, stop := 301412 ), rec( content := " and col\ ourings ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "A", next := 301448, root := ~, start := 301429, stop := 301447 ), rec( content := ", ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge_colours", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "A", next := 301469, root := ~, start := 301450, stop := 301468 ), rec( content := ".\n L\ et ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "C", next := 301487, root := ~, start := 301479, stop := 301486 ), rec( content := " be the \ number of vertices of ", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "A", next := 301531, root := ~, start := 301517, stop := 301530 ), rec( content := ". The co\ lourings\n must have the following forms:\n\n ", count := [ 7, 2, 6, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 6, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "vert_colours", count := [ 7, 2, 6, 5 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 5 ], mathmode := \ "Text", name := "A", next := 301635, root := ~, start := 301616, stop := 30163\ 4 ), rec( content := " must be ", count := [ 7, 2, 6, 5 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "fail", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "K", next :\ = 301655, root := ~, start := 301644, stop := 301654 ), rec( content := " or a\ list of ", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "n", count :\ = [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 2, 6, 5 ], mathmode := "Text", name := "C", next := 301677, root := ~, star\ t := 301669, stop := 301676 ), rec( content := " integers,\n where ", coun\ t := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "vert_colours", count := [ 7, \ 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, \ 2, 6, 5 ], mathmode := "Text", name := "A", next := 301721, root := ~, start :\ = 301702, stop := 301720 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "[i]", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "C", next :\ = 301731, root := ~, start := 301721, stop := 301730 ), rec( content := " is t\ he colour of vertex ", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "i", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ \ ) ], count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "C", next := 301764, root\ := ~, start := 301756, stop := 301763 ), rec( content := ",\n using th\ e colours ", count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "[1 .. m]", \ count := [ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 6, 5 ], mathmode := "Text", name := "C", next := 301807, root := \ ~, start := 301792, stop := 301806 ), rec( content := " for some ", count := [ \ 7, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "m <= n", count := [ 7, 2, 6, 5 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 5 ], math\ mode := "Text", name := "C", next := 301833, root := ~, start := 301817, stop \ := 301832 ), rec( content := ";\n ", count := [ 7, 2, 6, 5 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 5 ], name := "Ite\ m", next := 301848, root := ~, start := 301601, stop := 301847 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 6, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "edge_colours", count := [ 7, 2, 6, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], mathmode := \ "Text", name := "A", next := 301889, root := ~, start := 301870, stop := 30188\ 8 ), rec( content := " must be ", count := [ 7, 2, 6, 7 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "fail", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "K", next :\ = 301909, root := ~, start := 301898, stop := 301908 ), rec( content := " or a\ list of ", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "n", count :\ = [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 2, 6, 7 ], mathmode := "Text", name := "C", next := 301931, root := ~, star\ t := 301923, stop := 301930 ), rec( content := " lists of\n integers of\ the same shape as ", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PC\ DATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\ OutNeighbours(digraph)", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], mathmode := "Text", name := \ "C", next := 302008, root := ~, start := 301979, stop := 302007 ), rec( conten\ t := ", where\n ", count := [ 7, 2, 6, 7 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "edge_colours", count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 6, 7 ], mathmode := "Text", name := "A", next\ := 302043, root := ~, start := 302024, stop := 302042 ), rec( attributes := re\ c( ), content := [ rec( content := "[i][j]", count := [ 7, 2, 6, 7 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], mathmod\ e := "Text", name := "C", next := 302056, root := ~, start := 302043, stop := \ 302055 ), rec( content := " is the colour of the edge\n ", count := [ 7\ , 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "OutNeighbours(digraph)[i][j]", cou\ nt := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 6, 7 ], mathmode := "Text", name := "C", next := 302126, root := \ ~, start := 302091, stop := 302125 ), rec( content := ", using the colours ", coun\ t := [ 7, 2, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[1 .. k]", count := [ 7, 2, 6\ , 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6\ , 7 ], mathmode := "Text", name := "C", next := 302161, root := ~, start := 30\ 2146, stop := 302160 ), rec( content := "\n for some ", count := [ 7, 2\ , 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "k <= n", count := [ 7, 2, 6, 7 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], math\ mode := "Text", name := "C", next := 302195, root := ~, start := 302179, stop \ := 302194 ), rec( content := ";\n ", count := [ 7, 2, 6, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 7 ], name := "Ite\ m", next := 302210, root := ~, start := 301855, stop := 302209 ) ], count := [ 7, 2, 6, 4 ], mathmode := "Text", name := "List", next := 302222, root := ~, start := 301588, stop := 302221 ), rec( content := "\n\n \ Giving ", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ert_colours", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "A", next := 302254, root := ~, start := 302235, stop := 302253 ), rec( content := " [", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ dge_colours", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "A", next := 302275, root := ~, start := 302256, stop := 302274 ), rec( content := "] as ", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "C", next := 302291, root := ~, start := 302280, stop := 302290 ), rec( content := " is\n \ equivalent to setting all vertices [edges] to be the same colour. ", count := [ 7, 2, 6, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "P", next := 302369, root := ~, start := 302365, stop := 302368 ), rec( content := "\n\n \ Unlike ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "Ref", next := 302435, root := ~, start := 302382, stop := 302434 ), rec( content := ", it is\\ n possible to obtain the automorphism group of an edge-coloured multidigrap\ h\n (see ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "Ref", next := 302560, root := ~, start := 302531, stop := 302559 ), rec( content := ") when n\ o two edges share the same\n source, range, and colour.\n\n The ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism group", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "E", next := 302660, root := ~, start := 302635, stop := 302659 ), rec( content := " of a ve\ rtex/edge-coloured digraph\n ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 302713, root := ~, start := 302699, stop := 302712 ), rec( content := " with co\ louring ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 302737, root := ~, start := 302729, stop := 302736 ), rec( content := " is the \ group consisting of its\n vertex/edge-colour preserving automorphisms; an\n\ ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ utomorphism", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "E", next := 302844, root := ~, start := 302825, stop := 302843 ), rec( content := " of ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 302862, root := ~, start := 302848, stop := 302861 ), rec( content := " is an i\ somorphism of vertex/edge-coloured\n digraphs from ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 302937, root := ~, start := 302923, stop := 302936 ), rec( content := " to itse\ lf. This group is equal to the\n subgroup of ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ utomorphismGroup(", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 3030\ 28, root := ~, start := 303014, stop := 303027 ), rec( content := ")", count := [ 7, 2, 6, 10 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "C", next := 303033, root := ~, start := 302993, stop := 303032 ), rec( content := " consist\ ing of those\n automorphisms that preserve the colouring specified by ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "A", next := 303127, root := ~, start := 303113, stop := 303126 ), rec( content := ". See\n \ ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "Ref", next := 303191, root := ~, start := 303137, stop := 303190 ), rec( content := ", and se\ e ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "Ref", next := 303284, root := ~, start := 303201, stop := 303283 ), rec( content := "\n fo\ r more information about isomorphisms of coloured digraphs. ", count := [ 7, 2, 6, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "P", next := 303355, root := ~, start := 303351, stop := 303354 ), rec( content := "\n\n \ If ", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "A", next := 303378, root := ~, start := 303364, stop := 303377 ), rec( content := " is not \ a multidigraph then the automorphism group is\n returned as a group of perm\ utations on the set of vertices of\n ", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "A", next := 303516, root := ~, start := 303502, stop := 303515 ), rec( content := ". ", count := [ 7, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "P", next := 303522, root := ~, start := 303518, stop := 303521 ), rec( content := "\n\n \ If ", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "A", next := 303545, root := ~, start := 303531, stop := 303544 ), rec( content := " is a mu\ ltidigraph then the automorphism group is returned\n as the direct product \ of a group of permutations on the set of vertices of\n ", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "A", next := 303701, root := ~, start := 303687, stop := 303700 ), rec( content := " with a \ group of permutations on the set of edges of\n ", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "A", next := 303772, root := ~, start := 303758, stop := 303771 ), rec( content := ". These \ groups can be accessed using ", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Label := "for a domain and a positive integer", Oper := "Projection" ), content := 0, count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "Ref", next := 303898, root := ~, start := 303809, stop := 303897 ), rec( content := " on the\\ n returned group.", count := [ 7, 2, 6, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "P", next := 303929, root := ~, start := 303925, stop := 303928 ), rec( content := "\n\n \ By default, the automorphism group is found using ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 6, 13 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 6, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 6, 13 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 6, 13 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 6, 13 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 6, 13 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 6, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 13 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissAu\ tomorphismGroup", Label := "for a digraph, homogenous list, and list" ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Ref", next := 304249, root := ~, start := 304157, stop := 304248 ), rec( content := ",\n \ ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Attr := "NautyAu\ tomorphismGroup" ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Ref", next := 304292, root := ~, start := 304256, stop := 304291 ), rec( content := ", ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Ref", next := 304324, root := ~, start := 304294, stop := 304323 ), rec( content := ", and\n \ ", count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "Ref", next := 304365, root := ~, start := 304335, stop := 304364 ), rec( content := ".\n\n " , count := [ 7, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> cycle := CycleDigraph(12);\n\n\ gap> vert_colours := List([1 .. 12], x -> x mod 3 + 1);;\ngap> edge_colours :=\ List([1 .. 12], x -> [x mod 2 + 1]);;\ngap> Size(AutomorphismGroup(cycle));\n\ 12\ngap> Size(AutomorphismGroup(cycle, vert_colours));\n4\ngap> Size(Automorph\ ismGroup(cycle, fail, edge_colours));\n6\ngap> Size(AutomorphismGroup(cycle, v\ ert_colours, edge_colours));\n2\ngap> IsTrivial(AutomorphismGroup(cycle,\n> ve\ rt_colours, List([1 .. 12], x -> [x mod 4 + 1])));\ntrue\n", count := [ 7, 2, 6, 14 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 6, 14 ], mathmode := "Text", name := "Example", next := 304912, root := ~, start := 304370, stop := 304911 ), rec( content := "\n ", count := [ 7, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 6, 3 ], mathmode := "Text", name := "Description", next := 304929, root := ~, start := 301216, stop := 304928 ) ], count := [ 7, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 304943, root := ~, start := 301024, stop := 304942 ), rec( content := "\n\n ", count := [ 7, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "BlissCanonicalL\ abelling" ), content := 0, count := [ 7, 2, 7, 2 ], mathmode := "Text", name := "Attr", next := 305038, root := ~, start := 304964, stop := 305037 ), rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "NautyCanonicalL\ abelling" ), content := 0, count := [ 7, 2, 7, 4 ], mathmode := "Text", name := "Attr", next := 305115, root := ~, start := 305041, stop := 305114 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion, or a list of two permutations.", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "Returns", next := 305182, root := ~, start := 305118, stop := 305181 ), rec( attributes := rec( ), content := [ rec( content := "\n A f\ unction ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "M", next := 305225, root := ~, start := 305214, stop := 305224 ), rec( content := " that ma\ ps a digraph to a digraph is a ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical\n representative map", count := [ 7, 2, 7, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "E", next := 305305, root := ~, start := 305264, stop := 305304 ), rec( content := " if the \ following two conditions hold for all\n digraphs ", count := [ 7, 2, 7, 5 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "M", next := 305372, root := ~, start := 305364, stop := 305371 ), rec( content := " and ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "M", next := 305385, root := ~, start := 305377, stop := 305384 ), rec( content := ": ", count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 7, 6 ], mathmode := "Text", name := "P", next := 305391, root := ~, start := 305387, stop := 305390 ), rec( content := "\n\n \ ", count := [ 7, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 7, 8 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "\\rho(G)", count := [ 7, 2, 7, 8 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 8 ], mathmode := "Tex\ t", name := "M", next := 305439, root := ~, start := 305425, stop := 305438 ), rec( cont\ ent := " and ", count := [ 7, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count :\ = [ 7, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 2, 7, 8 ], mathmode := "Text", name := "M", next := 305452, root := ~, star\ t := 305444, stop := 305451 ), rec( content := " are isomorphic as digraphs; a\ nd\n ", count := [ 7, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 7, 8 ], name := "Item", next := 305498, root := ~, start := 305\ 410, stop := 305497 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\rho(G)=\\\ rho(H)", count := [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 7, 10 ], mathmode := "Text", name := "M", next := 3\ 05542, root := ~, start := 305520, stop := 305541 ), rec( content := " if and \ only if ", count := [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count :\ = [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 7, 2, 7, 10 ], mathmode := "Text", name := "M", next := 305566, root := ~ , star\ t := 305558, stop := 305565 ), rec( content := " and ", count := [ 7, 2, 7, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "H", count := [ 7, 2, 7, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 10 ], mathmode := \ "Text", name := "M", next := 305579, root := ~, start := 305571, stop := 30557\ 8 ), rec( content := " are\n isomorphic as digraphs.\n ", count :=\ [ 7, 2, 7, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 10 ], name := "Ite\ m", next := 305629, root := ~, start := 305505, stop := 305628 ) ], count := [ 7, 2, 7, 7 ], mathmode := "Text", name := "List", next := 305641, root := ~, start := 305397, stop := 305640 ), rec( content := "\n\n \ A ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical labelling", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "E", next := 305675, root := ~, start := 305649, stop := 305674 ), rec( content := " of a di\ graph ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "M", next := 305697, root := ~, start := 305689, stop := 305696 ), rec( content := " (under \ ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ \rho", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "M", next := 305716, root := ~, start := 305705, stop := 305715 ), rec( content := ") is an\\ n isomorphism of ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "M", next := 305751, root := ~, start := 305743, stop := 305750 ), rec( content := " onto it\ s ", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical representative", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "E", next := 305792, root := ~, start := 305761, stop := 305791 ), rec( content := ",\n " , count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho(G)", count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "M", next := 305812, root := ~, start := 305798, stop := 305811 ), rec( content := ". See " , count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 7, 12 ], mathmode := "Text", name := "Ref", next := 305874, root := ~, start := 305819, stop := 305873 ), rec( content := "\n fo\ r more information about isomorphisms of digraphs. ", count := [ 7, 2, 7, 12 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "P", next := 305936, root := ~, start := 305932, stop := 305935 ), rec( content := "\n\n \ ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "C", next := 305972, root := ~, start := 305942, stop := 305971 ), rec( content := " returns\ a canonical labelling of the digraph\n ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "A", next := 306036, root := ~, start := 306022, stop := 306035 ), rec( content := " found u\ sing ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 7, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila and Petteri Kaski.\n ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ autyCanonicalLabelling", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "C", next := 306128, root := ~, start := 306098, stop := 306127 ), rec( content := " returns\ a canonical labelling of the digraph\n ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "A", next := 306192, root := ~, start := 306178, stop := 306191 ), rec( content := " found u\ sing ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 7, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by Bren\ dan McKay and Adolfo Piperno.\n Note that the canonical labellings returned\ by ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 7, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " and ", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 7, 13 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 7, 13 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 13 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " are not\ \n usually the same (and may depend of the version used).", count := [ 7, 2, 7, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "P", next := 306391, root := ~, start := 306387, stop := 306390 ), rec( content := "\n\n \ ", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "C", next := 306427, root := ~, start := 306397, stop := 306426 ), rec( content := " can onl\ y be computed if ", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "A", next := 306466, root := ~, start := 306452, stop := 306465 ), rec( content := " has\n \ no multiple edges; see ", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "Ref", next := 306527, root := ~, start := 306498, stop := 306526 ), rec( content := ". ", count := [ 7, 2, 7, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 7, 15 ], mathmode := "Text", name := "P", next := 306533, root := ~, start := 306529, stop := 306532 ), rec( content := "\n\n \ ", count := [ 7, 2, 7, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := DigraphFromDiSparse6String(\".ImNS_AiB?qRN\");\n\ngap> BlissCanonicalLabelling(digraph1);\n(1\ ,9,5,7)(3,6,4,10)\ngap> p := (1, 2, 7, 5)(3, 9)(6, 10, 8);;\ngap> digraph2 := \ OnDigraphs(digraph1, p);\n\ngap> \ digraph1 = digraph2;\nfalse\ngap> OnDigraphs(digraph1, BlissCanonicalLabelling\ (digraph1)) =\n> OnDigraphs(digraph2, BlissCanonicalLabelling(digraph2));\n\ true", count := [ 7, 2, 7, 16 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 16 ], mathmode := "Text", name := "Example", next := 307029, root := ~, start := 306539, stop := 307028 ), rec( content := "\n ", count := [ 7, 2, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 7, 5 ], mathmode := "Text", name := "Description", next := 307046, root := ~, start := 305185, stop := 307045 ) ], count := [ 7, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 307060, root := ~, start := 304949, stop := 307059 ), rec( content := "\n\n ", count := [ 7, 2, 7, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, colours\ ", Label := "for a digraph and a list", Name := "BlissCanonicalLabelling" ), content := 0, count := [ 7, 2, 8, 2 ], mathmode := "Text", name := "Oper", next := 307179, root := ~, start := 307081, stop := 307178 ), rec( attributes := rec( Arg := "digraph, colours\ ", Label := "for a digraph and a list", Name := "NautyCanonicalLabelling" ), content := 0, count := [ 7, 2, 8, 4 ], mathmode := "Text", name := "Oper", next := 307280, root := ~, start := 307182, stop := 307279 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion.", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "Returns", next := 307316, root := ~, start := 307283, stop := 307315 ), rec( attributes := rec( ), content := [ rec( content := "\n A f\ unction ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "M", next := 307359, root := ~, start := 307348, stop := 307358 ), rec( content := " that ma\ ps a coloured digraph to a coloured digraph is\n a ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical representative map", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "E", next := 307455, root := ~, start := 307420, stop := 307454 ), rec( content := " if the \ following two conditions hold\n for all coloured digraphs ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "M", next := 307531, root := ~, start := 307523, stop := 307530 ), rec( content := " and ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "M", next := 307544, root := ~, start := 307536, stop := 307543 ), rec( content := ":\n\n \ ", count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 8, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "\\rho(G)", count := [ 7, 2, 8, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 7 ], mathmode := "Tex\ t", name := "M", next := 307593, root := ~, start := 307579, stop := 307592 ), rec( cont\ ent := " and ", count := [ 7, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count :\ = [ 7, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 2, 8, 7 ], mathmode := "Text", name := "M", next := 307606, root := ~, star\ t := 307598, stop := 307605 ), rec( content := " are isomorphic as coloured di\ graphs; and\n ", count := [ 7, 2, 8, 7 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 7, 2, 8, 7 ], name := "Item", next := 307661, root := ~, start := 307\ 564, stop := 307660 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\rho(G)=\\\ rho(H)", count := [ 7, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 7, 2, 8, 9 ], mathmode := "Text", name := "M", next := 307\ 705, root := ~, start := 307683, stop := 307704 ), rec( content := " if and on\ ly if ", count := [ 7, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ), rec( attributes := rec( ), content := [ rec( content := "G", count := \ [ 7, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 7, 2, 8, 9 ], mathmode := "Text", name := "M", next := 307729, root := ~, star\ t := 307721, stop := 307728 ), rec( content := " and ", count := [ 7, 2, 8, 9 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "H", count := [ 7, 2, 8, 9 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 9 ], mathmode := "Text", name\ := "M", next := 307742, root := ~, start := 307734, stop := 307741 ), rec( cont\ ent := " are\n isomorphic as coloured digraphs.\n ", count := [ 7,\ 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 9 ], name := "Ite\ m", next := 307801, root := ~, start := 307668, stop := 307800 ) ], count := [ 7, 2, 8, 6 ], mathmode := "Text", name := "List", next := 307813, root := ~, start := 307551, stop := 307812 ), rec( content := "\n\n \ A ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical labelling", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "E", next := 307847, root := ~, start := 307821, stop := 307846 ), rec( content := " of a co\ loured digraph ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "M", next := 307878, root := ~, start := 307870, stop := 307877 ), rec( content := " (under\\ n ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "M", next := 307901, root := ~, start := 307890, stop := 307900 ), rec( content := ") is an \ isomorphism of ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "M", next := 307932, root := ~, start := 307924, stop := 307931 ), rec( content := " onto it\ s ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ anonical\n representative", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "E", next := 307979, root := ~, start := 307942, stop := 307978 ), rec( content := ", ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ \rho(G)", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "M", next := 307995, root := ~, start := 307981, stop := 307994 ), rec( content := ". See " , count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "Ref", next := 308085, root := ~, start := 308002, stop := 308084 ), rec( content := " for mor\ e information about\n isomorphisms of coloured digraphs. ", count := [ 7, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "P", next := 308156, root := ~, start := 308152, stop := 308155 ), rec( content := "\n\n \ A coloured digraph can be specified by its underlying digraph ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "A", next := 308238, root := ~, start := 308224, stop := 308237 ), rec( content := "\n an\ d its colouring ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "A", next := 308275, root := ~, start := 308261, stop := 308274 ), rec( content := ". Let " , count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "C", next := 308290, root := ~, start := 308282, stop := 308289 ), rec( content := " be the \ number of vertices of\n ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "A", next := 308338, root := ~, start := 308324, stop := 308337 ), rec( content := ". The co\ louring ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "A", next := 308368, root := ~, start := 308354, stop := 308367 ), rec( content := " may hav\ e one of the following\n two forms: ", count := [ 7, 2, 8, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 8, 13 ], mathmode := "Text", name := "P", next := 308418, root := ~, start := 308414, stop := 308417 ), rec( content := "\n\n \ ", count := [ 7, 2, 8, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n a list of ", count := [ 7, 2\ , 8, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "n", count := [ 7, 2, 8, 15 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15 ], mathm\ ode := "Text", name := "C", next := 308470, root := ~, start := 308462, stop :\ = 308469 ), rec( content := " integers, where ", count := [ 7, 2, 8, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 8, 15 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15 ], mathmode := "Te\ xt", name := "A", next := 308501, root := ~, start := 308487, stop := 308500 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 8, 15 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15\ ], mathmode := "Text", name := "C", next := 308511, root := ~, start := 30850\ 1, stop := 308510 ), rec( content := " is the\n colour of vertex ", coun\ t := [ 7, 2, 8, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 8, 15 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15 ] , math\ mode := "Text", name := "C", next := 308552, root := ~, start := 308544, stop \ := 308551 ), rec( content := ", using the colours ", count := [ 7, 2, 8, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 .. m]", count := [ 7, 2, 8, 15 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 15 ], mathmode := "T\ ext", name := "C", next := 308587, root := ~, start := 308572, stop := 308586 ), rec( cont\ ent := " for some\n ", count := [ 7, 2, 8, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "m <= n", count := [ 7, 2, 8, 15 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 2, 8, 15 ], mathmode := "Text", name := "C", next\ := 308621, root := ~, start := 308605, stop := 308620 ), rec( content := "; o\ r\n ", count := [ 7, 2, 8, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 15 ], name := "Item", next := 308639, root := ~, start := 308\ 437, stop := 308638 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n a list of non-empty disjoint lists whose union is\n ", coun\ t := [ 7, 2, 8, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphVertices(", count := [\ 7, 2, 8, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 8, 1\ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, \ 17 ], mathmode := "Text", name := "A", next := 308752, root := ~, start := 308\ 738, stop := 308751 ), rec( content := ")", count := [ 7, 2, 8, 17 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], mathmod\ e := "Text", name := "C", next := 308757, root := ~, start := 308719, stop := \ 308756 ), rec( content := ", such that\n ", count := [ 7, 2, 8, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 8, 17 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], mathmode := "Te\ xt", name := "A", next := 308791, root := ~, start := 308777, stop := 308790 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 8, 17 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17\ ], mathmode := "Text", name := "C", next := 308801, root := ~, start := 30879\ 1, stop := 308800 ), rec( content := " is the list of all vertices with colour\ \n ", count := [ 7, 2, 8, 17 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "i", count :\ = [ 7, 2, 8, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 7, 2, 8, 17 ], mathmode := "Text", name := "C", next := 308858, root := ~ , star\ t := 308850, stop := 308857 ), rec( content := ".\n ", count := [ 7, 2, 8\ , 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 17 ], name := "Ite\ m", next := 308873, root := ~, start := 308646, stop := 308872 ) ], count := [ 7, 2, 8, 14 ], mathmode := "Text", name := "List", next := 308885, root := ~, start := 308424, stop := 308884 ), rec( content := "\n\n \ If ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "A", next := 308908, root := ~, start := 308894, stop := 308907 ), rec( content := " and ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "A", next := 308927, root := ~, start := 308913, stop := 308926 ), rec( content := " togethe\ r form a coloured digraph,\n ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "C", next := 308996, root := ~, start := 308966, stop := 308995 ), rec( content := " returns\ a canonical labelling of the digraph\n ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "A", next := 309060, root := ~, start := 309046, stop := 309059 ), rec( content := " found u\ sing ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 8, 19 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila and Petteri Kaski.\n Similarly, ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ autyCanonicalLabelling", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "C", next := 309163, root := ~, start := 309133, stop := 309162 ), rec( content := " returns\ a canonical labelling of\n the digraph ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "A", next := 309227, root := ~, start := 309213, stop := 309226 ), rec( content := " found u\ sing ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 8, 19 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by Bren\ dan McKay and Adolfo\n Piperno. Note that the canonical labellings returne\ d by ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 8, 19 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " and\n \ ", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 8, 19 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 8, 19 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 19 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " are not\ usually the same (and may depend of the version used).", count := [ 7, 2, 8, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "P", next := 309427, root := ~, start := 309423, stop := 309426 ), rec( content := "\n\n \ ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "C", next := 309463, root := ~, start := 309433, stop := 309462 ), rec( content := " can onl\ y be computed if ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "A", next := 309502, root := ~, start := 309488, stop := 309501 ), rec( content := " has\n \ no multiple edges; see ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMulti\ Digraph" ), content := 0, count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "Ref", next := 309563, root := ~, start := 309534, stop := 309562 ), rec( content := ". The ca\ nonical\n labelling of ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "A", next := 309610, root := ~, start := 309596, stop := 309609 ), rec( content := " is give\ n as a permutation of its vertices.\n The canonical representative of ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "A", next := 309704, root := ~, start := 309690, stop := 309703 ), rec( content := " can be \ created from\n ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "A", next := 309743, root := ~, start := 309729, stop := 309742 ), rec( content := " and its\ canonical labelling ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "C", next := 309780, root := ~, start := 309772, stop := 309779 ), rec( content := " by usin\ g the\n operation ", count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a perm", Oper := "OnDigraphs" ), content := 0, count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "Ref", next := 309866, root := ~, start := 309808, stop := 309865 ), rec( content := ":\n " , count := [ 7, 2, 8, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap> OnDigraphs(digraph, p);", count := [ 7, 2, 8, 21 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 21 ], mathmode := "Text", name := "Log", next := 309911, root := ~, start := 309872, stop := 309910 ), rec( content := "\n\n \ The colouring of the canonical representative can easily be\n constructed. \ A vertex ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 310011, root := ~, start := 310003, stop := 310010 ), rec( content := " (in ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "A", next := 310030, root := ~, start := 310016, stop := 310029 ), rec( content := ") has co\ lour ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 310051, root := ~, start := 310043, stop := 310050 ), rec( content := " if\n \ and only if the vertex ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ^ p", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 310094, root := ~, start := 310082, stop := 310093 ), rec( content := " (in the\ canonical representative) has\n colour ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 310152, root := ~, start := 310144, stop := 310151 ), rec( content := ", where \ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "C", next := 310168, root := ~, start := 310160, stop := 310167 ), rec( content := " is the \ permutation of the canonical\n labelling that acts on the vertices of ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "A", next := 310262, root := ~, start := 310248, stop := 310261 ), rec( content := ". In par\ ticular, if\n ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "A", next := 310300, root := ~, start := 310286, stop := 310299 ), rec( content := " has the\ first form that is described above, then the\n colouring of the canonical \ representative is given by:\n\n ", count := [ 7, 2, 8, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap> List(DigraphVertices(digraph), i -> colours[i / p]);", count := [ 7, 2, 8, 23 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 23 ], mathmode := "Text", name := "Log", next := 310486, root := ~, start := 310418, stop := 310485 ), rec( content := "\n\n \ On the other hand, if ", count := [ 7, 2, 8, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours", count := [ 7, 2, 8, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 24 ], mathmode := "Text", name := "A", next := 310528, root := ~, start := 310514, stop := 310527 ), rec( content := " has the\ second form above, then the\n canonical representative has colouring:\n\n \ ", count := [ 7, 2, 8, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ap> OnTuplesSets(colours, p);", count := [ 7, 2, 8, 25 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 25 ], mathmode := "Text", name := "Log", next := 310655, root := ~, start := 310614, stop := 310654 ), rec( content := "\n\n \ ", count := [ 7, 2, 8, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := 0, count := [ 7, 2, 8, 27 ], mathmode := "Text", name := "P", next := 310665, root := ~, start := 310661, stop := 310664 ), rec( content := "\n\n \ ", count := [ 7, 2, 8, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := DigraphFromDiSparse6String(\".ImNS_AiB?qRN\");\n\ngap> colours := [[1, 2, 8, 9, 10], [3, 4, 5,\ 6, 7]];;\ngap> p := BlissCanonicalLabelling(digraph, colours);\n(1,5,8,4,10,3\ ,9)(6,7)\ngap> OnDigraphs(digraph, p);\n\ngap> OnTuplesSets(colours, p);\n[ [ 1, 2, 3, 4, 5 ], [ 6, 7, 8, 9, 10\ ] ]\ngap> colours := [1, 1, 1, 1, 2, 3, 1, 3, 2, 1];;\ngap> p := BlissCanonic\ alLabelling(digraph, colours);\n(1,6,9,7)(3,4,5,8,10)\ngap> OnDigraphs(digraph\ , p);\n\ngap> List(DigraphVertice\ s(digraph), i -> colours[i / p]);\n[ 1, 1, 1, 1, 1, 1, 2, 2, 3, 3 ]", count := [ 7, 2, 8, 28 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 8, 28 ], mathmode := "Text", name := "Example", next := 311376, root := ~, start := 310671, stop := 311375 ), rec( content := "\n ", count := [ 7, 2, 8, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 8, 5 ], mathmode := "Text", name := "Description", next := 311393, root := ~, start := 307319, stop := 311392 ) ], count := [ 7, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 311407, root := ~, start := 307066, stop := 311406 ), rec( content := "\n\n ", count := [ 7, 2, 8, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "BlissCanonicalD\ igraph" ), content := 0, count := [ 7, 2, 9, 2 ], mathmode := "Text", name := "Attr", next := 311478, root := ~, start := 311428, stop := 311477 ), rec( attributes := rec( Arg := "digraph", Name := "NautyCanonicalD\ igraph" ), content := 0, count := [ 7, 2, 9, 4 ], mathmode := "Text", name := "Attr", next := 311531, root := ~, start := 311481, stop := 311530 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "Returns", next := 311563, root := ~, start := 311534, stop := 311562 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ attribute ", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ lissCanonicalLabelling", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "C", next := 311628, root := ~, start := 311598, stop := 311627 ), rec( content := " returns\ the canonical\n representative found by applying ", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "BlissCa\ nonicalLabelling", Label := "for a digraph" ), content := 0, count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "Ref", next := 311753, root := ~, start := 311688, stop := 311752 ), rec( content := ". The di\ graph returned is canonical in the sense\n that\n\n ", count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 9, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "BlissCanonicalDigraph(", count := [ 7, 2, 9, 7 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), c\ ontent := [ rec( content := "digraph", count := [ 7, 2, 9, 7 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 7 ], mathmode := "T\ ext", name := "A", next := 311883, root := ~, start := 311869, stop := 311882 \ ), rec( content := ")", count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "\ C", next := 311888, root := ~, start := 311844, stop := 311887 ), rec( content\ := " and ", count := [ 7, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", c\ ount := [ 7, 2, 9, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 9, 7 ], mathmode := "Text", name := "A", next := 311907, root := \ ~, start := 311893, stop := 311906 ), rec( content := " are\n isomorphi\ c as digraphs; and\n ", count := [ 7, 2, 9, 7 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 7 ], name := "Item", next := 3119\ 61, root := ~, start := 311829, stop := 311960 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n If ", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "gr", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "C", next := 31199\ 5, root := ~, start := 311986, stop := 311994 ), rec( content := " is any digr\ aph then ", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "BlissCanoni\ calDigraph(gr)", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "C", next\ := 312048, root := ~, start := 312016, stop := 312047 ), rec( content := "\n \ and ", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "BlissCanoni\ calDigraph(", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph"\ , count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ]\ , count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "A", next := 312100, ro\ ot := ~, start := 312086, stop := 312099 ), rec( content := ")", count := [ 7,\ 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7,\ 2, 9, 9 ], mathmode := "Text", name := "C", next := 312105, root := ~, start \ := 312061, stop := 312104 ), rec( content := " are equal if and only\n \ if ", count := [ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "gr", count := [ \ 7, 2, 9, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 7, 2, 9, 9 ], mathmode := "Text", name := "C", next := 312148, root := ~, star\ t := 312139, stop := 312147 ), rec( content := " and ", count := [ 7, 2, 9, 9 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "digraph", count := [ 7, 2, 9, 9 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 9 ], mathmode := "Text\ ", name := "A", next := 312167, root := ~, start := 312153, stop := 312166 ), rec( cont\ ent := " are isomorphic as digraphs.\n ", count := [ 7, 2, 9, 9 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 9 ], name := "Ite\ m", next := 312209, root := ~, start := 311968, stop := 312208 ) ], count := [ 7, 2, 9, 6 ], mathmode := "Text", name := "List", next := 312221, root := ~, start := 311816, stop := 312220 ), rec( content := "\n\n \ Analogously, the attribute ", count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ autyCanonicalLabelling", count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "C", next := 312284, root := ~, start := 312254, stop := 312283 ), rec( content := " returns\ the\n canonical representative found by applying ", count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "NautyCa\ nonicalLabelling", Label := "for a digraph" ), content := 0, count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "Ref", next := 312409, root := ~, start := 312344, stop := 312408 ), rec( content := ".\n " , count := [ 7, 2, 9, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "P", next := 312419, root := ~, start := 312415, stop := 312418 ), rec( content := "\n\n \ If the argument ", count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "A", next := 312455, root := ~, start := 312441, stop := 312454 ), rec( content := " is muta\ ble, then the return value of\n this attribute is recomputed every time it \ is called. \n\n ", count := [ 7, 2, 9, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := Digraph([[1], [2, 3], [3], [1, 2, 3]]);\n\ngap> canon := BlissCanonicalDigraph(digraph);\n\ngap> OutNeighbours(canon);\n[ [ 1 ], \ [ 2 ], [ 3, 2 ], [ 1, 3, 2 ] ]\n", count := [ 7, 2, 9, 13 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 13 ], mathmode := "Text", name := "Example", next := 312848, root := ~, start := 312557, stop := 312847 ), rec( content := "\n ", count := [ 7, 2, 9, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 9, 5 ], mathmode := "Text", name := "Description", next := 312865, root := ~, start := 311566, stop := 312864 ) ], count := [ 7, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 312879, root := ~, start := 311413, stop := 312878 ), rec( content := "\n\n ", count := [ 7, 2, 9, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphGroup" ) , content := 0, count := [ 7, 2, 10, 2 ], mathmode := "Text", name := "Attr", next := 312941, root := ~, start := 312900, stop := 312940 ), rec( attributes := rec( ), content := [ rec( content := "A permuta\ tion group.", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "Returns", next := 312983, root := ~, start := 312944, stop := 312982 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "A", next := 313021, root := ~, start := 313007, stop := 313020 ), rec( content := " is immu\ table and was created knowing a subgroup of\n its automorphism group, then \ this group is stored in the attribute\n ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGroup", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "C", next := 313167, root := ~, start := 313148, stop := 313166 ), rec( content := ". If ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "A", next := 313186, root := ~, start := 313172, stop := 313185 ), rec( content := " is muta\ ble, or was not created\n knowing a subgroup of its automorphism group, the\ n ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGroup", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "C", next := 313292, root := ~, start := 313273, stop := 313291 ), rec( content := "\n re\ turns the entire automorphism group of ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "A", next := 313352, root := ~, start := 313338, stop := 313351 ), rec( content := ". \n \ Note that if ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "A", next := 313389, root := ~, start := 313375, stop := 313388 ), rec( content := " is muta\ ble, then the automorphism group is\n recomputed every time this function i\ s called.\n ", count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "P", next := 313492, root := ~, start := 313488, stop := 313491 ), rec( content := "\n\n \ Note that certain other constructor operations such as ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CayleyD\ igraph" ), content := 0, count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "Ref", next := 313587, root := ~, start := 313554, stop := 313586 ), rec( content := ", ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Biparti\ teDoubleDigraph" ), content := 0, count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "Ref", next := 313625, root := ~, start := 313589, stop := 313624 ), rec( content := ", and ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DoubleD\ igraph" ), content := 0, count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "Ref", next := 313664, root := ~, start := 313631, stop := 313663 ), rec( content := ", may no\ t require a group as one of the arguments,\n but use the standard construct\ or method using a group, and hence set the\n ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGroup", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "C", next := 313815, root := ~, start := 313796, stop := 313814 ), rec( content := " attribu\ te for the resulting digraph.\n\n ", count := [ 7, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> n := 4;;\ngap> adj := function(x, y)\n> return (((x - y) mod n) = 1\ ) or (((x - y) mod n) = n - 1);\n> end;;\ngap> group := CyclicGroup(IsPermG\ roup, n);\nGroup([ (1,2,3,4) ])\ngap> D := Digraph(IsMutableDigraph, group, [1\ .. n], \\^, adj);\n\ngap> HasDigrap\ hGroup(D);\nfalse\ngap> DigraphGroup(D);\nGroup([ (2,4), (1,2)(3,4) ])\ngap> A\ utomorphismGroup(D);\nGroup([ (2,4), (1,2)(3,4) ])\ngap> D := Digraph(group, [\ 1 .. n], \\^, adj);\n\ngap> HasDig\ raphGroup(D);\ntrue\ngap> DigraphGroup(D);\nGroup([ (1,2,3,4) ])\ngap> D := Do\ ubleDigraph(D);\n\ngap> HasDigrap\ hGroup(D);\ntrue\ngap> DigraphGroup(D);\nGroup([ (1,2,3,4)(5,6,7,8), (1,5)(2,6\ )(3,7)(4,8) ])\ngap> AutomorphismGroup(D) =\n> Group([(6, 8), (5, 7), (4, 6), \ (3, 5), (2, 4),\n> (1, 2)(3, 4)(5, 6)(7, 8)]);\ntrue\ngap> D := Digraph\ ([[2, 3], [], []]);\n\ngap> HasDig\ raphGroup(D);\nfalse\ngap> HasAutomorphismGroup(D);\nfalse\ngap> DigraphGroup(\ D);\nGroup([ (2,3) ])\ngap> HasAutomorphismGroup(D);\ntrue\ngap> group := Dihe\ dralGroup(8);\n\ngap> D := CayleyDigraph\ (group);\n\ngap> HasDigraphGroup(\ D);\ntrue\ngap> GeneratorsOfGroup(DigraphGroup(D));\n[ (1,2)(3,5)(4,6)(7,8), (\ 1,7,4,3)(2,5,6,8), (1,4)(2,6)(3,7)(5,8) ]", count := [ 7, 2, 10, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 10, 5 ], mathmode := "Text", name := "Example", next := 315285, root := ~, start := 313858, stop := 315284 ), rec( content := "\n ", count := [ 7, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 315302, root := ~, start := 312986, stop := 315301 ) ], count := [ 7, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 315316, root := ~, start := 312885, stop := 315315 ), rec( content := "\n\n ", count := [ 7, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphOrbits" ), content := 0, count := [ 7, 2, 11, 2 ], mathmode := "Text", name := "Attr", next := 315379, root := ~, start := 315337, stop := 315378 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ immutable list of lists of integers.\n ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "Returns", next := 315448, root := ~, start := 315382, stop := 315447 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphOrbits", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "C", next := 315489, root := ~, start := 315469, stop := 315488 ), rec( content := " returns\ the orbits of the action of the\n ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "Ref", next := 315560, root := ~, start := 315534, stop := 315559 ), rec( content := " on the \ set of vertices of ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "A", next := 315601, root := ~, start := 315587, stop := 315600 ), rec( content := ".\n\n \ ", count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> G := Group([(2, 3)(7, 8, 9), (1, 2, 3)(4, 5, 6)(8, 9)]);;\ngap> D := Edg\ eOrbitsDigraph(G, [1, 2]);\n\ngap>\ DigraphOrbits(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]\ngap> D := Digra\ phMutableCopy(D);\n\ngap> DigraphOrb\ its(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6, 7, 8, 9 ] ]\n", count := [ 7, 2, 11, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 11, 4 ], mathmode := "Text", name := "Example", next := 315990, root := ~, start := 315608, stop := 315989 ), rec( content := "\n ", count := [ 7, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 11, 3 ], mathmode := "Text", name := "Description", next := 316007, root := ~, start := 315451, stop := 316006 ) ], count := [ 7, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 316021, root := ~, start := 315322, stop := 316020 ), rec( content := "\n\n ", count := [ 7, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphOrbitRep\ s" ), content := 0, count := [ 7, 2, 12, 2 ], mathmode := "Text", name := "Attr", next := 316087, root := ~, start := 316042, stop := 316086 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ immutable list of integers.\n ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "Returns", next := 316147, root := ~, start := 316090, stop := 316146 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphOrbitReps", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "C", next := 316191, root := ~, start := 316168, stop := 316190 ), rec( content := " returns\ a list of orbit representatives\n of the action of the ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "Ref", next := 316283, root := ~, start := 316257, stop := 316282 ), rec( content := " on the \ set of vertices of\n ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "A", next := 316328, root := ~, start := 316314, stop := 316327 ), rec( content := ".\n\n \ ", count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CayleyDigraph(AlternatingGroup(4));\n\ngap> DigraphOrbitReps(D);\n[ 1 ]\ngap> D := DigraphMutableCo\ py(D);\n\ngap> DigraphOrbitReps(D)\ ;\n[ 1 ]\ngap> D := DigraphFromDigraph6String(\"&IGO??S?`?_@?a?CK?O\");\n\ngap> DigraphOrbitReps(D);\n[ 1, 2, \ 3, 4, 5, 6, 7, 8, 9, 10 ]\ngap> DigraphOrbitReps(DigraphMutableCopy(D));\n[ 1,\ 2, 3, 4, 5, 6, 7, 8, 9, 10 ]\n", count := [ 7, 2, 12, 4 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 4 ], mathmode := "Text", name := "Example", next := 316849, root := ~, start := 316335, stop := 316848 ), rec( content := "\n ", count := [ 7, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 12, 3 ], mathmode := "Text", name := "Description", next := 316866, root := ~, start := 316150, stop := 316865 ) ], count := [ 7, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 316880, root := ~, start := 316027, stop := 316879 ), rec( content := "\n\n ", count := [ 7, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphSchreier\ Vector" ), content := 0, count := [ 7, 2, 13, 2 ], mathmode := "Text", name := "Attr", next := 316951, root := ~, start := 316901, stop := 316950 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ immutable list of integers.\n ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "Returns", next := 317011, root := ~, start := 316954, stop := 317010 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphSchreierVector", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "C", next := 317060, root := ~, start := 317032, stop := 317059 ), rec( content := " returns\ the so-called ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ chreier vector", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "E", next := 317105, root := ~, start := 317083, stop := 317104 ), rec( content := "\n of\ the action of the ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "Ref", next := 317157, root := ~, start := 317131, stop := 317156 ), rec( content := " on the \ set of vertices of\n ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "A", next := 317202, root := ~, start := 317188, stop := 317201 ), rec( content := ". The Sc\ hreier vector is a list ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ch", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "C", next := 317244, root := ~, start := 317234, stop := 317243 ), rec( content := " of inte\ gers with\n length ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphNrVertices(", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "A", next := 3173\ 08, root := ~, start := 317294, stop := 317307 ), rec( content := ")", count := [ 7, 2, 13, 3 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "C", next := 317313, root := ~, start := 317273, stop := 317312 ), rec( content := " where:\\ n ", count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "sch[i] < 0:", count := [ 7, 2, 13, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 13, 4 ], mathmode := "Text", name := "C", next\ := 317365, root := ~, start := 317344, stop := 317364 ) ], count := [ 7, 2, 13, 4 ], name := "Mar\ k", next := 317372, root := ~, start := 317338, stop := 317371 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n implies that ", count := [ \ 7, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 13, 5 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 5 ], math\ mode := "Text", name := "C", next := 317415, root := ~, start := 317407, stop \ := 317414 ), rec( content := " is an orbit representative and\n ", coun\ t := [ 7, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphOrbitReps(", count := \ [ 7, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attri\ butes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 13,\ 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13\ , 5 ], mathmode := "Text", name := "A", next := 317489, root := ~, start := 31\ 7475, stop := 317488 ), rec( content := ")[-sch[i]] = i", count := [ 7, 2, 13,\ 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13\ , 5 ], mathmode := "Text", name := "C", next := 317507, root := ~, start := 31\ 7455, stop := 317506 ), rec( content := ".\n ", count := [ 7, 2, 13, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 5 ], name := "Ite\ m", next := 317522, root := ~, start := 317379, stop := 317521 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "sch[i] > 0:", count := [ 7, 2, 13, 6 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 13, 6 ], mathmode := "Text", name := "C", next\ := 317557, root := ~, start := 317536, stop := 317556 ) ], count := [ 7, 2, 13, 6 ], name := "Mar\ k", next := 317564, root := ~, start := 317530, stop := 317563 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n implies that ", count := [ \ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "i / gens[sch[i]]", count := [ 7,\ 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7\ , 2, 13, 7 ], mathmode := "Text", name := "C", next := 317622, root := ~, star\ t := 317599, stop := 317621 ), rec( content := "\n is one step closer t\ o the root (or representative) of the tree, where\n ", count := [ 7, 2,\ 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "gens", count := [ 7, 2, 13, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 7 ], math\ mode := "Text", name := "C", next := 317720, root := ~, start := 317709, stop \ := 317719 ), rec( content := " is the generators of ", count := [ 7, 2, 13, 7 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphGroup(", count := [ 7, 2, 13, 7 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "digraph", count := [ 7, 2, 13, 7 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 7 ], mathmode := "Text", \ name := "A", next := 317772, root := ~, start := 317758, stop := 317771 ), rec\ ( content := ")", count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "C", next\ := 317777, root := ~, start := 317742, stop := 317776 ), rec( content := ".\n\ ", count := [ 7, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 2, 13, 7 ], name := "Item", next := 317792, root := ~, start := 317\ 571, stop := 317791 ) ], count := [ 7, 2, 13, 4 ], mathmode := "Text", name := "List", next := 317804, root := ~, start := 317325, stop := 317803 ), rec( content := "\n\n \ ", count := [ 7, 2, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> n := 4;;\ngap> adj := function(x, y)\n> return (((x - y) mod n) = 1\ ) or (((x - y) mod n) = n - 1);\n> end;;\ngap> group := CyclicGroup(IsPermG\ roup, n);\nGroup([ (1,2,3,4) ])\ngap> D := Digraph(IsMutableDigraph, group, [1\ .. n], \\^, adj);\n\ngap> sch := Di\ graphSchreierVector(D);\n[ -1, 2, 2, 1 ]\ngap> D := CayleyDigraph(AlternatingG\ roup(4));\n\ngap> sch := Digraph\ SchreierVector(D);\n[ -1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1 ]\ngap> DigraphOrbit\ Reps(D);\n[ 1 ]\ngap> gens := GeneratorsOfGroup(DigraphGroup(D));\n[ (1,7,5)(2\ ,10,9)(3,4,11)(6,8,12), (1,3,2)(4,5,6)(7,9,8)(10,11,12) ]\ngap> 10 / gens[sch[\ 10]];\n2\ngap> 7 / gens[sch[7]];\n1\ngap> 5 / gens[sch[5]];\n7", count := [ 7, 2, 13, 10 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 13, 10 ], mathmode := "Text", name := "Example", next := 318581, root := ~, start := 317810, stop := 318580 ), rec( content := "\n ", count := [ 7, 2, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 13, 3 ], mathmode := "Text", name := "Description", next := 318598, root := ~, start := 317014, stop := 318597 ) ], count := [ 7, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 318612, root := ~, start := 316886, stop := 318611 ), rec( content := "\n\n ", count := [ 7, 2, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, v", Name := "DigraphStabiliz\ er" ), content := 0, count := [ 7, 2, 14, 2 ], mathmode := "Text", name := "Oper", next := 318682, root := ~, start := 318633, stop := 318681 ), rec( attributes := rec( ), content := [ rec( content := "\n A p\ ermutation group.\n ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "Returns", next := 318732, root := ~, start := 318685, stop := 318731 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphStabilizer", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "C", next := 318777, root := ~, start := 318753, stop := 318776 ), rec( content := " returns\ the stabilizer of the vertex ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "A", next := 318823, root := ~, start := 318815, stop := 318822 ), rec( content := "\n un\ der of the action of the ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "Ref", next := 318881, root := ~, start := 318855, stop := 318880 ), rec( content := " on the \ set of\n vertices of ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "A", next := 318926, root := ~, start := 318912, stop := 318925 ), rec( content := ".\n\n \ ", count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphFromDigraph6String(\"&GYHPQgWTIIPW\");\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\ngap> Digr\ aphStabilizer(D, 2);\nGroup(())\ngap> D := DigraphMutableCopy(D);\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\nga\ p> DigraphStabilizer(D, 2);\nGroup(())\n", count := [ 7, 2, 14, 4 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 14, 4 ], mathmode := "Text", name := "Example", next := 319302, root := ~, start := 318933, stop := 319301 ), rec( content := "\n ", count := [ 7, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 14, 3 ], mathmode := "Text", name := "Description", next := 319319, root := ~, start := 318735, stop := 319318 ) ], count := [ 7, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 319333, root := ~, start := 318618, stop := 319332 ), rec( content := "\n\n ", count := [ 7, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Label := "for digraphs", Name := "IsIsomorphicDigraph" ), content := 0, count := [ 7, 2, 15, 2 ], mathmode := "Text", name := "Oper", next := 319434, root := ~, start := 319354, stop := 319433 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "K", next := 319457, root := ~, start := 319446, stop := 319456 ), rec( content := " or ", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "K", next := 319473, root := ~, start := 319461, stop := 319472 ), rec( content := ".", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Returns", next := 319484, root := ~, start := 319437, stop := 319483 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns ", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "K", next := 319539, root := ~, start := 319528, stop := 319538 ), rec( content := " if ther\ e exists an isomorphism from the\n digraph ", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "A", next := 319607, root := ~, start := 319592, stop := 319606 ), rec( content := " to the \ digraph ", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "A", next := 319638, root := ~, start := 319623, stop := 319637 ), rec( content := ". See " , count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Ref", next := 319706, root := ~, start := 319645, stop := 319705 ), rec( content := " for mor\ e information\n about isomorphisms of digraphs. ", count := [ 7, 2, 15, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "P", next := 319768, root := ~, start := 319764, stop := 319767 ), rec( content := "\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 15, 4 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 15, 4 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 15, 4 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 15, 4 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 15, 4 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 15, 4 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 15, 4 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 4 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Ref", next := 320076, root := ~, start := 320046, stop := 320075 ), rec( content := ", and ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "Ref", next := 320112, root := ~, start := 320082, stop := 320111 ), rec( content := ".\n\n \ ", count := [ 7, 2, 15, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := CycleDigraph(4);\n\ \ngap> digraph2 := CycleDigraph(5);\n\ \ngap> IsIsomorphicDigraph(digraph1, digraph2);\nfalse\ngap> digraph2 := Digra\ phReverse(digraph1);\n\ngap> IsIso\ morphicDigraph(digraph1, digraph2);\ntrue\ngap> digraph1 := Digraph([[3], [], \ []]);\n\ngap> digraph2 := Digraph([\ [], [], [2]]);\n\ngap> IsIsomorphic\ Digraph(digraph1, digraph2);\ntrue", count := [ 7, 2, 15, 5 ], name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 15, 5 ], mathmode := "Text", name := "Example", next := 320714, root := ~, start := 320119, stop := 320713 ), rec( content := "\n ", count := [ 7, 2, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 15, 3 ], mathmode := "Text", name := "Description", next := 320731, root := ~, start := 319487, stop := 320730 ) ], count := [ 7, 2, 15, 1 ], mathmode := "Text", name := "ManSection", next := 320745, root := ~, start := 319339, stop := 320744 ), rec( content := "\n\n ", count := [ 7, 2, 15, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2, colours1, colours2", Label := "for digraphs and homogeneous lists", Name := "IsIsomorphicDig\ raph" ), content := 0, count := [ 7, 2, 16, 2 ], mathmode := "Text", name := "Oper", next := 320892, root := ~, start := 320766, stop := 320891 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "K", next := 320915, root := ~, start := 320904, stop := 320914 ), rec( content := " or ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "K", next := 320931, root := ~, start := 320919, stop := 320930 ), rec( content := ".", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "Returns", next := 320942, root := ~, start := 320895, stop := 320941 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation tests for isomorphism of coloured digraphs. A coloured\n digra\ ph can be specified by its underlying digraph ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "A", next := 321104, root := ~, start := 321089, stop := 321103 ), rec( content := " and its\ \n colouring ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "A", next := 321142, root := ~, start := 321127, stop := 321141 ), rec( content := ". Let " , count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "C", next := 321157, root := ~, start := 321149, stop := 321156 ), rec( content := " be the \ number of vertices of\n ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "A", next := 321206, root := ~, start := 321191, stop := 321205 ), rec( content := ". The co\ louring ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "A", next := 321237, root := ~, start := 321222, stop := 321236 ), rec( content := " may hav\ e one of the following\n two forms:\n\n ", count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n a list of ", count := [ 7, 2\ , 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "n", count := [ 7, 2, 16, 5 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5 ], mathm\ ode := "Text", name := "C", next := 321334, root := ~, start := 321326, stop :\ = 321333 ), rec( content := " integers, where ", count := [ 7, 2, 16, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 16, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5 ], mathmode := "Te\ xt", name := "A", next := 321365, root := ~, start := 321351, stop := 321364 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 16, 5 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5\ ], mathmode := "Text", name := "C", next := 321375, root := ~, start := 32136\ 5, stop := 321374 ), rec( content := " is the\n colour of vertex ", coun\ t := [ 7, 2, 16, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 16, 5 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5 ] , math\ mode := "Text", name := "C", next := 321416, root := ~, start := 321408, stop \ := 321415 ), rec( content := ", using the colours ", count := [ 7, 2, 16, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 .. m]", count := [ 7, 2, 16, 5 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 5 ], mathmode := "T\ ext", name := "C", next := 321451, root := ~, start := 321436, stop := 321450 ), rec( cont\ ent := " for some\n ", count := [ 7, 2, 16, 5 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "m <= n", count := [ 7, 2, 16, 5 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 2, 16, 5 ], mathmode := "Text", name := "C", next\ := 321485, root := ~, start := 321469, stop := 321484 ), rec( content := "; o\ r\n ", count := [ 7, 2, 16, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 16, 5 ], name := "Item", next := 321503, root := ~, start := 321\ 301, stop := 321502 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n a list of non-empty disjoint lists whose union is\n ", coun\ t := [ 7, 2, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphVertices(", count := [\ 7, 2, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 16, \ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16,\ 7 ], mathmode := "Text", name := "A", next := 321616, root := ~, start := 321\ 602, stop := 321615 ), rec( content := ")", count := [ 7, 2, 16, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], mathmod\ e := "Text", name := "C", next := 321621, root := ~, start := 321583, stop := \ 321620 ), rec( content := ", such that\n ", count := [ 7, 2, 16, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 16, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], mathmode := "Te\ xt", name := "A", next := 321655, root := ~, start := 321641, stop := 321654 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 16, 7 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7\ ], mathmode := "Text", name := "C", next := 321665, root := ~, start := 32165\ 5, stop := 321664 ), rec( content := " is the list of all vertices with colour\ \n ", count := [ 7, 2, 16, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "i", count :\ = [ 7, 2, 16, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 7, 2, 16, 7 ], mathmode := "Text", name := "C", next := 321722, root := ~ , star\ t := 321714, stop := 321721 ), rec( content := ".\n ", count := [ 7, 2, 1\ 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 7 ], name := "Ite\ m", next := 321737, root := ~, start := 321510, stop := 321736 ) ], count := [ 7, 2, 16, 4 ], mathmode := "Text", name := "List", next := 321749, root := ~, start := 321288, stop := 321748 ), rec( content := "\n\n \ If ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 321773, root := ~, start := 321758, stop := 321772 ), rec( content := " and ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 321793, root := ~, start := 321778, stop := 321792 ), rec( content := " are dig\ raphs without multiple edges,\n and ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 321854, root := ~, start := 321839, stop := 321853 ), rec( content := " and ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours2", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 321874, root := ~, start := 321859, stop := 321873 ), rec( content := " are col\ ourings of ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 321908, root := ~, start := 321893, stop := 321907 ), rec( content := "\n an\ d ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "A", next := 321932, root := ~, start := 321917, stop := 321931 ), rec( content := ", respec\ tively, then this operation returns ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "K", next := 321987, root := ~, start := 321976, stop := 321986 ), rec( content := "\n if\ there exists an isomorphism between these two coloured digraphs. See\n ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs and homogeneous lists", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "Ref", next := 322152, root := ~, start := 322069, stop := 322151 ), rec( content := " for mor\ e information about isomorphisms of coloured digraphs. ", count := [ 7, 2, 16, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "P", next := 322219, root := ~, start := 322215, stop := 322218 ), rec( content := "\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "URL\ ", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 16, 10 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 16, 10 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 16, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 16, 10 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 16, 10 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 16, 10 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 16, 10 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 10 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Ref", next := 322527, root := ~, start := 322497, stop := 322526 ), rec( content := ", and ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "Ref", next := 322563, root := ~, start := 322533, stop := 322562 ), rec( content := ".\n\n \ ", count := [ 7, 2, 16, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := ChainDigraph(4);\n\ \ngap> digraph2 := ChainDigraph(3);\n\ \ngap> IsIsomorphicDigraph(digraph1, digraph2,\n> [[1, 4], [2, 3]], [[1, 2], \ [3]]);\nfalse\ngap> digraph2 := DigraphReverse(digraph1);\n\ngap> IsIsomorphicDigraph(digraph1, digraph2,\n> [1\ , 1, 1, 1], [1, 1, 1, 1]);\ntrue\ngap> IsIsomorphicDigraph(digraph1, digraph2,\ \n> [1, 2, 2, 1], [1, 2, 2, 1]);\ntrue\ngap> IsIsomorphicDigraph(digraph1, di\ graph2,\n> [1, 1, 2, 2], [1, 1, 2, 2]);\nfalse", count := [ 7, 2, 16, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 16, 11 ], mathmode := "Text", name := "Example", next := 323176, root := ~, start := 322570, stop := 323175 ), rec( content := "\n ", count := [ 7, 2, 16, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 16, 3 ], mathmode := "Text", name := "Description", next := 323193, root := ~, start := 320945, stop := 323192 ) ], count := [ 7, 2, 16, 1 ], mathmode := "Text", name := "ManSection", next := 323207, root := ~, start := 320751, stop := 323206 ), rec( content := "\n\n ", count := [ 7, 2, 16, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Label := "for digraphs", Name := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 17, 2 ], mathmode := "Text", name := "Oper", next := 323308, root := ~, start := 323228, stop := 323307 ), rec( attributes := rec( ), content := [ rec( content := " A permut\ ation, or a pair of permutations, or ", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "K", next := 323377, root := ~, start := 323366, stop := 323376 ), rec( content := ".", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "Returns", next := 323388, root := ~, start := 323311, stop := 323387 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation returns an isomorphism between the digraphs ", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "A", next := 323483, root := ~, start := 323468, stop := 323482 ), rec( content := "\n an\ d ", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "A", next := 323507, root := ~, start := 323492, stop := 323506 ), rec( content := " if one \ exists, else this operation returns ", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "K", next := 323562, root := ~, start := 323551, stop := 323561 ), rec( content := ".\n " , count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "P", next := 323572, root := ~, start := 323568, stop := 323571 ), rec( content := "\n\n \ An ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ somorphism", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "E", next := 323599, root := ~, start := 323581, stop := 323598 ), rec( content := " from a \ digraph ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 323630, root := ~, start := 323615, stop := 323629 ), rec( content := " to a di\ graph\n ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 323663, root := ~, start := 323648, stop := 323662 ), rec( content := " is a bi\ jection ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 323687, root := ~, start := 323679, stop := 323686 ), rec( content := " from th\ e vertices of\n ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 323728, root := ~, start := 323713, stop := 323727 ), rec( content := " to the \ vertices of ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 323763, root := ~, start := 323748, stop := 323762 ), rec( content := " with th\ e following\n property: for all vertices ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 323822, root := ~, start := 323814, stop := 323821 ), rec( content := " and ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 323835, root := ~, start := 323827, stop := 323834 ), rec( content := " of ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 323854, root := ~, start := 323839, stop := 323853 ), rec( content := ",\n " , count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 323873, root := ~, start := 323860, stop := 323872 ), rec( content := " is an e\ dge of ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 323903, root := ~, start := 323888, stop := 323902 ), rec( content := " if and \ only if ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i ^ p, j\n ^ p]", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "C", next := 323946, root := ~, start := 323919, stop := 323945 ), rec( content := " is an e\ dge of ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "A", next := 323976, root := ~, start := 323961, stop := 323975 ), rec( content := ". ", count := [ 7, 2, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "P", next := 323982, root := ~, start := 323978, stop := 323981 ), rec( content := "\n\n \ If there exists such an isomorphism, then this operation returns one.\n The\ form of this isomorphism is a permutation ", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "C", next := 324116, root := ~, start := 324108, stop := 324115 ), rec( content := " of the \ vertices of\n ", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "A", next := 324155, root := ~, start := 324140, stop := 324154 ), rec( content := " such th\ at ", count := [ 7, 2, 17, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "P", next := 324170, root := ~, start := 324166, stop := 324169 ), rec( content := "\n\n \ ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "O\ nDigraphs(", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph1", count := [ 7, 2, 17, 6 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := \ "Text", name := "A", next := 324205, root := ~, start := 324190, stop := 32420\ 4 ), rec( content := ", p) = digraph2", count := [ 7, 2, 17, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "C", next := 324224, root := ~, start := 324176, stop := 324223 ), rec( content := ".\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "URL", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 17, 6 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 17, 6 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 17, 6 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 17, 6 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 17, 6 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 17, 6 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 17, 6 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 6 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Ref", next := 324533, root := ~, start := 324503, stop := 324532 ), rec( content := ", and ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "Ref", next := 324569, root := ~, start := 324539, stop := 324568 ), rec( content := ".\n\n \ ", count := [ 7, 2, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := CycleDigraph(4);\n\ \ngap> digraph2 := CycleDigraph(5);\n\ \ngap> IsomorphismDigraphs(digraph1, digraph2);\nfail\ngap> digraph1 := Comple\ teBipartiteDigraph(10, 5);\n\ngap> digraph2 := CompleteBipartiteDigraph(5, 10);\n\ngap> p := I\ somorphismDigraphs(digraph1, digraph2);\n(1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10\ ,15)\ngap> OnDigraphs(digraph1, p) = digraph2;\ntrue\n", count := [ 7, 2, 17, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 17, 7 ], mathmode := "Text", name := "Example", next := 325192, root := ~, start := 324576, stop := 325191 ), rec( content := "\n ", count := [ 7, 2, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 17, 3 ], mathmode := "Text", name := "Description", next := 325209, root := ~, start := 323391, stop := 325208 ) ], count := [ 7, 2, 17, 1 ], mathmode := "Text", name := "ManSection", next := 325223, root := ~, start := 323213, stop := 325222 ), rec( content := "\n\n ", count := [ 7, 2, 17, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2, colours1, colours2", Label := "for digraphs and homogeneous lists", Name := "IsomorphismDigr\ aphs" ), content := 0, count := [ 7, 2, 18, 2 ], mathmode := "Text", name := "Oper", next := 325370, root := ~, start := 325244, stop := 325369 ), rec( attributes := rec( ), content := [ rec( content := " A permut\ ation, or ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "K", next := 325412, root := ~, start := 325401, stop := 325411 ), rec( content := ".", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "Returns", next := 325423, root := ~, start := 325373, stop := 325422 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s operation searches for an isomorphism between coloured digraphs. A\n col\ oured digraph can be specified by its underlying digraph ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "A", next := 325596, root := ~, start := 325581, stop := 325595 ), rec( content := "\n an\ d its colouring ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "A", next := 325634, root := ~, start := 325619, stop := 325633 ), rec( content := ". Let " , count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "C", next := 325649, root := ~, start := 325641, stop := 325648 ), rec( content := " be the \ number of vertices\n of ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "A", next := 325698, root := ~, start := 325683, stop := 325697 ), rec( content := ". The co\ louring ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "A", next := 325729, root := ~, start := 325714, stop := 325728 ), rec( content := " may hav\ e one of the\n following two forms:\n\n ", count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n a list of ", count := [ 7, 2\ , 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "n", count := [ 7, 2, 18, 5 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5 ], mathm\ ode := "Text", name := "C", next := 325826, root := ~, start := 325818, stop :\ = 325825 ), rec( content := " integers, where ", count := [ 7, 2, 18, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 18, 5 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5 ], mathmode := "Te\ xt", name := "A", next := 325857, root := ~, start := 325843, stop := 325856 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 18, 5 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5\ ], mathmode := "Text", name := "C", next := 325867, root := ~, start := 32585\ 7, stop := 325866 ), rec( content := " is the\n colour of vertex ", coun\ t := [ 7, 2, 18, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 18, 5 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5 ] , math\ mode := "Text", name := "C", next := 325908, root := ~, start := 325900, stop \ := 325907 ), rec( content := ", using the colours ", count := [ 7, 2, 18, 5 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 .. m]", count := [ 7, 2, 18, 5 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 5 ], mathmode := "T\ ext", name := "C", next := 325943, root := ~, start := 325928, stop := 325942 ), rec( cont\ ent := " for some\n ", count := [ 7, 2, 18, 5 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "m <= n", count := [ 7, 2, 18, 5 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 2, 18, 5 ], mathmode := "Text", name := "C", next\ := 325977, root := ~, start := 325961, stop := 325976 ), rec( content := "; o\ r\n ", count := [ 7, 2, 18, 5 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 18, 5 ], name := "Item", next := 325995, root := ~, start := 325\ 793, stop := 325994 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n a list of non-empty disjoint lists whose union is\n ", coun\ t := [ 7, 2, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphVertices(", count := [\ 7, 2, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "digraph", count := [ 7, 2, 18, \ 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18,\ 7 ], mathmode := "Text", name := "A", next := 326108, root := ~, start := 326\ 094, stop := 326107 ), rec( content := ")", count := [ 7, 2, 18, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], mathmod\ e := "Text", name := "C", next := 326113, root := ~, start := 326075, stop := \ 326112 ), rec( content := ", such that\n ", count := [ 7, 2, 18, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "colours", count := [ 7, 2, 18, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], mathmode := "Te\ xt", name := "A", next := 326147, root := ~, start := 326133, stop := 326146 ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 2, 18, 7 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7\ ], mathmode := "Text", name := "C", next := 326157, root := ~, start := 32614\ 7, stop := 326156 ), rec( content := " is the list of all vertices with colour\ \n ", count := [ 7, 2, 18, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "i", count :\ = [ 7, 2, 18, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count \ := [ 7, 2, 18, 7 ], mathmode := "Text", name := "C", next := 326214, root := ~ , star\ t := 326206, stop := 326213 ), rec( content := ".\n ", count := [ 7, 2, 1\ 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 7 ], name := "Ite\ m", next := 326229, root := ~, start := 326002, stop := 326228 ) ], count := [ 7, 2, 18, 4 ], mathmode := "Text", name := "List", next := 326241, root := ~, start := 325780, stop := 326240 ), rec( content := "\n\n \ An ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ somorphism", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "E", next := 326268, root := ~, start := 326250, stop := 326267 ), rec( content := " between\ coloured digraphs is an isomorphism between\n the underlying digraphs that\ preserves the colourings. See ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "Ref", next := 326445, root := ~, start := 326385, stop := 326444 ), rec( content := " for mor\ e information\n about isomorphisms of digraphs. More precisely, let ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 326531, root := ~, start := 326523, stop := 326530 ), rec( content := " be an\n\ isomorphism of digraphs from the digraph ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 326598, root := ~, start := 326583, stop := 326597 ), rec( content := " (with c\ olouring\n ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 326634, root := ~, start := 326619, stop := 326633 ), rec( content := ") to the\ digraph ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 326666, root := ~, start := 326651, stop := 326665 ), rec( content := " (with c\ olouring\n ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olours2", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 326702, root := ~, start := 326687, stop := 326701 ), rec( content := "), and l\ et ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 326721, root := ~, start := 326713, stop := 326720 ), rec( content := " be the \ permutation of the vertices of\n ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 326779, root := ~, start := 326764, stop := 326778 ), rec( content := " that co\ rresponds to ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 326808, root := ~, start := 326800, stop := 326807 ), rec( content := ". Then \ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 326824, root := ~, start := 326816, stop := 326823 ), rec( content := " preserv\ es the\n colourings of ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 326872, root := ~, start := 326857, stop := 326871 ), rec( content := " and ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 326892, root := ~, start := 326877, stop := 326891 ), rec( content := " – and\ hence is an\n isomorphism of coloured digraphs – if ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "colours1", count := [ 7, 2, 18, 9 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := \ "Text", name := "A", next := 326975, root := ~, start := 326960, stop := 32697\ 4 ), rec( content := "[i] =\n ", count := [ 7, 2, 18, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "colours2", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 3270\ 02, root := ~, start := 326987, stop := 327001 ), rec( content := "[i ^ p]", count := [ 7, 2, 18, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 327013, root := ~, start := 326957, stop := 327012 ), rec( content := " for all\ vertices ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "C", next := 327039, root := ~, start := 327031, stop := 327038 ), rec( content := " in ", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "A", next := 327058, root := ~, start := 327043, stop := 327057 ), rec( content := ".\n " , count := [ 7, 2, 18, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "P", next := 327068, root := ~, start := 327064, stop := 327067 ), rec( content := "\n\n \ This operation returns such an isomorphism if one exists, else it returns\n \ ", count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "K", next := 327163, root := ~, start := 327152, stop := 327162 ), rec( content := ". ", count := [ 7, 2, 18, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "P", next := 327169, root := ~, start := 327165, stop := 327168 ), rec( content := "\n\n \ By default, an isomorphism is found using the canonical labellings of the\n \ digraphs obtained from ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "bliss" ), content := [ rec( content := "http://www.tcs.tkk.fi/S\ oftware/bliss/", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "URL\ ", next := 81, root := ~, start := 17, stop := 80 ) ], count := [ 7, 2, 18, 11 ], name := "Alt", next := 87, root := ~, start := 1, stop := 86 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "bliss", count := [ 7, 2, 18, 11 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], name := "Package", next := 131, root := ~, start := 107 , stop := 130 ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Alt", next := 137, root := ~, start := 87, stop := 136 ), rec( content := " by Tomm\ i Junttila\n and Petteri Kaski. If ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "NautyTracesInterface" ), content := [ rec( content := "https://\ github.com/gap-packages/NautyTracesInterface", count := [ 7, 2, 18, 11 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], mathmode := \ "Text", name := "URL", next := 111, root := ~, start := 17, stop := 110 ) ], count := [ 7, 2, 18, 11 ], name := "Alt", next := 117, root := ~, start := 1, stop := 116 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "NautyTracesInterface", count := [ 7, 2\ , 18, 11 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], name := "Pac\ kage", next := 176, root := ~, start := 137, stop := 175 ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Alt", next := 182, root := ~, start := 117, stop := 181 ), rec( content := " is avai\ lable, then ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "nauty" ), content := [ rec( content := "http://pallini.di.uniro\ ma1.it/", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "URL", next := 74, root := ~, start := 17, stop := 73 ) ], count := [ 7, 2, 18, 11 ], name := "Alt", next := 80, root := ~, start := 1, stop := 79 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "nauty", count := [ 7, 2, 18, 11 ], name\ := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 11 ], name := "Package", next := 124, root := ~, start := 100 , stop := 123 ) ], count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Alt", next := 130, root := ~, start := 80, stop := 129 ), rec( content := " by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseBliss" ), content := 0, count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Ref", next := 327477, root := ~, start := 327447, stop := 327476 ), rec( content := ", and ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ sUseNauty" ), content := 0, count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "Ref", next := 327513, root := ~, start := 327483, stop := 327512 ), rec( content := ".\n\n \ ", count := [ 7, 2, 18, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph1 := ChainDigraph(4);\n\ \ngap> digraph2 := ChainDigraph(3);\n\ \ngap> IsomorphismDigraphs(digraph1, digraph2,\n> [[1, 4], [2, 3]], [[1, 2], \ [3]]);\nfail\ngap> digraph2 := DigraphReverse(digraph1);\n\ngap> colours1 := [1, 1, 1, 1];;\ngap> colours2 := [1\ , 1, 1, 1];;\ngap> p := IsomorphismDigraphs(digraph1, digraph2, colours1, colo\ urs2);\n(1,4)(2,3)\ngap> OnDigraphs(digraph1, p) = digraph2;\ntrue\ngap> List(\ DigraphVertices(digraph1), i -> colours1[i ^ p]) = colours2;\ntrue\ngap> colou\ rs1 := [1, 1, 2, 2];;\ngap> colours2 := [2, 2, 1, 1];;\ngap> p := IsomorphismD\ igraphs(digraph1, digraph2, colours1, colours2);\n(1,4)(2,3)\ngap> OnDigraphs(\ digraph1, p) = digraph2;\ntrue\ngap> List(DigraphVertices(digraph1), i -> colo\ urs1[i ^ p]) = colours2;\ntrue\ngap> IsomorphismDigraphs(digraph1, digraph2,\n\ > [1, 1, 2, 2], [1, 1, 2, 2]);\nfail", count := [ 7, 2, 18, 12 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 18, 12 ], mathmode := "Text", name := "Example", next := 328496, root := ~, start := 327520, stop := 328495 ), rec( content := "\n ", count := [ 7, 2, 18, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 18, 3 ], mathmode := "Text", name := "Description", next := 328513, root := ~, start := 325426, stop := 328512 ) ], count := [ 7, 2, 18, 1 ], mathmode := "Text", name := "ManSection", next := 328527, root := ~, start := 325229, stop := 328526 ), rec( content := "\n\n ", count := [ 7, 2, 18, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "RepresentativeO\ utNeighbours" ), content := 0, count := [ 7, 2, 19, 2 ], mathmode := "Text", name := "Attr", next := 328604, root := ~, start := 328548, stop := 328603 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of lists.", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "Returns", next := 328653, root := ~, start := 328607, stop := 328652 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function returns the list ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "C", next := 328715, root := ~, start := 328705, stop := 328714 ), rec( content := " of ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut-neighbours", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "E", next := 328740, root := ~, start := 328719, stop := 328739 ), rec( content := " of each\ \n representative of the orbits of the action of ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "Ref", next := 328825, root := ~, start := 328799, stop := 328824 ), rec( content := " on\n \ the vertex set of the digraph ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "A", next := 328877, root := ~, start := 328863, stop := 328876 ), rec( content := ". ", count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "P", next := 328883, root := ~, start := 328879, stop := 328882 ), rec( content := "\n\n \ More specifically, if ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ eps", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 328922, root := ~, start := 328911, stop := 328921 ), rec( content := " is the \ list of orbit representatives,\n then a vertex ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 328987, root := ~, start := 328979, stop := 328986 ), rec( content := " appears\ in ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ut[i]", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 329012, root := ~, start := 328999, stop := 329011 ), rec( content := " each ti\ me\n there exists an edge with source ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ eps[i]", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 329074, root := ~, start := 329060, stop := 329073 ), rec( content := " and ran\ ge ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "C", next := 329093, root := ~, start := 329085, stop := 329092 ), rec( content := " in\n \ ", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "A", next := 329115, root := ~, start := 329101, stop := 329114 ), rec( content := ".\n " , count := [ 7, 2, 19, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "P", next := 329125, root := ~, start := 329121, stop := 329124 ), rec( content := "\n\n \ If ", count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Group" ), content := 0, count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "Ref", next := 329160, root := ~, start := 329134, stop := 329159 ), rec( content := " is triv\ ial, then ", count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutNeig\ hbours" ), content := 0, count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "Ref", next := 329205, root := ~, start := 329178, stop := 329204 ), rec( content := "\n is\ returned.\n\n ", count := [ 7, 2, 19, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2, 3,\ 4]]);\n\ngap> DigraphGroup(D);\n\ Group(())\ngap> RepresentativeOutNeighbours(D);\n[ [ 2, 1, 3, 4, 5 ], [ 3, 5 ]\ , [ 2 ], [ 1, 2, 3, 5 ], [ 1, 2, 3, 4 ] ]\ngap> D := Digraph(IsMutableDigraph,\ [\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2, 3, 4]]);\n\ngap> DigraphGroup(D);\nGroup(())\ngap> Repr\ esentativeOutNeighbours(D);\n[ [ 2, 1, 3, 4, 5 ], [ 3, 5 ], [ 2 ], [ 1, 2, 3, \ 5 ], [ 1, 2, 3, 4 ] ]\ngap> D := DigraphFromDigraph6String(\"&GYHPQgWTIIPW\");\ \n\ngap> G := DigraphGroup(D);;\n\ gap> GeneratorsOfGroup(G);\n[ (1,2)(3,4)(5,6)(7,8), (1,3,2,4)(5,7,6,8), (1,5)(\ 2,6)(3,8)(4,7) ]\ngap> Set(RepresentativeOutNeighbours(D), Set);\n[ [ 2, 3, 5 \ ] ]", count := [ 7, 2, 19, 6 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 6 ], mathmode := "Text", name := "Example", next := 330098, root := ~, start := 329228, stop := 330097 ), rec( content := "\n ", count := [ 7, 2, 19, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 19, 3 ], mathmode := "Text", name := "Description", next := 330115, root := ~, start := 328656, stop := 330114 ) ], count := [ 7, 2, 19, 1 ], mathmode := "Text", name := "ManSection", next := 330129, root := ~, start := 328533, stop := 330128 ), rec( content := "\n\n ", count := [ 7, 2, 19, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd transformation or permutation", Name := "IsDigraphIsomorphism" ), content := 0, count := [ 7, 2, 20, 2 ], mathmode := "Text", name := "Oper", next := 330267, root := ~, start := 330150, stop := 330266 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphIsomorphism" ), content := 0, count := [ 7, 2, 20, 4 ], mathmode := "Text", name := "Oper", next := 330335, root := ~, start := 330270, stop := 330334 ), rec( attributes := rec( Arg := "digraph, x", Label := "for a digraph \ and a transformation or permutation", Name := "IsDigraphAutomorphism" ), content := 0, count := [ 7, 2, 20, 6 ], mathmode := "Text", name := "Oper", next := 330458, root := ~, start := 330338, stop := 330457 ), rec( attributes := rec( Arg := "digraph, x, col" , Name := "IsDigraphAutomo\ rphism" ), content := 0, count := [ 7, 2, 20, 8 ], mathmode := "Text", name := "Oper", next := 330519, root := ~, start := 330461, stop := 330518 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "K", next := 330542, root := ~, start := 330531, stop := 330541 ), rec( content := " or ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "K", next := 330558, root := ~, start := 330546, stop := 330557 ), rec( content := ".", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "Returns", next := 330569, root := ~, start := 330522, stop := 330568 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphIsomorphism", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "C", next := 330617, root := ~, start := 330590, stop := 330616 ), rec( content := " returns\ ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "K", next := 330637, root := ~, start := 330626, stop := 330636 ), rec( content := " if the \ permutation or\n transformation ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "A", next := 330687, root := ~, start := 330679, stop := 330686 ), rec( content := " is an i\ somorphism from the digraph ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "A", next := 330733, root := ~, start := 330723, stop := 330732 ), rec( content := " to\n \ the digraph ", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "A", next := 330763, root := ~, start := 330753, stop := 330762 ), rec( content := ".\n " , count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "P", next := 330773, root := ~, start := 330769, stop := 330772 ), rec( content := "\n\n \ ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphAutomorphism", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "C", next := 330807, root := ~, start := 330779, stop := 330806 ), rec( content := " returns\ ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "K", next := 330827, root := ~, start := 330816, stop := 330826 ), rec( content := " if the \ permutation or\n transformation ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "A", next := 330877, root := ~, start := 330869, stop := 330876 ), rec( content := " is an a\ utomorphism of the digraph ", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "A", next := 330926, root := ~, start := 330912, stop := 330925 ), rec( content := ".\n " , count := [ 7, 2, 20, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "P", next := 330936, root := ~, start := 330932, stop := 330935 ), rec( content := "\n\n \ A permutation or transformation ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "A", next := 330982, root := ~, start := 330974, stop := 330981 ), rec( content := " is an " , count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ somorphism", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "E", next := 331007, root := ~, start := 330989, stop := 331006 ), rec( content := " from a\\ n digraph ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "A", next := 331037, root := ~, start := 331027, stop := 331036 ), rec( content := " to a di\ graph ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "A", next := 331061, root := ~, start := 331051, stop := 331060 ), rec( content := " if the \ following hold:\n ", count := [ 7, 2, 20, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 20, 13 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "x", count := [ 7, 2, 20, 13 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 13 ], mathmode := "Text", name\ := "A", next := 331125, root := ~, start := 331117, stop := 331124 ), rec( cont\ ent := " is a bijection from the vertices of ", count := [ 7, 2, 20, 13 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "src", count := [ 7, 2, 20, 13 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 13 ], mathmode := "Text\ ", name := "A", next := 331172, root := ~, start := 331162, stop := 331171 ), rec( cont\ ent := " to those of\n ", count := [ 7, 2, 20, 13 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "ran", count := [ 7, 2, 20, 13 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 7, 2, 20, 13 ], mathmode := "Text", name := "A", next\ := 331203, root := ~, start := 331193, stop := 331202 ), rec( content := ";\n\ ", count := [ 7, 2, 20, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 20, 13 ], name := "Item", next := 331218, root := ~, start := 331\ 102, stop := 331217 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[u ^\ ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "x", count := [ 7\ , 2, 20, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [\ 7, 2, 20, 15 ], mathmode := "Text", name := "A", next := 331256, root := ~, s\ tart := 331248, stop := 331255 ), rec( content := ", v ^ ", count := [ 7, 2, 2\ 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "x", count := [ 7, 2, 20, 15 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 15 ], mathm\ ode := "Text", name := "A", next := 331270, root := ~, start := 331262, stop :\ = 331269 ), rec( content := "]", count := [ 7, 2, 20, 15 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Tex\ t", name := "C", next := 331275, root := ~, start := 331240, stop := 331274 ), rec( cont\ ent := " is an edge of\n ", count := [ 7, 2, 20, 15 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "ran", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Text", name := \ "A", next := 331308, root := ~, start := 331298, stop := 331307 ), rec( conten\ t := " if and only if ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content\ := "[u, v]", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "C", next\ := 331337, root := ~, start := 331324, stop := 331336 ), rec( content := " is\ an\n edge of ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "src", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "A", next :\ = 331370, root := ~, start := 331360, stop := 331369 ), rec( content := "; and\ \n ", count := [ 7, 2, 20, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 2, 20, 15 ], name := "Item", next := 331389, root := ~, start := 331\ 225, stop := 331388 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 2, 20, 17 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x", \ count := [ 7, 2, 20, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 7, 2, 20, 17 ], mathmode := "Text", name := "A", next := 331419, root :\ = ~, start := 331411, stop := 331418 ), rec( content := " fixes every ", count\ := [ 7, 2, 20, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 2, 20, 17 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 17\ ], mathmode := "Text", name := "C", next := 331440, root := ~, start := 33143\ 2, stop := 331439 ), rec( content := " which is not a vertex of ", count := [ \ 7, 2, 20, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "src", count := [ 7, 2, 20, 17 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 17\ ], mathmode := "Text", name := "A", next := 331476, root := ~, start := 33146\ 6, stop := 331475 ), rec( content := ".\n ", count := [ 7, 2, 20, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 17 ], name := "Ite\ m", next := 331491, root := ~, start := 331396, stop := 331490 ) ], count := [ 7, 2, 20, 12 ], mathmode := "Text", name := "List", next := 331503, root := ~, start := 331089, stop := 331502 ), rec( content := "\n Se\ e also ", count := [ 7, 2, 20, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 2, 20, 19 ], mathmode := "Text", name := "Ref", next := 331577, root := ~, start := 331517, stop := 331576 ), rec( content := ".", count := [ 7, 2, 20, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "P", next := 331582, root := ~, start := 331578, stop := 331581 ), rec( content := "\n \n\ If ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "A", next := 331606, root := ~, start := 331595, stop := 331605 ), rec( content := " and ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol2", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "A", next := 331622, root := ~, start := 331611, stop := 331621 ), rec( content := ", or ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "A", next := 331637, root := ~, start := 331627, stop := 331636 ), rec( content := ", are gi\ ven, then they must\n represent vertex colourings; see \n ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a homogeneous list", Oper := "AutomorphismGroup" ), content := 0, count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "Ref", next := 331783, root := ~, start := 331707, stop := 331782 ), rec( content := " \n f\ or details of the permissible values for\n these arguments. The homomorphis\ m must then also have the property:\n \n ", count := [ 7, 2, 20, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 20, 22 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "col1[i] = col2[i ^ x]", count := [ 7, 2, 20, 22 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 22 ], math\ mode := "Text", name := "C", next := 331966, root := ~, start := 331938, stop \ := 331965 ), rec( content := " for all vertices ", count := [ 7, 2, 20, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 7, 2, 20, 22 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 22 ], mathmode := "Text", name\ := "C", next := 331992, root := ~, start := 331984, stop := 331991 ), rec( cont\ ent := " of ", count := [ 7, 2, 20, 22 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "src", count\ := [ 7, 2, 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 20, 22 ], mathmode := "Text", name := "A", next := 332006, root :\ = ~, start := 331996, stop := 332005 ), rec( content := ",\n for ", coun\ t := [ 7, 2, 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "IsDigraphIsomorphism", count \ := [ 7, 2, 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 20, 22 ], mathmode := "Text", name := "C", next := 332047, root :\ = ~, start := 332020, stop := 332046 ), rec( content := ". ", count := [ 7, 2,\ 20, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 22 ] , name := "Ite\ m", next := 332056, root := ~, start := 331923, stop := 332055 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 7, 2, 20, 24 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "col[i] = col[i ^ x]", count := [ 7, 2, 20, 24 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 24 ], math\ mode := "Text", name := "C", next := 332104, root := ~, start := 332078, stop \ := 332103 ), rec( content := " for all vertices ", count := [ 7, 2, 20, 24 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 7, 2, 20, 24 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 24 ], mathmode := "Text", name\ := "C", next := 332130, root := ~, start := 332122, stop := 332129 ), rec( cont\ ent := " of ", count := [ 7, 2, 20, 24 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", c\ ount := [ 7, 2, 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 20, 24 ], mathmode := "Text", name := "A", next := 332148, root :\ = ~, start := 332134, stop := 332147 ), rec( content := ",\n for ", coun\ t := [ 7, 2, 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "IsDigraphAutomorphism", count\ := [ 7, 2, 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 2, 20, 24 ], mathmode := "Text", name := "C", next := 332190, root :\ = ~, start := 332162, stop := 332189 ), rec( content := ". ", count := [ 7, 2,\ 20, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 24 ] , name := "Ite\ m", next := 332199, root := ~, start := 332063, stop := 332198 ) ], count := [ 7, 2, 20, 21 ], mathmode := "Text", name := "List", next := 332211, root := ~, start := 331910, stop := 332210 ), rec( content := "\n\n \ For some digraphs, it can be faster to use ", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphAutomorphism", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "C", next := 332288, root := ~, start := 332260, stop := 332287 ), rec( content := "\n th\ an to test membership in the automorphism group of ", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "A", next := 332360, root := ~, start := 332346, stop := 332359 ), rec( content := ".\n\n \ ", count := [ 7, 2, 20, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> src := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> IsDigraphAutomorphism(src, (1, 2, 3));\nfalse\ngap> IsDigra\ phAutomorphism(src, (2, 3));\ntrue\ngap> IsDigraphAutomorphism(src, (2, 3), [2\ , 1, 1]);\ntrue\ngap> IsDigraphAutomorphism(src, (2, 3), [2, 2, 1]);\nfalse\ng\ ap> IsDigraphAutomorphism(src, (2, 3)(4, 5));\nfalse\ngap> IsDigraphAutomorphi\ sm(src, (1, 4));\nfalse\ngap> IsDigraphAutomorphism(src, ());\ntrue\ngap> ran \ := Digraph([[2, 1], [2], [2, 3]]);\n\ngap> IsDigraphIsomorphism(src, ran, (1, 2));\ntrue\ngap> IsDigraphIsomorph\ ism(ran, src, (1, 2));\ntrue\ngap> IsDigraphIsomorphism(ran, src, (1, 2));\ntr\ ue\ngap> IsDigraphIsomorphism(src, Digraph([[3], [1, 3], [2]]), (1, 2, 3));\nf\ alse\ngap> IsDigraphIsomorphism(src, ran, (1, 2), [1, 2, 3], [2, 1, 3]);\ntrue\ \ngap> IsDigraphIsomorphism(src, ran, (1, 2), [1, 2, 2], [2, 1, 3]);\nfalse\n" , count := [ 7, 2, 20, 27 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 20, 27 ], mathmode := "Text", name := "Example", next := 333303, root := ~, start := 332367, stop := 333302 ), rec( content := "\n ", count := [ 7, 2, 20, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 20, 9 ], mathmode := "Text", name := "Description", next := 333320, root := ~, start := 330572, stop := 333319 ) ], count := [ 7, 2, 20, 1 ], mathmode := "Text", name := "ManSection", next := 333334, root := ~, start := 330135, stop := 333333 ), rec( content := "\n\n ", count := [ 7, 2, 20, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, list", Name := "IsDigraphColour\ ing" ), content := 0, count := [ 7, 2, 21, 2 ], mathmode := "Text", name := "Oper", next := 333412, root := ~, start := 333359, stop := 333411 ), rec( attributes := rec( Arg := "digraph, t", Label := "for a transfor\ mation", Name := "IsDigraphColouring" ), content := 0, count := [ 7, 2, 21, 4 ], mathmode := "Text", name := "Oper", next := 333503, root := ~, start := 333417, stop := 333502 ), rec( attributes := rec( ), content := [ rec( content := " ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "K", next := 333529, root := ~, start := 333518, stop := 333528 ), rec( content := " or ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "K", next := 333545, root := ~, start := 333533, stop := 333544 ), rec( content := ". ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "Returns", next := 333557, root := ~, start := 333508, stop := 333556 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ he operation ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphColouring", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "C", next := 333621, root := ~, start := 333596, stop := 333620 ), rec( content := " verifie\ s whether or not \n the list ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "A", next := 333673, root := ~, start := 333662, stop := 333672 ), rec( content := " describ\ es a proper colouring of the digraph \n ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "A", next := 333739, root := ~, start := 333725, stop := 333738 ), rec( content := ". \n \ ", count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "P", next := 333752, root := ~, start := 333748, stop := 333751 ), rec( content := "\n\n \ A list ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 333778, root := ~, start := 333767, stop := 333777 ), rec( content := " describ\ es a ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper colouring", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "E", next := 333814, root := ~, start := 333791, stop := 333813 ), rec( content := " of a di\ graph\n ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 333848, root := ~, start := 333834, stop := 333847 ), rec( content := " if ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 333863, root := ~, start := 333852, stop := 333862 ), rec( content := " consist\ s of positive integers, the length of\n ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 333926, root := ~, start := 333915, stop := 333925 ), rec( content := " equals \ the number of vertices in ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 333974, root := ~, start := 333960, stop := 333973 ), rec( content := ", and fo\ r any\n vertices ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ , v", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 334014, root := ~, start := 334003, stop := 334013 ), rec( content := " of ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 334032, root := ~, start := 334018, stop := 334031 ), rec( content := " if ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 334044, root := ~, start := 334036, stop := 334043 ), rec( content := " and ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 334057, root := ~, start := 334049, stop := 334056 ), rec( content := " are adj\ acent,\n then ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "list", count := [ 7, 2, 21, 6 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := \ "Text", name := "A", next := 334097, root := ~, start := 334086, stop := 33409\ 6 ), rec( content := "[u] >< ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ist", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "A", next := 3341\ 21, root := ~, start := 334110, stop := 334120 ), rec( content := "[v]", count := [ 7, 2, 21, 6 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "C", next := 334128, root := ~, start := 334083, stop := 334127 ), rec( content := ".\n \ ", count := [ 7, 2, 21, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "P", next := 334140, root := ~, start := 334136, stop := 334139 ), rec( content := "\n\n \ A transformation ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "A", next := 334173, root := ~, start := 334165, stop := 334172 ), rec( content := " describ\ es a proper colouring of a digraph\n ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "A", next := 334236, root := ~, start := 334222, stop := 334235 ), rec( content := ", if ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ mageListOfTransformation(", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := \ "Text", name := "A", next := 334278, root := ~, start := 334270, stop := 33427\ 7 ), rec( content := ",\n DigraphNrVertices(", count := [ 7, 2, 21, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "A", next := 3343\ 18, root := ~, start := 334304, stop := 334317 ), rec( content := "))", count := [ 7, 2, 21, 7 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "C", next := 334324, root := ~, start := 334241, stop := 334323 ), rec( content := " is a pr\ oper colouring of \n ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "A", next := 334371, root := ~, start := 334357, stop := 334370 ), rec( content := ". ", count := [ 7, 2, 21, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 2, 21, 8 ], mathmode := "Text", name := "P", next := 334377, root := ~, start := 334373, stop := 334376 ), rec( content := "\n\n \ See also ", count := [ 7, 2, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsDigra\ phHomomorphism" ), content := 0, count := [ 7, 2, 21, 8 ], mathmode := "Text", name := "Ref", next := 334429, root := ~, start := 334394, stop := 334428 ), rec( content := ".\n \ \n ", count := [ 7, 2, 21, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := JohnsonDigraph(5, 3);\n\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 4, 5, 6, 7]);\ntru\ e\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, 7]);\nfalse\ngap> Is\ DigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, -1]);\nfalse\ngap> IsDigraphCo\ louring(D, [1, 2, 3]);\nfalse\ngap> IsDigraphColouring(D, IdentityTransformati\ on);\ntrue\n", count := [ 7, 2, 21, 9 ], name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 9 ], mathmode := "Text", name := "Example", next := 334865, root := ~, start := 334444, stop := 334864 ), rec( content := "\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 21, 5 ], mathmode := "Text", name := "Description", next := 334884, root := ~, start := 333562, stop := 334883 ) ], count := [ 7, 2, 21, 1 ], mathmode := "Text", name := "ManSection", next := 334900, root := ~, start := 333342, stop := 334899 ), rec( content := "\n\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 334914, root := ~, start := 290992, stop := 334913 ) , rec( content := "\n\n ", count := [ 7, 2, 21, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Homomorphisms of \ digraphs", count := [ 7, 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 334971, root := ~, start := 334927, stop := 334970 ), rec( content := "\n\n The following me\ thods exist to find homomorphisms between digraphs.\n If an argument to one\ of these methods is a digraph with multiple edges,\n then the multiplicity\ of edges will be ignored in order to perform the\n calculation; the digrap\ h will be treated as if it has no multiple edges.\n\n ", count := [ 7, 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2, hook, us\ er_param, max_results, hint, injective, image, partial_map, colors1, colors2[,\ order, aut_grp]", Name := "HomomorphismDigraphsFinder" ), content := 0, count := [ 7, 3, 1, 2 ], mathmode := "Text", name := "Func", next := 335452, root := ~, start := 335294, stop := 335451 ), rec( attributes := rec( ), content := [ rec( content := "The argum\ ent ", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ser_param", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "A", next := 335494, root := ~, start := 335477, stop := 335493 ), rec( content := ".", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "Returns", next := 335505, root := ~, start := 335455, stop := 335504 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function finds homomorphisms from the digraph ", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "A", next := 335586, root := ~, start := 335577, stop := 335585 ), rec( content := " to the \ digraph\n ", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "A", next := 335615, root := ~, start := 335606, stop := 335614 ), rec( content := " subject\ to the conditions imposed by the other arguments as\n described below.", count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "P", next := 335700, root := ~, start := 335696, stop := 335699 ), rec( content := "\n\n \ If ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 335717, root := ~, start := 335709, stop := 335716 ), rec( content := " and ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 335730, root := ~, start := 335722, stop := 335729 ), rec( content := " are hom\ omorphisms found by\n ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ omomorphismDigraphsFinder", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 335795, root := ~, start := 335762, stop := 335794 ), rec( content := ", then " , count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 335810, root := ~, start := 335802, stop := 335809 ), rec( content := " cannot \ be obtained from\n ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "C", next := 335847, root := ~, start := 335839, stop := 335846 ), rec( content := " by righ\ t multiplying by an automorphism of ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "A", next := 335900, root := ~, start := 335891, stop := 335899 ), rec( content := " in\n \ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "a\ ut_grp", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "A", next := 335922, root := ~, start := 335908, stop := 335921 ), rec( content := ".\n\n \ ", count := [ 7, 3, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "hook", count := [ 7, 3, 1, 5 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 3, 1, 5 ], mathmode := "Text", name := "A", next := 33\ 5959, root := ~, start := 335948, stop := 335958 ) ], count := [ 7, 3, 1, 5 ], name := "Mar\ k", next := 335966, root := ~, start := 335942, stop := 335965 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be a f\ unction or ", count := [ 7, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 7, 3, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 1, 6 ], mathmode := "Text", name := "K", next := 336037, root := \ ~, start := 336026, stop := 336036 ), rec( content := ".", count := [ 7, 3, 1,\ 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := 0, count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "P", next\ := 336042, root := ~, start := 336038, stop := 336041 ), rec( content := "\n\ \n If ", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "hook", coun\ t := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 1, 7 ], mathmode := "Text", name := "A", next := 336066, root := \ ~, start := 336055, stop := 336065 ), rec( content := " is a function, then it\ must have two arguments\n ", count := [ 7, 3, 1, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "user_param", count := [ 7, 3, 1, 7 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name \ := "A", next := 336139, root := ~, start := 336122, stop := 336138 ), rec( cont\ ent := " (see below) and a transformation ", count := [ 7, 3, 1, 7 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "t", count := [ 7, 3, 1, 7 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name :\ = "C", next := 336181, root := ~, start := 336173, stop := 336180 ), rec( cont\ ent := ". The\n function ", count := [ 7, 3, 1, 7 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attri\ butes := rec( ), content := [ rec( content := "hook", count := [ 7, 3, 1, 7 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ]\ , mathmode := "Text", name := "A", next := 336218, root := ~, start := 336207,\ stop := 336217 ), rec( content := "(", count := [ 7, 3, 1, 7 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [\ rec( content := "user_param", count := [ 7, 3, 1, 7 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", na\ me := "A", next := 336236, root := ~, start := 336219, stop := 336235 ), rec( \ content := ", t)", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "C", next\ := 336244, root := ~, start := 336204, stop := 336243 ), rec( content := " is\ called every time\n a new homomorphism ", count := [ 7, 3, 1, 7 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "t", count := [ 7, 3, 1, 7 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name\ := "C", next := 336301, root := ~, start := 336293, stop := 336300 ), rec( cont\ ent := " is found by\n ", count := [ 7, 3, 1, 7 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "HomomorphismDigraphsFinder", count := [ 7, 3, 1, 7 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text\ ", name := "C", next := 336355, root := ~, start := 336322, stop := 336354 ), rec( cont\ ent := ". If the function returns ", count := [ 7, 3, 1, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "true", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "K" , next\ := 336392, root := ~, start := 336381, stop := 336391 ), rec( content := ",\n\ then ", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "Homomorphis\ mDigraphsFinder", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "C", next\ := 336440, root := ~, start := 336407, stop := 336439 ), rec( content := " st\ ops and does not find any\n further homomorphisms. This feature might b\ e useful if you are\n searching for a homomorphism that satisfies some \ condition that you\n cannot specify via the other arguments to\n \ ", count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "HomomorphismDigraphsFinder", \ count := [ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 1, 7 ], mathmode := "Text", name := "C", next := 336707, root := \ ~, start := 336674, stop := 336706 ), rec( content := ".\n ", count := [ \ 7, 3, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := 0, count := [ 7, 3, 1, 8 ], mathmode := "Text", name\ := "P", next := 336721, root := ~, start := 336717, stop := 336720 ), rec( cont\ ent := "\n\n If ", count := [ 7, 3, 1, 8 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "hook", count := [ 7, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 1, 8 ], mathmode := "Text", name := "A", next := 3\ 36745, root := ~, start := 336734, stop := 336744 ), rec( content := " is ", coun\ t := [ 7, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 7, 3, 1, 8 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 8 ] , math\ mode := "Text", name := "K", next := 336760, root := ~, start := 336749, stop \ := 336759 ), rec( content := ", then a default function is used which\n \ simply adds every new homomorphism found by\n ", count := [ 7, 3, 1, 8\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "HomomorphismDigraphsFinder", count := [ 7, \ 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, \ 3, 1, 8 ], mathmode := "Text", name := "C", next := 336893, root := ~, start :\ = 336860, stop := 336892 ), rec( content := " to ", count := [ 7, 3, 1, 8 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "user_param", count := [ 7, 3, 1, 8 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 8 ], mathmode := "T\ ext", name := "A", next := 336914, root := ~, start := 336897, stop := 336913 ), rec( cont\ ent := ", which must be a\n mutable list in this case.\n ", count \ := [ 7, 3, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 6 ], name := "Ite\ m", next := 336980, root := ~, start := 335973, stop := 336979 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "user_param", count := [ 7, 3, 1, 9 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 3, 1, 9 ], mathmode := "Text", name := "A", next\ := 337011, root := ~, start := 336994, stop := 337010 ) ], count := [ 7, 3, 1, 9 ], name := "Mar\ k", next := 337018, root := ~, start := 336988, stop := 337017 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If ", count := [ 7, 3, 1, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "hook", count := [ 7, 3, 1, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode\ := "Text", name := "A", next := 337054, root := ~, start := 337043, stop := 3\ 37053 ), rec( content := " is a function, then ", count := [ 7, 3, 1, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "user_param", count := [ 7, 3, 1, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := \ "Text", name := "A", next := 337092, root := ~, start := 337075, stop := 33709\ 1 ), rec( content := " can be any ", count := [ 7, 3, 1, 10 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "GAP", count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "\ Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := "\n\ object. The object ", count := [ 7, 3, 1, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "user_param", count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "A" , next\ := 337154, root := ~, start := 337137, stop := 337153 ), rec( content := " is\ used as the first argument of\n the function ", count := [ 7, 3, 1, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "hook", count := [ 7, 3, 1, 10 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode \ := "Text", name := "A", next := 337220, root := ~, start := 337209, stop := 33\ 7219 ), rec( content := ". For example, ", count := [ 7, 3, 1, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content \ := [ rec( content := "user_param", count := [ 7, 3, 1, 10 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := "Tex\ t", name := "A", next := 337252, root := ~, start := 337235, stop := 337251 ), rec( cont\ ent := " might be a\n transformation semigroup, and ", count := [ 7, 3,\ 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "hook", count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := "Text", name := "A", next :=\ 337316, root := ~, start := 337305, stop := 337315 ), rec( content := "(", co\ unt := [ 7, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec\ ( attributes := rec( ), content := [ rec( content := "user_param", count := [\ 7, 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := \ [ 7, 3, 1, 10 ], mathmode := "Text", name := "A", next := 337334, root := ~, s\ tart := 337317, stop := 337333 ), rec( content := ", t)", count := [ 7, 3, 1, \ 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1,\ 10 ], mathmode := "Text", name := "C", next := 337342, root := ~, start := 33\ 7302, stop := 337341 ), rec( content := "\n might set ", count := [ 7, \ 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "user_param", count := [ 7, 3, 1, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 1\ 0 ], mathmode := "Text", name := "A", next := 337378, root := ~, start := 3373\ 61, stop := 337377 ), rec( content := " to be the closure of ", count := [ 7, \ 3, 1, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "user_param", count := [ 7, 3, 1, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 1\ 0 ], mathmode := "Text", name := "A", next := 337417, root := ~, start := 3374\ 00, stop := 337416 ), rec( content := " and\n ", count := [ 7, 3, 1, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "t", count := [ 7, 3, 1, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], mathmode := \ "Text", name := "C", next := 337438, root := ~, start := 337430, stop := 33743\ 7 ), rec( content := ". ", count := [ 7, 3, 1, 10 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ \ 7, 3, 1, 11 ], mathmode := "Text", name := "P", next := 337444, root := ~, star\ t := 337440, stop := 337443 ), rec( content := "\n\n If the value of ", coun\ t := [ 7, 3, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "hook", count := [ 7, 3, 1, 11\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 1\ 1 ], mathmode := "Text", name := "A", next := 337481, root := ~, start := 3374\ 70, stop := 337480 ), rec( content := " is ", count := [ 7, 3, 1, 11 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "fail", count := [ 7, 3, 1, 11 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 11 ], mathmode := "Text", name\ := "K", next := 337496, root := ~, start := 337485, stop := 337495 ), rec( cont\ ent := ", then the value of\n ", count := [ 7, 3, 1, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "user_param", count := [ 7, 3, 1, 11 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 11 ], mathmode := "Text", name\ := "A", next := 337541, root := ~, start := 337524, stop := 337540 ), rec( cont\ ent := " must be a mutable list.\n ", count := [ 7, 3, 1, 11 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 10 ], name := "Ite\ m", next := 337579, root := ~, start := 337025, stop := 337578 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "max_results", count := [ 7, 3, 1, 12 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 12 ], mathmode := "Text", name := "A", next\ := 337611, root := ~, start := 337593, stop := 337610 ) ], count := [ 7, 3, 1, 12 ], name := "Mar\ k", next := 337618, root := ~, start := 337587, stop := 337617 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be a p\ ositive integer or ", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "infinity", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "K", next :\ = 337701, root := ~, start := 337686, stop := 337700 ), rec( content := ".\n \ ", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "PCDATA", root \ := ~ ), rec( attributes := rec( ), content := [ rec( content := "Homomorphism\ DigraphsFinder", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "C", next\ := 337744, root := ~, start := 337711, stop := 337743 ), rec( content := " wi\ ll return after it has found\n ", count := [ 7, 3, 1, 13 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "max_results", count := [ 7, 3, 1, 13 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 13 ], mathmode := "Text\ ", name := "A", next := 337802, root := ~, start := 337784, stop := 337801 ), rec( cont\ ent := " homomorphisms or the search is complete, whichever\n happens f\ irst.\n ", count := [ 7, 3, 1, 13 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 3, 1, 13 ], name := "Item", next := 337890, root := ~, start := 337\ 625, stop := 337889 ), rec( attributes := rec( ), content := [ rec( attribute\ s := rec( ), content := [ rec( content := "hint", count := [ 7, 3, 1, 14 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 14 ], math\ mode := "Text", name := "A", next := 337915, root := ~, start := 337904, stop \ := 337914 ) ], count := [ 7, 3, 1, 14 ], name := "Mark", next := 337922, root := ~, start := 337\ 898, stop := 337921 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n This argument should be a positive integer or ", count := [ 7, 3,\ 1, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "fail", count := [ 7, 3, 1, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 15 ], math\ mode := "Text", name := "K", next := 338001, root := ~, start := 337990, stop \ := 338000 ), rec( content := ". ", count := [ 7, 3, 1, 15 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, coun\ t := [ 7, 3, 1, 16 ], mathmode := "Text", name := "P", next := 338007, root :=\ ~, start := 338003, stop := 338006 ), rec( content := "\n\n If ", coun\ t := [ 7, 3, 1, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "hint", count := [ 7, 3, 1, 16\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 1\ 6 ], mathmode := "Text", name := "A", next := 338031, root := ~, start := 3380\ 20, stop := 338030 ), rec( content := " is a positive integer, then only homor\ phisms of rank\n ", count := [ 7, 3, 1, 16 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "hint", count := [ 7, 3, 1, 16 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 3, 1, 16 ], mathmode := "Text", name := "A", next \ := 338104, root := ~, start := 338093, stop := 338103 ), rec( content := " are\ found.", count := [ 7, 3, 1, 16 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 1, 17 ], math\ mode := "Text", name := "P", next := 338119, root := ~, start := 338115, stop \ := 338118 ), rec( content := "\n\n If ", count := [ 7, 3, 1, 17 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "hint", count := [ 7, 3, 1, 17 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 17 ], mathmode := "Text" , name\ := "A", next := 338143, root := ~, start := 338132, stop := 338142 ), rec( cont\ ent := " is ", count := [ 7, 3, 1, 17 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 7, 3, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 1, 17 ], mathmode := "Text", name := "K", next := 338158, root :=\ ~, start := 338147, stop := 338157 ), rec( content := ", then no restriction \ is put on the rank\n of homomorphisms found.\n ", count := [ 7, 3,\ 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 15 ], name := "Ite\ m", next := 338244, root := ~, start := 337929, stop := 338243 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "injective", count := [ 7, 3, 1, 18 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 7, 3, 1, 18 ], mathmode := "Text", name := "A", next\ := 338274, root := ~, start := 338258, stop := 338273 ) ], count := [ 7, 3, 1, 18 ], name := "Mar\ k", next := 338281, root := ~, start := 338252, stop := 338280 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be ", coun\ t := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "0", count := [ 7, 3, 1, 19 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 19 ] , math\ mode := "Text", name := "C", next := 338335, root := ~, start := 338327, stop \ := 338334 ), rec( content := ", ", count := [ 7, 3, 1, 19 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "1", count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "C", next\ := 338345, root := ~, start := 338337, stop := 338344 ), rec( content := ", o\ r ", count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "2", count := [ 7\ , 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 7, 3, 1, 19 ], mathmode := "Text", name := "C", next := 338358, root := ~, star\ t := 338350, stop := 338357 ), rec( content := ". If it is\n ", count \ := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "2", count := [ 7, 3, 1, 19 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 19 ] , math\ mode := "Text", name := "C", next := 338386, root := ~, start := 338378, stop \ := 338385 ), rec( content := ", then only embeddings are found, if it is ", coun\ t := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "1", count := [ 7, 3, 1, 19 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 19 ] , math\ mode := "Text", name := "C", next := 338437, root := ~, start := 338429, stop \ := 338436 ), rec( content := ", then only\n injective homomorphisms are\ found, and if it is ", count := [ 7, 3, 1, 19 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "0", count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ) ], count := [ 7, 3, 1, 19 ], mathmode := "Text", name := "C", next := 338\ 513, root := ~, start := 338505, stop := 338512 ), rec( content := " there are\ no\n restrictions imposed by this argument.\n ", count := [ 7, \ 3, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := 0, count := [ 7, 3, 1, 20 ], mathmode := "Text", name \ := "P", next := 338586, root := ~, start := 338582, stop := 338585 ), rec( cont\ ent := "\n\n For backwards compatibility, ", count := [ 7, 3, 1, 20 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "injective", count := [ 7, 3, 1, 20 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 20 ], mathmode := "\ Text", name := "A", next := 338641, root := ~, start := 338625, stop := 338640\ ), rec( content := " can also be ", count := [ 7, 3, 1, 20 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "false", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 20 ], mathmode := "Text", name :=\ "K", next := 338666, root := ~, start := 338654, stop := 338665 ), rec( conte\ nt := "\n or ", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "true", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "K", next := 3\ 38689, root := ~, start := 338678, stop := 338688 ), rec( content := " which c\ orrespond to the values ", count := [ 7, 3, 1, 20 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "0", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 1, 20 ], mathmode := "Text", name := "C", next := \ 338729, root := ~, start := 338721, stop := 338728 ), rec( content := " and ", coun\ t := [ 7, 3, 1, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "1", count := [ 7, 3, 1, 20 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 20 ] , math\ mode := "Text", name := "C", next := 338742, root := ~, start := 338734, stop \ := 338741 ), rec( content := "\n described in the previous paragraph, r\ espectively. \n ", count := [ 7, 3, 1, 20 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 19 ], name := "Item", next := 3388\ 16, root := ~, start := 338288, stop := 338815 ), rec( attributes := rec( ), content := [ rec( attr\ ibutes := rec( ), content := [ rec( content := "image", count := [ 7, 3, 1, 2\ 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, \ 21 ], mathmode := "Text", name := "A", next := 338842, root := ~, start := 338\ 830, stop := 338841 ) ], count := [ 7, 3, 1, 21 ], name := "Mark", next := 3388\ 49, root := ~, start := 338824, stop := 338848 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n This argument should be a subset of the vertices of the grap\ h ", count := [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "D2", count := [ \ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 7, 3, 1, 22 ], mathmode := "Text", name := "A", next := 338942, root := ~, star\ t := 338933, stop := 338941 ), rec( content := ".\n ", count := [ 7, 3,\ 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "HomomorphismDigraphsFinder", count :=\ [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 7, 3, 1, 22 ], mathmode := "Text", name := "C", next := 338985, root := ~, star\ t := 338952, stop := 338984 ), rec( content := " only finds homomorphisms from\ \n ", count := [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "D1", count \ := [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 7, 3, 1, 22 ], mathmode := "Text", name := "A", next := 339033, root := \ ~, start := 339024, stop := 339032 ), rec( content := " to the subgraph of ", coun\ t := [ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "D2", count := [ 7, 3, 1, 22 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 22 ] , math\ mode := "Text", name := "A", next := 339062, root := ~, start := 339053, stop \ := 339061 ), rec( content := " induced by the vertices\n ", count := [ \ 7, 3, 1, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "image", count := [ 7, 3, 1, 22 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 22 ] , math\ mode := "Text", name := "A", next := 339107, root := ~, start := 339095, stop \ := 339106 ), rec( content := ".\n ", count := [ 7, 3, 1, 22 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 22 ], name := "Ite\ m", next := 339122, root := ~, start := 338856, stop := 339121 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "partial_map", count := [ 7, 3, 1, 23 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 1, 23 ], mathmode := "Text", name := "A", next\ := 339154, root := ~, start := 339136, stop := 339153 ) ], count := [ 7, 3, 1, 23 ], name := "Mar\ k", next := 339161, root := ~, start := 339130, stop := 339160 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be a p\ artial map from ", count := [ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D1\ ", count := [ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 7, 3, 1, 24 ], mathmode := "Text", name := "A", next := 339235, root\ := ~, start := 339226, stop := 339234 ), rec( content := " to ", count := [ 7\ , 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "D2", count := [ 7, 3, 1, 24 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], math\ mode := "Text", name := "A", next := 339248, root := ~, start := 339239, stop \ := 339247 ), rec( content := ",\n that is, a (not necessarily dense) li\ st of vertices of the digraph\n ", count := [ 7, 3, 1, 24 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "D2", count := [ 7, 3, 1, 24 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], mathmode := "Text", name \ := "A", next := 339342, root := ~, start := 339333, stop := 339341 ), rec( cont\ ent := " of length no greater than the number vertices in the digraph\n \ ", count := [ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "D1", count := [ 7, 3, 1, 24 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ] , math\ mode := "Text", name := "A", next := 339421, root := ~, start := 339412, stop \ := 339420 ), rec( content := ". ", count := [ 7, 3, 1, 24 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "HomomorphismDigraphsFinder", count := [ 7, 3, 1, 24 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], mathmode\ := "Text", name := "C", next := 339456, root := ~, start := 339423, stop := 3\ 39455 ), rec( content := " only finds homomorphisms\n extending ", coun\ t := [ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "partial_map", count := [ 7, 3\ , 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, \ 3, 1, 24 ], mathmode := "Text", name := "A", next := 339518, root := ~, start \ := 339500, stop := 339517 ), rec( content := " (if any).\n ", count := [ \ 7, 3, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 24 ], name := "Ite\ m", next := 339542, root := ~, start := 339168, stop := 339541 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "colors1", count := [ 7, 3, 1, 25 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 7, 3, 1, 25 ], mathmode := "Text", name := "A", next\ := 339576, root := ~, start := 339562, stop := 339575 ) ], count := [ 7, 3, 1, 25 ], name := "Mar\ k", next := 339583, root := ~, start := 339556, stop := 339582 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This should be a list repre\ senting possible colours of vertices in the\n digraph ", count := [ 7, \ 3, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "D1", count := [ 7, 3, 1, 26 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 26 ], math\ mode := "Text", name := "A", next := 339701, root := ~, start := 339692, stop \ := 339700 ), rec( content := "; see \n ", count := [ 7, 3, 1, 26 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label :\ = "for a digraph and a homogeneous list", Oper := "AutomorphismGroup" ), conte\ nt := 0, count := [ 7, 3, 1, 26 ], mathmode := "Text", name := "Ref", next := \ 339803, root := ~, start := 339716, stop := 339802 ), rec( content := " \n \ for details of the permissible values for this argument.\n ", count :\ = [ 7, 3, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 26 ], name := "Ite\ m", next := 339883, root := ~, start := 339590, stop := 339882 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "colors2", count := [ 7, 3, 1, 27 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 7, 3, 1, 27 ], mathmode := "Text", name := "A", next\ := 339917, root := ~, start := 339903, stop := 339916 ) ], count := [ 7, 3, 1, 27 ], name := "Mar\ k", next := 339924, root := ~, start := 339897, stop := 339923 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This should be a list repre\ senting possible colours of vertices in the\n digraph ", count := [ 7, \ 3, 1, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "D2", count := [ 7, 3, 1, 28 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 28 ], math\ mode := "Text", name := "A", next := 340042, root := ~, start := 340033, stop \ := 340041 ), rec( content := "; see \n ", count := [ 7, 3, 1, 28 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label :\ = "for a digraph and a homogeneous list", Oper := "AutomorphismGroup" ), conte\ nt := 0, count := [ 7, 3, 1, 28 ], mathmode := "Text", name := "Ref", next := \ 340144, root := ~, start := 340057, stop := 340143 ), rec( content := " \n \ for details of the permissible values for this argument.\n ", count :\ = [ 7, 3, 1, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 28 ], name := "Ite\ m", next := 340224, root := ~, start := 339931, stop := 340223 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "order", count := [ 7, 3, 1, 29 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 7, 3, 1, 29 ], mathmode := "Text", name := "A", next :\ = 340249, root := ~, start := 340237, stop := 340248 ) ], count := [ 7, 3, 1, 29 ], name := "Mar\ k", next := 340256, root := ~, start := 340231, stop := 340255 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n The optional argument ", coun\ t := [ 7, 3, 1, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "order", count := [ 7, 3, 1, 3\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, \ 30 ], mathmode := "Text", name := "A", next := 340312, root := ~, start := 340\ 300, stop := 340311 ), rec( content := " specifies the order the\n vert\ ices in ", count := [ 7, 3, 1, 30 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "D1", count \ := [ 7, 3, 1, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 7, 3, 1, 30 ], mathmode := "Text", name := "A", next := 340366, root := \ ~, start := 340357, stop := 340365 ), rec( content := " appear in the search f\ or homomorphisms. \n The value of this parameter can have a large impac\ t\n on the runtime of the function. It seems in many cases to be a good\ \n idea for this to be the ", count := [ 7, 3, 1, 30 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DigraphWe\ lshPowellOrder" ), content := 0, count := [ 7, 3, 1, 30 ], mathmode := "Text", name\ := "Ref", next := 340613, root := ~, start := 340576, stop := 340612 ), rec( cont\ ent := ", i.e.\n vertices ordered from highest to lowest degree.\n \ ", count := [ 7, 3, 1, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 30 ], name := "Ite\ m", next := 340689, root := ~, start := 340263, stop := 340688 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n The optional argument ", coun\ t := [ 7, 3, 1, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "aut_grp", count := [ 7, 3, 1,\ 32 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1\ , 32 ], mathmode := "Text", name := "A", next := 340747, root := ~, start := 3\ 40733, stop := 340746 ), rec( content := " should be a subgroup of the\n \ automorphism group of ", count := [ 7, 3, 1, 32 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "D2", count := [ 7, 3, 1, 32 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 7, 3, 1, 32 ], mathmode := "Text", name := "A", next :=\ 340815, root := ~, start := 340806, stop := 340814 ), rec( content := ". This\ function returns unique\n representatives of the homomorphisms found u\ p to right multiplication\n by ", count := [ 7, 3, 1, 32 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "aut_grp", count := [ 7, 3, 1, 32 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 32 ], mathmode := "Text", name\ := "A", next := 340949, root := ~, start := 340935, stop := 340948 ), rec( cont\ ent := ". If this argument is not specific, it defaults to the\n full a\ utomorphism group of ", count := [ 7, 3, 1, 32 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "D2", count := [ 7, 3, 1, 32 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 7, 3, 1, 32 ], mathmode := "Text", name := "A", next := 34\ 1048, root := ~, start := 341039, stop := 341047 ), rec( content := ", which m\ ay be costly to calculate.\n ", count := [ 7, 3, 1, 32 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 32 ], name := "Item" , next := 3410\ 97, root := ~, start := 340696, stop := 341096 ) ], count := [ 7, 3, 1, 5 ], mathmode := "Text", name := "List", next := 341109, root := ~, start := 335929, stop := 341108 ), rec( content := "\n\n \ ", count := [ 7, 3, 1, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := ChainDigraph(10);\n\ngap>\ D := DigraphSymmetricClosure(D);\n\ngap> HomomorphismDigraphsFinder(D, D, fail, [], infinity, 2, 0,\ \n> [3, 4], [], fail, fail);\n[ Transformation( [ 3, 4, 3, 4, 3, 4, 3, 4, 3, 4\ ] ), \n Transformation( [ 4, 3, 4, 3, 4, 3, 4, 3, 4, 3 ] ) ]\ngap> D2 := Com\ pleteDigraph(6);;\ngap> HomomorphismDigraphsFinder(D, D2, fail, [], 1, fail, 0\ ,\n> [1 .. 6], [1, 2, 1], fail, fail);\n[ Transformation( [ 1, 2, 1, 3, 4, 5, \ 6, 1, 2, 1 ] ) ]\ngap> func := function(user_param, t)\n> Add(user_param, t * \ user_param[1]);\n> end;;\ngap> HomomorphismDigraphsFinder(D, D2, func, [Transf\ ormation([2, 2])],\n> 3, fail, 0, [1 .. 6], [1, 2, 1], fail, fail);\n[ Transfo\ rmation( [ 2, 2 ] ), \n Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 2 ] ), \ \n Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 3 ] ), \n Transformation( [ \ 2, 2, 2, 3, 4, 5, 6, 2, 2, 4 ] ) ]\ngap> HomomorphismDigraphsFinder(NullDigrap\ h(2), NullDigraph(3), fail,\n> [], infinity, fail, 1, [1, 2, 3], fail, fail, f\ ail, fail,\n> Group(()));\n[ IdentityTransformation, Transformation( [ 1, 3, 3\ ] ), \n Transformation( [ 2, 1 ] ), Transformation( [ 2, 3, 3 ] ), \n Trans\ formation( [ 3, 1, 3 ] ), Transformation( [ 3, 2, 3 ] ) ]\ngap> HomomorphismDi\ graphsFinder(NullDigraph(2), NullDigraph(3), fail,\n> [], infinity, fail, 1, [\ 1, 2, 3], fail, fail, fail, fail,\n> Group((1, 2)));\n[ IdentityTransformation\ , Transformation( [ 1, 3, 3 ] ), \n Transformation( [ 3, 1, 3 ] ) ]", count := [ 7, 3, 1, 35 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 1, 35 ], mathmode := "Text", name := "Example", next := 342664, root := ~, start := 341115, stop := 342663 ), rec( content := "\n ", count := [ 7, 3, 1, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 1, 3 ], mathmode := "Text", name := "Description", next := 342681, root := ~, start := 335508, stop := 342680 ) ], count := [ 7, 3, 1, 1 ], mathmode := "Text", name := "ManSection", next := 342695, root := ~, start := 335279, stop := 342694 ), rec( content := "\n\n ", count := [ 7, 3, 1, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "DigraphHomomorphism" ), content := 0, count := [ 7, 3, 2, 2 ], mathmode := "Text", name := "Oper", next := 342775, root := ~, start := 342716, stop := 342774 ), rec( attributes := rec( ), content := [ rec( content := " A transf\ ormation, or ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "K", next := 342820, root := ~, start := 342809, stop := 342819 ), rec( content := ".", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "Returns", next := 342831, root := ~, start := 342778, stop := 342830 ), rec( attributes := rec( ), content := [ rec( content := "\n A h\ omomorphism from ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "A", next := 342887, root := ~, start := 342872, stop := 342886 ), rec( content := " to ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "A", next := 342906, root := ~, start := 342891, stop := 342905 ), rec( content := " is a ma\ pping from\n the vertex set of ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "A", next := 342962, root := ~, start := 342947, stop := 342961 ), rec( content := " to a su\ bset of the vertices of\n ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "A", next := 343013, root := ~, start := 342998, stop := 343012 ), rec( content := ", such t\ hat every pair of vertices ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i,j]", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "C", next := 343060, root := ~, start := 343048, stop := 343059 ), rec( content := " which h\ as\n an edge ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ->j", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "C", next := 343094, root := ~, start := 343083, stop := 343093 ), rec( content := " is mapp\ ed to a pair of vertices ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ a,b]", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "C", next := 343139, root := ~, start := 343127, stop := 343138 ), rec( content := " which\n\ has an edge ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ ->b", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "C", next := 343173, root := ~, start := 343162, stop := 343172 ), rec( content := ". Note \ that non-adjacent vertices can still be\n mapped to adjacent vertices. ", count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "P", next := 343258, root := ~, start := 343254, stop := 343257 ), rec( content := "\n\n \ ", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphHomomorphism", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "C", next := 343290, root := ~, start := 343264, stop := 343289 ), rec( content := " returns\ a single homomorphism between\n ", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "A", next := 343348, root := ~, start := 343333, stop := 343347 ), rec( content := " and ", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "A", next := 343368, root := ~, start := 343353, stop := 343367 ), rec( content := " if it e\ xists, otherwise it returns\n ", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "K", next := 343419, root := ~, start := 343408, stop := 343418 ), rec( content := ".\n", count := [ 7, 3, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2], [3, 1], [], \ [4]]);\n\ngap> DigraphHomomorphism\ (gr1, gr1);\nIdentityTransformation\ngap> map := DigraphHomomorphism(gr1, gr2)\ ;\nTransformation( [ 3, 1, 5, 4, 5 ] )\ngap> IsDigraphHomomorphism(gr1, gr2, m\ ap);\ntrue\n", count := [ 7, 3, 2, 5 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 5 ], mathmode := "Text", name := "Example", next := 343767, root := ~, start := 343421, stop := 343766 ), rec( content := "\n ", count := [ 7, 3, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 2, 3 ], mathmode := "Text", name := "Description", next := 343784, root := ~, start := 342834, stop := 343783 ) ], count := [ 7, 3, 2, 1 ], mathmode := "Text", name := "ManSection", next := 343798, root := ~, start := 342701, stop := 343797 ), rec( content := "\n\n ", count := [ 7, 3, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "HomomorphismsDigraphs" ), content := 0, count := [ 7, 3, 3, 2 ], mathmode := "Text", name := "Oper", next := 343880, root := ~, start := 343819, stop := 343879 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "HomomorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 3, 4 ], mathmode := "Text", name := "Oper", next := 343959, root := ~, start := 343883, stop := 343958 ), rec( attributes := rec( ), content := [ rec( content := " A list o\ f transformations.", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "Returns", next := 344008, root := ~, start := 343962, stop := 344007 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ omomorphismsDigraphsRepresentatives", count := [ 7, 3, 3, 5 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 344072, root := ~, start := 344029, stop := 344071 ), rec( content := " finds e\ very\n ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Homomorphism" ), content := 0, count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "Ref", next := 344122, root := ~, start := 344089, stop := 344121 ), rec( content := " between\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 344146, root := ~, start := 344131, stop := 344145 ), rec( content := " and\n \ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 344170, root := ~, start := 344155, stop := 344169 ), rec( content := ", up to \ right multiplication by an element of the\n ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 7, 3, 3, 5 ] , mathmode := "Text", name := "Ref", next := 344277, root := ~, start := 344224, stop := 344276 ), rec( content := " of ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 344296, root := ~, start := 344281, stop := 344295 ), rec( content := ".\n I\ n other words, every homomorphism ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 344345, root := ~, start := 344337, stop := 344344 ), rec( content := " between\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 344369, root := ~, start := 344354, stop := 344368 ), rec( content := " and\n \ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 344393, root := ~, start := 344378, stop := 344392 ), rec( content := " can be \ written as the composition ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ = g * x", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 344444, root := ~, start := 344428, stop := 344443 ), rec( content := ",\n w\ here ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "g\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 344464, root := ~, start := 344456, stop := 344463 ), rec( content := " is one \ of the ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ omomorphismsDigraphsRepresentatives", count := [ 7, 3, 3, 5 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 344522, root := ~, start := 344479, stop := 344521 ), rec( content := "\n an\ d ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "C", next := 344539, root := ~, start := 344531, stop := 344538 ), rec( content := " is an a\ utomorphism of ", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "A", next := 344577, root := ~, start := 344562, stop := 344576 ), rec( content := ".\n " , count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "P", next := 344587, root := ~, start := 344583, stop := 344586 ), rec( content := "\n\n \ ", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "H\ omomorphismsDigraphs", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "C", next := 344621, root := ~, start := 344593, stop := 344620 ), rec( content := " returns\ all homomorphisms between\n ", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "A", next := 344675, root := ~, start := 344660, stop := 344674 ), rec( content := " and ", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "A", next := 344695, root := ~, start := 344680, stop := 344694 ), rec( content := ".\n " , count := [ 7, 3, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2], [3, 1], [], \ [4]]);\n\ngap> HomomorphismsDigrap\ hs(gr1, gr2);\n[ Transformation( [ 1, 3, 1 ] ), Transformation( [ 1, 3, 3 ] ),\ \n Transformation( [ 1, 5, 4, 4, 5 ] ), Transformation( [ 2, 2, 2 ] ), \n T\ ransformation( [ 3, 1, 3 ] ), Transformation( [ 3, 1, 5, 4, 5 ] ), \n Transfo\ rmation( [ 3, 3, 1 ] ), Transformation( [ 3, 3, 3 ] ) ]\ngap> HomomorphismsDig\ raphsRepresentatives(gr1, CompleteDigraph(3));\n[ Transformation( [ 2, 1 ] ), \ Transformation( [ 2, 1, 2 ] ) ]\n", count := [ 7, 3, 3, 7 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 7 ], mathmode := "Text", name := "Example", next := 345301, root := ~, start := 344701, stop := 345300 ), rec( content := "\n ", count := [ 7, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 3, 5 ], mathmode := "Text", name := "Description", next := 345318, root := ~, start := 344011, stop := 345317 ) ], count := [ 7, 3, 3, 1 ], mathmode := "Text", name := "ManSection", next := 345332, root := ~, start := 343804, stop := 345331 ), rec( content := "\n\n ", count := [ 7, 3, 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "DigraphMonomorphism" ), content := 0, count := [ 7, 3, 4, 2 ], mathmode := "Text", name := "Oper", next := 345412, root := ~, start := 345353, stop := 345411 ), rec( attributes := rec( ), content := [ rec( content := "A transfo\ rmation, or ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "K", next := 345456, root := ~, start := 345445, stop := 345455 ), rec( content := ".", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Returns", next := 345467, root := ~, start := 345415, stop := 345466 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphMonomorphism", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "C", next := 345514, root := ~, start := 345488, stop := 345513 ), rec( content := " returns\ a single ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ njective", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "E", next := 345548, root := ~, start := 345532, stop := 345547 ), rec( content := " ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Homomorphism" ), content := 0, count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Ref", next := 345588, root := ~, start := 345549, stop := 345587 ), rec( content := " between\ ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "A", next := 345612, root := ~, start := 345597, stop := 345611 ), rec( content := " and ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "A", next := 345632, root := ~, start := 345617, stop := 345631 ), rec( content := "\n if\ one exists, otherwise it returns ", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "K", next := 345684, root := ~, start := 345673, stop := 345683 ), rec( content := ".\n " , count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2], [3, 1], [], \ [4]]);\n\ngap> DigraphMonomorphism\ (gr1, gr1);\nIdentityTransformation\ngap> DigraphMonomorphism(gr1, gr2);\nTran\ sformation( [ 3, 1, 5, 4, 5 ] )\n", count := [ 7, 3, 4, 4 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 4 ], mathmode := "Text", name := "Example", next := 345981, root := ~, start := 345690, stop := 345980 ), rec( content := "\n ", count := [ 7, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 4, 3 ], mathmode := "Text", name := "Description", next := 345998, root := ~, start := 345470, stop := 345997 ) ], count := [ 7, 3, 4, 1 ], mathmode := "Text", name := "ManSection", next := 346012, root := ~, start := 345338, stop := 346011 ), rec( content := "\n\n ", count := [ 7, 3, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "MonomorphismsDigraphs" ), content := 0, count := [ 7, 3, 5, 2 ], mathmode := "Text", name := "Oper", next := 346094, root := ~, start := 346033, stop := 346093 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "MonomorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 5, 4 ], mathmode := "Text", name := "Oper", next := 346173, root := ~, start := 346097, stop := 346172 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Returns", next := 346221, root := ~, start := 346176, stop := 346220 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations behave the same as ", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphs" ), content := 0, count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Ref", next := 346313, root := ~, start := 346278, stop := 346312 ), rec( content := " and\n \ ", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Ref", next := 346372, root := ~, start := 346322, stop := 346371 ), rec( content := ", except\ they only return\n ", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ njective", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "E", next := 346418, root := ~, start := 346402, stop := 346417 ), rec( content := " homomor\ phisms.\n ", count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph([[3, 5], [2], [3, 1], [], \ [4]]);\n\ngap> MonomorphismsDigrap\ hs(gr1, gr2);\n[ Transformation( [ 1, 5, 4, 4, 5 ] ), \n Transformation( [ 3,\ 1, 5, 4, 5 ] ) ]\ngap> MonomorphismsDigraphsRepresentatives(gr1, CompleteDigr\ aph(3));\n[ Transformation( [ 2, 1 ] ) ]\n", count := [ 7, 3, 5, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 5, 6 ], mathmode := "Text", name := "Example", next := 346815, root := ~, start := 346438, stop := 346814 ), rec( content := "\n ", count := [ 7, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 5, 5 ], mathmode := "Text", name := "Description", next := 346832, root := ~, start := 346224, stop := 346831 ) ], count := [ 7, 3, 5, 1 ], mathmode := "Text", name := "ManSection", next := 346846, root := ~, start := 346018, stop := 346845 ), rec( content := "\n\n ", count := [ 7, 3, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "DigraphEpimorphism" ), content := 0, count := [ 7, 3, 6, 2 ], mathmode := "Text", name := "Oper", next := 346925, root := ~, start := 346867, stop := 346924 ), rec( attributes := rec( ), content := [ rec( content := "A transfo\ rmation, or ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "K", next := 346969, root := ~, start := 346958, stop := 346968 ), rec( content := ".", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Returns", next := 346980, root := ~, start := 346928, stop := 346979 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEpimorphism", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "C", next := 347026, root := ~, start := 347001, stop := 347025 ), rec( content := " returns\ a single ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ urjective", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "E", next := 347061, root := ~, start := 347044, stop := 347060 ), rec( content := " ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Homomorphism" ), content := 0, count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Ref", next := 347101, root := ~, start := 347062, stop := 347100 ), rec( content := " between\ ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "A", next := 347125, root := ~, start := 347110, stop := 347124 ), rec( content := " and ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "A", next := 347145, root := ~, start := 347130, stop := 347144 ), rec( content := "\n if\ one exists, otherwise it returns ", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "K", next := 347197, root := ~, start := 347186, stop := 347196 ), rec( content := ".\n " , count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := DigraphReverse(ChainDigraph(4));\n\ngap> gr2 := DigraphRemoveEdge(CompleteDigraph(3), [1, 2]);\n\ngap> DigraphEpimorphism(gr2, gr1);\ \nfail\ngap> DigraphEpimorphism(gr1, gr2);\nTransformation( [ 3, 1, 2, 3 ] )\n\ ", count := [ 7, 3, 6, 4 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 4 ], mathmode := "Text", name := "Example", next := 347537, root := ~, start := 347203, stop := 347536 ), rec( content := "\n ", count := [ 7, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 6, 3 ], mathmode := "Text", name := "Description", next := 347554, root := ~, start := 346983, stop := 347553 ) ], count := [ 7, 3, 6, 1 ], mathmode := "Text", name := "ManSection", next := 347568, root := ~, start := 346852, stop := 347567 ), rec( content := "\n\n ", count := [ 7, 3, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "EpimorphismsDigraphs" ), content := 0, count := [ 7, 3, 7, 2 ], mathmode := "Text", name := "Oper", next := 347649, root := ~, start := 347589, stop := 347648 ), rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "EpimorphismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 7, 4 ], mathmode := "Text", name := "Oper", next := 347727, root := ~, start := 347652, stop := 347726 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Returns", next := 347775, root := ~, start := 347730, stop := 347774 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations behave the same as ", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphs" ), content := 0, count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Ref", next := 347867, root := ~, start := 347832, stop := 347866 ), rec( content := " and\n \ ", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Ref", next := 347926, root := ~, start := 347876, stop := 347925 ), rec( content := ", except\ they only return\n ", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ urjective", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "E", next := 347973, root := ~, start := 347956, stop := 347972 ), rec( content := " homomor\ phisms.\n ", count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr1 := DigraphReverse(ChainDigraph(4));\n\ngap> gr2 := DigraphSymmetricClosure(CycleDigraph(3));\n\ngap> EpimorphismsDigraphsRepre\ sentatives(gr1, gr2);\n[ Transformation( [ 3, 1, 2, 1 ] ), Transformation( [ 3\ , 1, 2, 3 ] ), \n Transformation( [ 2, 1, 2, 3 ] ) ]\ngap> EpimorphismsDigrap\ hs(gr1, gr2);\n[ Transformation( [ 1, 2, 1, 3 ] ), Transformation( [ 1, 2, 3, \ 1 ] ), \n Transformation( [ 1, 2, 3, 2 ] ), Transformation( [ 1, 3, 1, 2 ] ),\ \n Transformation( [ 1, 3, 2, 1 ] ), Transformation( [ 1, 3, 2, 3 ] ), \n T\ ransformation( [ 2, 1, 2, 3 ] ), Transformation( [ 2, 1, 3, 1 ] ), \n Transfo\ rmation( [ 2, 1, 3, 2 ] ), Transformation( [ 2, 3, 1, 2 ] ), \n Transformatio\ n( [ 2, 3, 1, 3 ] ), Transformation( [ 2, 3, 2, 1 ] ), \n Transformation( [ 3\ , 1, 2, 1 ] ), Transformation( [ 3, 1, 2, 3 ] ), \n Transformation( [ 3, 1, 3\ , 2 ] ), Transformation( [ 3, 2, 1, 2 ] ), \n Transformation( [ 3, 2, 1, 3 ] \ ), Transformation( [ 3, 2, 3, 1 ] ) ]\n", count := [ 7, 3, 7, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 7, 6 ], mathmode := "Text", name := "Example", next := 349060, root := ~, start := 347993, stop := 349059 ), rec( content := "\n ", count := [ 7, 3, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 7, 5 ], mathmode := "Text", name := "Description", next := 349077, root := ~, start := 347778, stop := 349076 ) ], count := [ 7, 3, 7, 1 ], mathmode := "Text", name := "ManSection", next := 349091, root := ~, start := 347574, stop := 349090 ), rec( content := "\n\n ", count := [ 7, 3, 7, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph1, digrap\ h2", Name := "DigraphEmbedding" ), content := 0, count := [ 7, 3, 8, 2 ], mathmode := "Text", name := "Oper", next := 349168, root := ~, start := 349112, stop := 349167 ), rec( attributes := rec( ), content := [ rec( content := " A transf\ ormation, or ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "K", next := 349213, root := ~, start := 349202, stop := 349212 ), rec( content := ".", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Returns", next := 349224, root := ~, start := 349171, stop := 349223 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ embedding of a digraph ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 349286, root := ~, start := 349271, stop := 349285 ), rec( content := " into an\ other digraph\n ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 349327, root := ~, start := 349312, stop := 349326 ), rec( content := " is a ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Monomorphism" ), content := 0, count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Ref", next := 349366, root := ~, start := 349333, stop := 349365 ), rec( content := " from ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 349387, root := ~, start := 349372, stop := 349386 ), rec( content := "\n to\ ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 349410, root := ~, start := 349395, stop := 349409 ), rec( content := " which h\ as the additional property that a pair of\n vertices ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "C", next := 349486, root := ~, start := 349473, stop := 349485 ), rec( content := " which h\ ave no edge ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "C", next := 349519, root := ~, start := 349506, stop := 349518 ), rec( content := " in ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 349538, root := ~, start := 349523, stop := 349537 ), rec( content := " are\n\n\ mapped to a pair of vertices ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ a, b]", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "C", next := 349590, root := ~, start := 349577, stop := 349589 ), rec( content := " which h\ ave no edge ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "a\ ->b", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "C", next := 349621, root := ~, start := 349610, stop := 349620 ), rec( content := " in\n \ ", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "A", next := 349644, root := ~, start := 349629, stop := 349643 ), rec( content := ".", count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "P", next := 349649, root := ~, start := 349645, stop := 349648 ), rec( content := "\n\n \ In other words, an embedding ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "C", next := 349692, root := ~, start := 349684, stop := 349691 ), rec( content := " is an i\ somorphism from\n ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph1", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "A", next := 349735, root := ~, start := 349720, stop := 349734 ), rec( content := " to the \ ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Induced\ Subdigraph" ), content := 0, count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "Ref", next := 349774, root := ~, start := 349743, stop := 349773 ), rec( content := " of ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph2", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "A", next := 349793, root := ~, start := 349778, stop := 349792 ), rec( content := "\n on\ the image of ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "C", next := 349822, root := ~, start := 349814, stop := 349821 ), rec( content := ". ", count := [ 7, 3, 8, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "P", next := 349828, root := ~, start := 349824, stop := 349827 ), rec( content := "\n\n \ ", count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphEmbedding", count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "C", next := 349857, root := ~, start := 349834, stop := 349856 ), rec( content := " returns\ a single embedding if one exists, otherwise\n it returns ", count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "K", next := 349936, root := ~, start := 349925, stop := 349935 ), rec( content := ".\n " , count := [ 7, 3, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := ChainDigraph(3);\n\ngap> \ DigraphEmbedding(gr, CompleteDigraph(4));\nfail\ngap> DigraphEmbedding(gr, Dig\ raph([[3], [1, 4], [1], [3]]));\nTransformation( [ 2, 4, 3, 4 ] )\n", count := [ 7, 3, 8, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 8, 6 ], mathmode := "Text", name := "Example", next := 350190, root := ~, start := 349942, stop := 350189 ), rec( content := "\n ", count := [ 7, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 8, 3 ], mathmode := "Text", name := "Description", next := 350207, root := ~, start := 349227, stop := 350206 ) ], count := [ 7, 3, 8, 1 ], mathmode := "Text", name := "ManSection", next := 350221, root := ~, start := 349097, stop := 350220 ), rec( content := "\n\n ", count := [ 7, 3, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D1, D2", Name := "EmbeddingsDigra\ phs" ), content := 0, count := [ 7, 3, 9, 2 ], mathmode := "Text", name := "Oper", next := 350288, root := ~, start := 350242, stop := 350287 ), rec( attributes := rec( Arg := "D1, D2", Name := "EmbeddingsDigra\ phsRepresentatives" ), content := 0, count := [ 7, 3, 9, 4 ], mathmode := "Text", name := "Oper", next := 350352, root := ~, start := 350291, stop := 350351 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ transformations.", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Returns", next := 350400, root := ~, start := 350355, stop := 350399 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations behave the same as ", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphs" ), content := 0, count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Ref", next := 350492, root := ~, start := 350457, stop := 350491 ), rec( content := " and\n \ ", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Homomor\ phismsDigraphsRepresentatives" ), content := 0, count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Ref", next := 350551, root := ~, start := 350501, stop := 350550 ), rec( content := ", except\ they only return\n embeddings of ", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 1", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "A", next := 350604, root := ~, start := 350595, stop := 350603 ), rec( content := " into ", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ 2", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "A", next := 350619, root := ~, start := 350610, stop := 350618 ), rec( content := ".", count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 9, 6 ], mathmode := "Text", name := "P", next := 350624, root := ~, start := 350620, stop := 350623 ), rec( content := "\n\n \ See also ", count := [ 7, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsDigra\ phEmbedding" ), content := 0, count := [ 7, 3, 9, 6 ], mathmode := "Text", name := "Ref", next := 350671, root := ~, start := 350639, stop := 350670 ), rec( content := ".\n " , count := [ 7, 3, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D1 := NullDigraph(2);\n\ngap> D\ 2 := CycleDigraph(5);\n\ngap> Embeddi\ ngsDigraphsRepresentatives(D1, D2);\n[ Transformation( [ 1, 3, 3 ] ), Transfor\ mation( [ 1, 4, 3, 4 ] ) ]\ngap> EmbeddingsDigraphs(D1, D2);\n[ Transformation\ ( [ 1, 3, 3 ] ), Transformation( [ 1, 4, 3, 4 ] ), \n Transformation( [ 2, 4,\ 4, 5, 1 ] ), \n Transformation( [ 2, 5, 4, 5, 1 ] ), \n Transformation( [ 3\ , 1, 5, 1, 2 ] ), \n Transformation( [ 3, 5, 5, 1, 2 ] ), \n Transformation(\ [ 4, 1, 1, 2, 3 ] ), \n Transformation( [ 4, 2, 1, 2, 3 ] ), \n Transformat\ ion( [ 5, 2, 2, 3, 4 ] ), \n Transformation( [ 5, 3, 2, 3, 4 ] ) ]\n", count := [ 7, 3, 9, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 9, 7 ], mathmode := "Text", name := "Example", next := 351385, root := ~, start := 350677, stop := 351384 ), rec( content := "\n ", count := [ 7, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 9, 5 ], mathmode := "Text", name := "Description", next := 351402, root := ~, start := 350403, stop := 351401 ) ], count := [ 7, 3, 9, 1 ], mathmode := "Text", name := "ManSection", next := 351416, root := ~, start := 350227, stop := 351415 ), rec( content := "\n\n ", count := [ 7, 3, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphHomomorphism" ), content := 0, count := [ 7, 3, 10, 2 ], mathmode := "Text", name := "Oper", next := 351557, root := ~, start := 351437, stop := 351556 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphHomomorphism" ), content := 0, count := [ 7, 3, 10, 4 ], mathmode := "Text", name := "Oper", next := 351626, root := ~, start := 351560, stop := 351625 ), rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphEpimorphism" ), content := 0, count := [ 7, 3, 10, 6 ], mathmode := "Text", name := "Oper", next := 351747, root := ~, start := 351629, stop := 351746 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphEpimorphism" ), content := 0, count := [ 7, 3, 10, 8 ], mathmode := "Text", name := "Oper", next := 351815, root := ~, start := 351750, stop := 351814 ), rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphMonomorphism" ), content := 0, count := [ 7, 3, 10, 10 ], mathmode := "Text", name := "Oper", next := 351937, root := ~, start := 351818, stop := 351936 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphMonomorphism" ), content := 0, count := [ 7, 3, 10, 12 ], mathmode := "Text", name := "Oper", next := 352006, root := ~, start := 351940, stop := 352005 ), rec( attributes := rec( Arg := "digraph, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphEndomorphism" ), content := 0, count := [ 7, 3, 10, 14 ], mathmode := "Text", name := "Oper", next := 352127, root := ~, start := 352009, stop := 352126 ), rec( attributes := rec( Arg := "digraph, x, col" , Name := "IsDigraphEndomo\ rphism" ), content := 0, count := [ 7, 3, 10, 16 ], mathmode := "Text", name := "Oper", next := 352188, root := ~, start := 352130, stop := 352187 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "K", next := 352211, root := ~, start := 352200, stop := 352210 ), rec( content := " or ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "K", next := 352227, root := ~, start := 352215, stop := 352226 ), rec( content := ".", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "Returns", next := 352238, root := ~, start := 352191, stop := 352237 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphHomomorphism", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "C", next := 352287, root := ~, start := 352259, stop := 352286 ), rec( content := " returns\ ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "K", next := 352307, root := ~, start := 352296, stop := 352306 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "A", next := 352357, root := ~, start := 352349, stop := 352356 ), rec( content := " is a ho\ momorphism from the digraph\n ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "A", next := 352407, root := ~, start := 352397, stop := 352406 ), rec( content := " to the \ digraph ", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "A", next := 352433, root := ~, start := 352423, stop := 352432 ), rec( content := ".\n " , count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "P", next := 352443, root := ~, start := 352439, stop := 352442 ), rec( content := "\n\n \ ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphEpimorphism", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "C", next := 352476, root := ~, start := 352449, stop := 352475 ), rec( content := " returns\ ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "K", next := 352496, root := ~, start := 352485, stop := 352495 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "A", next := 352546, root := ~, start := 352538, stop := 352545 ), rec( content := " is a su\ rjective homomorphism from the digraph\n ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "A", next := 352607, root := ~, start := 352597, stop := 352606 ), rec( content := " to the \ digraph ", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "A", next := 352633, root := ~, start := 352623, stop := 352632 ), rec( content := ".\n " , count := [ 7, 3, 10, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "P", next := 352643, root := ~, start := 352639, stop := 352642 ), rec( content := "\n\n \ ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphMonomorphism", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "C", next := 352677, root := ~, start := 352649, stop := 352676 ), rec( content := " returns\ ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "K", next := 352697, root := ~, start := 352686, stop := 352696 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "A", next := 352747, root := ~, start := 352739, stop := 352746 ), rec( content := " is an i\ njective homomorphism from the digraph\n ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "A", next := 352808, root := ~, start := 352798, stop := 352807 ), rec( content := " to the \ digraph ", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "A", next := 352834, root := ~, start := 352824, stop := 352833 ), rec( content := ".\n " , count := [ 7, 3, 10, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "P", next := 352844, root := ~, start := 352840, stop := 352843 ), rec( content := "\n\n \ ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphEndomorphism", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "C", next := 352878, root := ~, start := 352850, stop := 352877 ), rec( content := " returns\ ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "K", next := 352898, root := ~, start := 352887, stop := 352897 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "A", next := 352948, root := ~, start := 352940, stop := 352947 ), rec( content := " is an e\ ndomorphism of the digraph\n ", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "A", next := 353001, root := ~, start := 352987, stop := 353000 ), rec( content := ".\n " , count := [ 7, 3, 10, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "P", next := 353011, root := ~, start := 353007, stop := 353010 ), rec( content := "\n\n \ A permutation or transformation ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "A", next := 353057, root := ~, start := 353049, stop := 353056 ), rec( content := " is a ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ omomorphism", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "E", next := 353082, root := ~, start := 353063, stop := 353081 ), rec( content := " from a\\ n digraph ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "A", next := 353112, root := ~, start := 353102, stop := 353111 ), rec( content := " to a di\ graph ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "A", next := 353136, root := ~, start := 353126, stop := 353135 ), rec( content := " if the \ following hold:\n ", count := [ 7, 3, 10, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 10, 23 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[u ^ ", count := [ 7, 3, 10, 23 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "x", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "A"\ , next := 353208, root := ~, start := 353200, stop := 353207 ), rec( content :\ = ", v ^ ", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ), rec( attributes := rec( ), content := [ rec( content := "x", coun\ t := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], co\ unt := [ 7, 3, 10, 23 ], mathmode := "Text", name := "A", next := 353222, root\ := ~, start := 353214, stop := 353221 ), rec( content := "]", count := [ 7, 3\ , 10, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7,\ 3, 10, 23 ], mathmode := "Text", name := "C", next := 353227, root := ~, star\ t := 353192, stop := 353226 ), rec( content := " is an edge of\n ", coun\ t := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "ran", count := [ 7, 3, 10, 23\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, \ 23 ], mathmode := "Text", name := "A", next := 353260, root := ~, start := 353\ 250, stop := 353259 ), rec( content := " whenever ", count := [ 7, 3, 10, 23 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[u, v]", count := [ 7, 3, 10, 23 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 23 ], mathmode := "T\ ext", name := "C", next := 353283, root := ~, start := 353270, stop := 353282 ), rec( cont\ ent := " is an\n edge of ", count := [ 7, 3, 10, 23 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "src", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 7, 3, 10, 23 ], mathmode := "Text", name := \ "A", next := 353316, root := ~, start := 353306, stop := 353315 ), rec( conten\ t := "; and ", count := [ 7, 3, 10, 23 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 7, 3, 10, 23 ], name := "Item", next := 353329, root := ~, start := 353\ 177, stop := 353328 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n ", count := [ 7, 3, 10, 25 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x", \ count := [ 7, 3, 10, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 7, 3, 10, 25 ], mathmode := "Text", name := "A", next := 353359, root :\ = ~, start := 353351, stop := 353358 ), rec( content := " fixes every ", count\ := [ 7, 3, 10, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "i", count := [ 7, 3, 10, 25 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 25\ ], mathmode := "Text", name := "C", next := 353380, root := ~, start := 35337\ 2, stop := 353379 ), rec( content := " which is not a vertex of ", count := [ \ 7, 3, 10, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "src", count := [ 7, 3, 10, 25 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 25\ ], mathmode := "Text", name := "A", next := 353416, root := ~, start := 35340\ 6, stop := 353415 ), rec( content := ".\n ", count := [ 7, 3, 10, 25 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 25 ], name := "Ite\ m", next := 353431, root := ~, start := 353336, stop := 353430 ) ], count := [ 7, 3, 10, 22 ], mathmode := "Text", name := "List", next := 353443, root := ~, start := 353164, stop := 353442 ), rec( content := "\n Se\ e also ", count := [ 7, 3, 10, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Generat\ orsOfEndomorphismMonoid" ), content := 0, count := [ 7, 3, 10, 27 ], mathmode := "Text", name := "Ref", next := 353501, root := ~, start := 353457, stop := 353500 ), rec( content := ".", count := [ 7, 3, 10, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "P", next := 353506, root := ~, start := 353502, stop := 353505 ), rec( content := "\n\n\n \ If ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "A", next := 353527, root := ~, start := 353516, stop := 353526 ), rec( content := " and ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol2", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "A", next := 353543, root := ~, start := 353532, stop := 353542 ), rec( content := ", or ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "A", next := 353558, root := ~, start := 353548, stop := 353557 ), rec( content := ", are gi\ ven, then they must\n represent vertex colourings; see \n ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a homogeneous list", Oper := "AutomorphismGroup" ), content := 0, count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "Ref", next := 353704, root := ~, start := 353628, stop := 353703 ), rec( content := " \n f\ or details of the permissible values for these argument. The\n homomorphism\ must then also have the property:\n \n ", count := [ 7, 3, 10, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 10, 30 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "col[i] = col[i ^ x]", count := [ 7, 3, 10, 30 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 30 ], math\ mode := "Text", name := "C", next := 353884, root := ~, start := 353858, stop \ := 353883 ), rec( content := " for all vertices ", count := [ 7, 3, 10, 30 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 7, 3, 10, 30 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 30 ], mathmode := "Text", name\ := "C", next := 353910, root := ~, start := 353902, stop := 353909 ), rec( cont\ ent := " of ", count := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", c\ ount := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 10, 30 ], mathmode := "Text", name := "A", next := 353928, root :\ = ~, start := 353914, stop := 353927 ), rec( content := ",\n in the cas\ e of ", count := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", root \ := ~ ), rec( attributes := rec( ), content := [ rec( content := "IsDigraphEnd\ omorphism", count := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 7, 3, 10, 30 ], mathmode := "Text", name := "C", next\ := 353981, root := ~, start := 353953, stop := 353980 ), rec( content := ".", coun\ t := [ 7, 3, 10, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 30 ] , name := "Ite\ m", next := 353989, root := ~, start := 353843, stop := 353988 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 10, 32 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "col1[i] = col2[i ^ x]", count := [ 7, 3, 10, 32 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 32 ], math\ mode := "Text", name := "C", next := 354039, root := ~, start := 354011, stop \ := 354038 ), rec( content := " for all vertices ", count := [ 7, 3, 10, 32 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "i", count := [ 7, 3, 10, 32 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 32 ], mathmode := "Text", name\ := "C", next := 354065, root := ~, start := 354057, stop := 354064 ), rec( cont\ ent := " of ", count := [ 7, 3, 10, 32 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "src", count\ := [ 7, 3, 10, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 7, 3, 10, 32 ], mathmode := "Text", name := "A", next := 354079, root :\ = ~, start := 354069, stop := 354078 ), rec( content := ",\n in the cas\ es of the other operations.", count := [ 7, 3, 10, 32 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 32 ], name := "Item", next := 3541\ 33, root := ~, start := 353996, stop := 354132 ) ], count := [ 7, 3, 10, 29 ], mathmode := "Text", name := "List", next := 354145, root := ~, start := 353830, stop := 354144 ), rec( content := "\n\n \ See also ", count := [ 7, 3, 10, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ sRespectsColouring" ), content := 0, count := [ 7, 3, 10, 34 ], mathmode := "Text", name := "Ref", next := 354199, root := ~, start := 354160, stop := 354198 ), rec( content := ".\n \\ n ", count := [ 7, 3, 10, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> src := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> ran := Digraph([[1], [1, 2]]);\n\ngap> IsDigraphHomomorphism(src, ran, Transformation([1, 2, \ 2]));\ntrue\ngap> IsDigraphHomomorphism(src, ran, Transformation([2, 1, 2]));\ \nfalse\ngap> IsDigraphHomomorphism(src, ran, Transformation([3, 3, 3]));\nfal\ se\ngap> IsDigraphHomomorphism(src, src, Transformation([3, 3, 3]));\ntrue\nga\ p> IsDigraphHomomorphism(src, ran, Transformation([1, 2, 2]),\n> \ [1, 2, 2], [1, 2]);\ntrue\ngap> IsDigraphHomomorphism(src, ran, Tr\ ansformation([1, 2, 2]),\n> [2, 1, 1], [1, 2]);\nfals\ e\ngap> IsDigraphEndomorphism(src, Transformation([3, 3, 3]));\ntrue\ngap> IsD\ igraphEndomorphism(src, Transformation([3, 3, 3]), [1, 1, 1]);\ntrue\ngap> IsD\ igraphEndomorphism(src, Transformation([3, 3, 3]), [1, 1, 2]);\nfalse\ngap> Is\ DigraphEpimorphism(src, ran, Transformation([3, 3, 3]));\nfalse\ngap> IsDigrap\ hMonomorphism(src, ran, Transformation([1, 2, 2]));\nfalse\ngap> IsDigraphEpim\ orphism(src, ran, Transformation([1, 2, 2]));\ntrue\ngap> IsDigraphMonomorphis\ m(ran, src, ());\ntrue", count := [ 7, 3, 10, 35 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 35 ], mathmode := "Text", name := "Example", next := 355401, root := ~, start := 354210, stop := 355400 ), rec( content := "\n ", count := [ 7, 3, 10, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 10, 17 ], mathmode := "Text", name := "Description", next := 355418, root := ~, start := 352241, stop := 355417 ) ], count := [ 7, 3, 10, 1 ], mathmode := "Text", name := "ManSection", next := 355432, root := ~, start := 351422, stop := 355431 ), rec( content := "\n\n ", count := [ 7, 3, 10, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "src, ran, x", Label := "for digraphs a\ nd a permutation or transformation", Name := "IsDigraphEmbedding" ), content := 0, count := [ 7, 3, 11, 2 ], mathmode := "Text", name := "Oper", next := 355569, root := ~, start := 355453, stop := 355568 ), rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "IsDigraphEmbedding" ), content := 0, count := [ 7, 3, 11, 4 ], mathmode := "Text", name := "Oper", next := 355635, root := ~, start := 355572, stop := 355634 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "K", next := 355658, root := ~, start := 355647, stop := 355657 ), rec( content := " or ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "K", next := 355674, root := ~, start := 355662, stop := 355673 ), rec( content := ".", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "Returns", next := 355685, root := ~, start := 355638, stop := 355684 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDigraphEmbedding", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "C", next := 355731, root := ~, start := 355706, stop := 355730 ), rec( content := " returns\ ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "K", next := 355751, root := ~, start := 355740, stop := 355750 ), rec( content := " if the \ permutation\n or transformation ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 355801, root := ~, start := 355793, stop := 355800 ), rec( content := " is a em\ bedding of the digraph\n ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 355846, root := ~, start := 355836, stop := 355845 ), rec( content := " into th\ e digraph ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 355874, root := ~, start := 355864, stop := 355873 ), rec( content := ", while \ respecting the colourings\n ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 355923, root := ~, start := 355912, stop := 355922 ), rec( content := " and ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol2", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "A", next := 355939, root := ~, start := 355928, stop := 355938 ), rec( content := " if give\ n.\n ", count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "P", next := 355958, root := ~, start := 355954, stop := 355957 ), rec( content := "\n\n \ A permutation or transformation ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 356004, root := ~, start := 355996, stop := 356003 ), rec( content := " is a ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ mbedding", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "E", next := 356026, root := ~, start := 356010, stop := 356025 ), rec( content := " of a di\ graph\n ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 356054, root := ~, start := 356044, stop := 356053 ), rec( content := " into a \ digraph ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 356080, root := ~, start := 356070, stop := 356079 ), rec( content := " if ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 356092, root := ~, start := 356084, stop := 356091 ), rec( content := " is a mo\ nomorphism from\n ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 356130, root := ~, start := 356120, stop := 356129 ), rec( content := " to ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 356144, root := ~, start := 356134, stop := 356143 ), rec( content := " and the\ inverse of ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 356172, root := ~, start := 356164, stop := 356171 ), rec( content := " is a mo\ nomorphism from\n the subdigraph of ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ an", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 356228, root := ~, start := 356218, stop := 356227 ), rec( content := " induced\ by the image of ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 356261, root := ~, start := 356253, stop := 356260 ), rec( content := " to\n \ ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "A", next := 356279, root := ~, start := 356269, stop := 356278 ), rec( content := ".\n\n \ See also ", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsDigra\ phHomomorphism" ), content := 0, count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "Ref", next := 356330, root := ~, start := 356295, stop := 356329 ), rec( content := ".", count := [ 7, 3, 11, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 11, 7 ], mathmode := "Text", name := "P", next := 356335, root := ~, start := 356331, stop := 356334 ), rec( content := "\n\n \ ", count := [ 7, 3, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> src := Digraph([[1], [1, 2]]);\n\ngap> ran := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> IsDigraphMonomorphism(src, ran, ());\ntrue\ngap> IsDig\ raphEmbedding(src, ran, ());\ntrue\ngap> IsDigraphEmbedding(src, ran, (), [2, \ 1], [2, 1, 1]);\ntrue\ngap> IsDigraphEmbedding(src, ran, (), [2, 1], [1, 2, 1]\ );\nfalse\ngap> ran := Digraph([[1, 2], [1, 2], [1, 3]]);\n\ngap> IsDigraphMonomorphism(src, ran, IdentityTransf\ ormation);\ntrue\ngap> IsDigraphEmbedding(src, ran, IdentityTransformation);\n\ false", count := [ 7, 3, 11, 8 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 8 ], mathmode := "Text", name := "Example", next := 356984, root := ~, start := 356341, stop := 356983 ), rec( content := "\n ", count := [ 7, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 11, 5 ], mathmode := "Text", name := "Description", next := 357001, root := ~, start := 355688, stop := 357000 ) ], count := [ 7, 3, 11, 1 ], mathmode := "Text", name := "ManSection", next := 357015, root := ~, start := 355438, stop := 357014 ), rec( content := "\n\n ", count := [ 7, 3, 11, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "src, ran, x, col\ 1, col2", Name := "DigraphsRespectsColouring" ), content := 0, count := [ 7, 3, 12, 2 ], mathmode := "Text", name := "Oper", next := 357110, root := ~, start := 357040, stop := 357109 ), rec( attributes := rec( ), content := [ rec( content := " ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "K", next := 357136, root := ~, start := 357125, stop := 357135 ), rec( content := " or ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "K", next := 357152, root := ~, start := 357140, stop := 357151 ), rec( content := ". ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "Returns", next := 357164, root := ~, start := 357115, stop := 357163 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ he operation ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphsRespectsColouring", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "C", next := 357235, root := ~, start := 357203, stop := 357234 ), rec( content := " verifie\ s whether or not \n the permutation or transformation ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "x\ ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 357309, root := ~, start := 357301, stop := 357308 ), rec( content := " respect\ s the vertex colourings\n ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 357358, root := ~, start := 357347, stop := 357357 ), rec( content := " and ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol2", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 357374, root := ~, start := 357363, stop := 357373 ), rec( content := " of the \ digraphs ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 357401, root := ~, start := 357391, stop := 357400 ), rec( content := " and ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "r\ ange", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 357418, root := ~, start := 357406, stop := 357417 ), rec( content := ".\n \ That is, ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphsRespectsColouring", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "C", next := 357467, root := ~, start := 357435, stop := 357466 ), rec( content := " returns\ ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "K", next := 357487, root := ~, start := 357476, stop := 357486 ), rec( content := " if and \ only if for\n all vertices ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "C", next := 357534, root := ~, start := 357526, stop := 357533 ), rec( content := " of ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ rc", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "A", next := 357548, root := ~, start := 357538, stop := 357547 ), rec( content := ", ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ol1[i] = col2[i ^ x]", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "C", next := 357578, root := ~, start := 357550, stop := 357577 ), rec( content := ".\n \ ", count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 12, 4 ], mathmode := "Text", name := "P", next := 357590, root := ~, start := 357586, stop := 357589 ), rec( content := "\n\n \ ", count := [ 7, 3, 12, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> src := Digraph([[1], [1, 2]]);\n\ngap> ran := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> DigraphsRespectsColouring(src, ran, (1, 2), [2, 1], [1\ , 2, 1]);\ntrue\ngap> DigraphsRespectsColouring(src, ran, (1, 2), [2, 1], [2, \ 1, 1]);\nfalse\n", count := [ 7, 3, 12, 5 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 5 ], mathmode := "Text", name := "Example", next := 357949, root := ~, start := 357598, stop := 357948 ), rec( content := "\n ", count := [ 7, 3, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 12, 3 ], mathmode := "Text", name := "Description", next := 357968, root := ~, start := 357169, stop := 357967 ) ], count := [ 7, 3, 12, 1 ], mathmode := "Text", name := "ManSection", next := 357984, root := ~, start := 357023, stop := 357983 ), rec( content := "\n\n ", count := [ 7, 3, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, colors\ ][, limit]", Name := "GeneratorsOfEndomorphismMonoid" ), content := 0, count := [ 7, 3, 13, 2 ], mathmode := "Text", name := "Func", next := 358083, root := ~, start := 358005, stop := 358082 ), rec( attributes := rec( Arg := "digraph", Name := "GeneratorsOfEnd\ omorphismMonoidAttr" ), content := 0, count := [ 7, 3, 13, 4 ], mathmode := "Text", name := "Attr", next := 358149, root := ~, start := 358086, stop := 358148 ), rec( attributes := rec( ), content := [ rec( content := " A list o\ f transformations.", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Returns", next := 358198, root := ~, start := 358152, stop := 358197 ), rec( attributes := rec( ), content := [ rec( content := "\n An \ endomorphism of ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "A", next := 358252, root := ~, start := 358238, stop := 358251 ), rec( content := " is a ho\ momorphism\n ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Homomorphism" ), content := 0, count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Ref", next := 358308, root := ~, start := 358275, stop := 358307 ), rec( content := " from ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "A", next := 358328, root := ~, start := 358314, stop := 358327 ), rec( content := " back to\ itself.\n\n ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eneratorsOfEndomorphismMonoid", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "C", next := 358387, root := ~, start := 358350, stop := 358386 ), rec( content := ", called\ with a single argument,\n returns a generating set for the monoid of all e\ ndomorphisms of\n ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "A", next := 358506, root := ~, start := 358492, stop := 358505 ), rec( content := ". If ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "A", next := 358525, root := ~, start := 358511, stop := 358524 ), rec( content := " belong\ s to ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Ref", next := 358576, root := ~, start := 358538, stop := 358575 ), rec( content := ", then t\ he value of\n ", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eneratorsOfEndomorphismMonoid", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "C", next := 358637, root := ~, start := 358600, stop := 358636 ), rec( content := " will no\ t be recomputed on future\n calls.", count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "P", next := 358685, root := ~, start := 358681, stop := 358684 ), rec( content := "\n\n \ If the ", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olors", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "A", next := 358711, root := ~, start := 358698, stop := 358710 ), rec( content := " argumen\ t is specified, then\n ", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eneratorsOfEndomorphismMonoid", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "C", next := 358781, root := ~, start := 358744, stop := 358780 ), rec( content := " will re\ turn a generating\n set for the monoid of endomorphisms which respect the g\ iven colouring.\n The colouring ", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olors", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "A", next := 358913, root := ~, start := 358900, stop := 358912 ), rec( content := " can be \ in one of two forms: ", count := [ 7, 3, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 13, 7 ], mathmode := "Text", name := "P", next := 358946, root := ~, start := 358942, stop := 358945 ), rec( content := "\n\n \ ", count := [ 7, 3, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n A list of positive integers \ of size the number of vertices of\n ", count := [ 7, 3, 13, 9 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "digraph", count := [ 7, 3, 13, 9 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], mathmode := "Tex\ t", name := "A", next := 359064, root := ~, start := 359050, stop := 359063 ), rec( cont\ ent := ", where ", count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "co\ lors", count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ) ], count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "A", next := 359\ 085, root := ~, start := 359072, stop := 359084 ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[i]", count := [ 7, 3, 13, 9 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], mathmode := "Text", name\ := "C", next := 359095, root := ~, start := 359085, stop := 359094 ), rec( cont\ ent := " is the colour of vertex\n ", count := [ 7, 3, 13, 9 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "i", count := [ 7, 3, 13, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], mathmode := "Text", name\ := "C", next := 359136, root := ~, start := 359128, stop := 359135 ), rec( cont\ ent := ".\n ", count := [ 7, 3, 13, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 9 ], name := "Item", next := 359151, root := ~, start := 358\ 965, stop := 359150 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n A list of lists, such that ", count := [ 7, 3, 13, 11 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "colors", count := [ 7, 3, 13, 11 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 11 ], mathmode := "Text\ ", name := "A", next := 359213, root := ~, start := 359200, stop := 359212 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "[i]", count := [ 7, 3, 13, 11\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, \ 11 ], mathmode := "Text", name := "C", next := 359223, root := ~, start := 359\ 213, stop := 359222 ), rec( content := " is a list of all\n vertices wi\ th colour ", count := [ 7, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "i", count :\ = [ 7, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 7, 3, 13, 11 ], mathmode := "Text", name := "C", next := 359278, root :=\ ~, start := 359270, stop := 359277 ), rec( content := ".\n ", count := [ \ 7, 3, 13, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 11 ] , name := "Ite\ m", next := 359293, root := ~, start := 359158, stop := 359292 ) ], count := [ 7, 3, 13, 8 ], mathmode := "Text", name := "List", next := 359305, root := ~, start := 358952, stop := 359304 ), rec( content := "\n\n \ If the ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "A", next := 359330, root := ~, start := 359318, stop := 359329 ), rec( content := " argumen\ t is specified, then it will return only the\n first ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "A", next := 359405, root := ~, start := 359393, stop := 359404 ), rec( content := " homomor\ phisms, where ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "A", next := 359439, root := ~, start := 359427, stop := 359438 ), rec( content := " must be\ a positive\n integer or ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "C", next := 359489, root := ~, start := 359474, stop := 359488 ), rec( content := ". ", count := [ 7, 3, 13, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 13, 14 ], mathmode := "Text", name := "P", next := 359495, root := ~, start := 359491, stop := 359494 ), rec( content := "\n\n \ ", count := [ 7, 3, 13, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph(List([1 .. 3], x -> [1 .. 3]));;\ngap> GeneratorsOfEndomor\ phismMonoid(gr);\n[ Transformation( [ 1, 3, 2 ] ), Transformation( [ 2, 1 ] ),\ \n IdentityTransformation, Transformation( [ 1, 2, 1 ] ), \n Transformation\ ( [ 1, 2, 2 ] ), Transformation( [ 1, 1, 2 ] ), \n Transformation( [ 1, 1, 1 \ ] ) ]\ngap> GeneratorsOfEndomorphismMonoid(gr, 3);\n[ Transformation( [ 1, 3, \ 2 ] ), Transformation( [ 2, 1 ] ), \n IdentityTransformation ]\ngap> gr := Co\ mpleteDigraph(3);;\ngap> GeneratorsOfEndomorphismMonoid(gr);\n[ Transformation\ ( [ 2, 3, 1 ] ), Transformation( [ 2, 1 ] ), \n IdentityTransformation ]\ngap\ > GeneratorsOfEndomorphismMonoid(gr, [1, 2, 2]);\n[ Transformation( [ 1, 3, 2 \ ] ), IdentityTransformation ]\ngap> GeneratorsOfEndomorphismMonoid(gr, [[1], [\ 2, 3]]);\n[ Transformation( [ 1, 3, 2 ] ), IdentityTransformation ]\n", count := [ 7, 3, 13, 15 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 13, 15 ], mathmode := "Text", name := "Example", next := 360364, root := ~, start := 359501, stop := 360363 ), rec( content := "\n ", count := [ 7, 3, 13, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 13, 5 ], mathmode := "Text", name := "Description", next := 360381, root := ~, start := 358201, stop := 360380 ) ], count := [ 7, 3, 13, 1 ], mathmode := "Text", name := "ManSection", next := 360395, root := ~, start := 357990, stop := 360394 ), rec( content := "\n\n ", count := [ 7, 3, 13, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, n", Label := "for a digraph \ and a number of colours", Name := "DigraphColouring" ), content := 0, count := [ 7, 3, 14, 2 ], mathmode := "Text", name := "Oper", next := 360514, root := ~, start := 360416, stop := 360513 ), rec( attributes := rec( ), content := [ rec( content := " A transf\ ormation, or ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "K", next := 360559, root := ~, start := 360548, stop := 360558 ), rec( content := ".", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Returns", next := 360570, root := ~, start := 360517, stop := 360569 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper colouring", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "E", next := 360616, root := ~, start := 360593, stop := 360615 ), rec( content := " of a di\ graph is a labelling of its vertices in\n such a way that adjacent vertices\ have different labels. A ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper\n ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "n", count := [ 7, 3, 14, 3 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := \ "Text", name := "C", next := 360751, root := ~, start := 360743, stop := 36075\ 0 ), rec( content := "-colouring", count := [ 7, 3, 14, 3 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "E", next := 360765, root := ~, start := 360727, stop := 360764 ), rec( content := " is a pr\ oper colouring that uses exactly ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 360814, root := ~, start := 360806, stop := 360813 ), rec( content := "\n co\ lours. Equivalently, a proper (", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 360861, root := ~, start := 360853, stop := 360860 ), rec( content := "-)colour\ ing of a digraph can be\n defined to be a ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Epimorphism" ), content := 0, count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Ref", next := 360945, root := ~, start := 360913, stop := 360944 ), rec( content := " from a \ digraph onto the\n complete digraph (with ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 361005, root := ~, start := 360997, stop := 361004 ), rec( content := " vertice\ s); see ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Complet\ eDigraph" ), content := 0, count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Ref", next := 361056, root := ~, start := 361021, stop := 361055 ), rec( content := ". Note \ that a digraph with loops (", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Ref", next := 361126, root := ~, start := 361091, stop := 361125 ), rec( content := ") does n\ ot have a proper ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 361159, root := ~, start := 361151, stop := 361158 ), rec( content := "-colouri\ ng for\n any value ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "C", next := 361196, root := ~, start := 361188, stop := 361195 ), rec( content := ".\n " , count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "P", next := 361206, root := ~, start := 361202, stop := 361205 ), rec( content := "\n\n \ If ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 361229, root := ~, start := 361215, stop := 361228 ), rec( content := " is a di\ graph and ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 361255, root := ~, start := 361247, stop := 361254 ), rec( content := " is a no\ n-negative integer, then\n ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphColouring(", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "di\ graph", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 3613\ 26, root := ~, start := 361312, stop := 361325 ), rec( content := ", ", count := [ 7, 3, 14, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "n", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 3613\ 36, root := ~, start := 361328, stop := 361335 ), rec( content := ")", count := [ 7, 3, 14, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "C", next := 361341, root := ~, start := 361292, stop := 361340 ), rec( content := " returns\ an epimorphism\n from ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 361388, root := ~, start := 361374, stop := 361387 ), rec( content := " onto th\ e complete digraph with ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "A", next := 361428, root := ~, start := 361420, stop := 361427 ), rec( content := " vertice\ s if one\n exists, else it returns ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "K", next := 361484, root := ~, start := 361473, stop := 361483 ), rec( content := ". ", count := [ 7, 3, 14, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 14, 5 ], mathmode := "Text", name := "P", next := 361490, root := ~, start := 361486, stop := 361489 ), rec( content := "\n\n \ See also ", count := [ 7, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ GreedyColouring", Label := "for a digraph" ), content := 0, count := [ 7, 3, 14, 5 ], mathmode := "Text", name := "Ref", next := 361563, root := ~, start := 361505, stop := 361562 ), rec( content := " and \n \ ", count := [ 7, 3, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 14, 6 ], mathmode := "Text", name := "P", next := 361577, root := ~, start := 361573, stop := 361576 ), rec( content := "\n\n \ Note that a digraph with at least two vertices has a 2-colouring if and only\n\ if it is bipartite, see ", count := [ 7, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipar\ titeDigraph" ), content := 0, count := [ 7, 3, 14, 6 ], mathmode := "Text", name := "Ref", next := 361720, root := ~, start := 361688, stop := 361719 ), rec( content := ".\n\n \ ", count := [ 7, 3, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphColouring(CompleteDigraph(5), 4);\nfail\ngap> DigraphColouring(Ch\ ainDigraph(10), 1);\nfail\ngap> D := ChainDigraph(10);;\ngap> t := DigraphColo\ uring(D, 2);\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\ngap> IsDigra\ phColouring(D, t); \ntrue\ngap> DigraphGreedyColouring(D);\nTransformation( [ \ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1 ] )\n", count := [ 7, 3, 14, 7 ], name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 14, 7 ], mathmode := "Text", name := "Example", next := 362093, root := ~, start := 361727, stop := 362092 ), rec( content := "\n ", count := [ 7, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 14, 3 ], mathmode := "Text", name := "Description", next := 362110, root := ~, start := 360573, stop := 362109 ) ], count := [ 7, 3, 14, 1 ], mathmode := "Text", name := "ManSection", next := 362124, root := ~, start := 360401, stop := 362123 ), rec( content := "\n\n ", count := [ 7, 3, 14, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, order", Label := "for a digraph \ and vertex order", Name := "DigraphGreedyColouring" ), content := 0, count := [ 7, 3, 15, 2 ], mathmode := "Text", name := "Oper", next := 362260, root := ~, start := 362145, stop := 362259 ), rec( attributes := rec( Arg := "digraph, func", Label := "for a digraph \ and vertex order function", Name := "DigraphGreedyColouring" ), content := 0, count := [ 7, 3, 15, 4 ], mathmode := "Text", name := "Oper", next := 362386, root := ~, start := 362263, stop := 362385 ), rec( attributes := rec( Arg := "digraph", Label := "for a digraph" , Name := "DigraphGreedyCo\ louring" ), content := 0, count := [ 7, 3, 15, 6 ], mathmode := "Text", name := "Attr", next := 362480, root := ~, start := 362389, stop := 362479 ), rec( attributes := rec( ), content := [ rec( content := " A transf\ ormation, or ", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "K", next := 362525, root := ~, start := 362514, stop := 362524 ), rec( content := ".", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Returns", next := 362536, root := ~, start := 362483, stop := 362535 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper colouring", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "E", next := 362582, root := ~, start := 362559, stop := 362581 ), rec( content := " of a di\ graph is a labelling of its vertices in\n such a way that adjacent vertices\ have different labels. Note that a digraph \n with loops (", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Ref", next := 362757, root := ~, start := 362728, stop := 362756 ), rec( content := ") does n\ ot have any proper \n colouring.\n ", count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "P", next := 362808, root := ~, start := 362804, stop := 362807 ), rec( content := "\n \n\ If ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "A", next := 362835, root := ~, start := 362821, stop := 362834 ), rec( content := " is a di\ graph and ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rder", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "A", next := 362865, root := ~, start := 362853, stop := 362864 ), rec( content := " is a de\ nse list consisting \n of all of the vertices of ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "A", next := 362938, root := ~, start := 362924, stop := 362937 ), rec( content := " (in any\ order), then \n ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGreedyColouring", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "C", next := 362995, root := ~, start := 362966, stop := 362994 ), rec( content := "\n us\ es a greedy algorithm with the specified order to obtain some proper\n colo\ uring of ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "A", next := 363102, root := ~, start := 363088, stop := 363101 ), rec( content := ", which \ may not use the minimal number of\n colours. ", count := [ 7, 3, 15, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "P", next := 363161, root := ~, start := 363157, stop := 363160 ), rec( content := "\n \n\ If ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 363188, root := ~, start := 363174, stop := 363187 ), rec( content := " is a di\ graph and ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ unc", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 363217, root := ~, start := 363206, stop := 363216 ), rec( content := " is a fu\ nction whose argument \n is a digraph, and that returns a dense list ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ rder", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 363308, root := ~, start := 363296, stop := 363307 ), rec( content := ", then \\ n ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGreedyColouring(", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3633\ 60, root := ~, start := 363346, stop := 363359 ), rec( content := ", ", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "func", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3633\ 73, root := ~, start := 363362, stop := 363372 ), rec( content := ")", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "C", next := 363378, root := ~, start := 363320, stop := 363377 ), rec( content := " returns\ \n ", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphGreedyColouring(", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3634\ 31, root := ~, start := 363417, stop := 363430 ), rec( content := ", ", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "func", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3634\ 44, root := ~, start := 363433, stop := 363443 ), rec( content := "(", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "A", next := 3634\ 59, root := ~, start := 363445, stop := 363458 ), rec( content := "))", count := [ 7, 3, 15, 9 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "C", next := 363465, root := ~, start := 363391, stop := 363464 ), rec( content := ".\n " , count := [ 7, 3, 15, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 10 ], mathmode := "Text", name := "P", next := 363475, root := ~, start := 363471, stop := 363474 ), rec( content := "\n\n \ If the optional second argument (either a list or a function), is not \n sp\ ecified, then ", count := [ 7, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ WelshPowellOrder" ), content := 0, count := [ 7, 3, 15, 10 ], mathmode := "Text", name := "Ref", next := 363609, root := ~, start := 363572, stop := 363608 ), rec( content := " is used\ by default. \n ", count := [ 7, 3, 15, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 11 ], mathmode := "Text", name := "P", next := 363639, root := ~, start := 363635, stop := 363638 ), rec( content := "\n \n\ See also\n ", count := [ 7, 3, 15, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a number of colours", Oper := "DigraphColouring" ), content := 0, count := [ 7, 3, 15, 11 ], mathmode := "Text", name := "Ref", next := 363748, root := ~, start := 363662, stop := 363747 ), rec( content := ".\n " , count := [ 7, 3, 15, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 15, 12 ], mathmode := "Text", name := "P", next := 363758, root := ~, start := 363754, stop := 363757 ), rec( content := "\n\n \ ", count := [ 7, 3, 15, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphGreedyColouring(ChainDigraph(10));\nTransformation( [ 2, 1, 2, 1,\ 2, 1, 2, 1, 2, 1 ] )\ngap> DigraphGreedyColouring(ChainDigraph(10), [1 .. 10]\ );\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\n", count := [ 7, 3, 15, 13 ] , name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 15, 13 ], mathmode := "Text", name := "Example", next := 364003, root := ~, start := 363764, stop := 364002 ), rec( content := "\n ", count := [ 7, 3, 15, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 15, 7 ], mathmode := "Text", name := "Description", next := 364020, root := ~, start := 362539, stop := 364019 ) ], count := [ 7, 3, 15, 1 ], mathmode := "Text", name := "ManSection", next := 364034, root := ~, start := 362130, stop := 364033 ), rec( content := "\n\n ", count := [ 7, 3, 15, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DigraphWelshPow\ ellOrder" ), content := 0, count := [ 7, 3, 16, 2 ], mathmode := "Text", name := "Attr", next := 364107, root := ~, start := 364055, stop := 364106 ), rec( attributes := rec( ), content := [ rec( content := " A list o\ f the vertices.", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "Returns", next := 364153, root := ~, start := 364110, stop := 364152 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphWelshPowellOrder", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "C", next := 364204, root := ~, start := 364174, stop := 364203 ), rec( content := " returns\ a list of all of the vertices of \n the digraph ", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "A", next := 364277, root := ~, start := 364263, stop := 364276 ), rec( content := " ordered\ according to the sum of the number of \n out- and in-neighbours, from high\ est to lowest. \n ", count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 16, 4 ], mathmode := "Text", name := "P", next := 364386, root := ~, start := 364382, stop := 364385 ), rec( content := "\n\n \ ", count := [ 7, 3, 16, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphWelshPowellOrder(Digraph([[4], [9], [9], [], \n> \ [4, 6, 9], [1], [], [], \n> \ [4, 5], [4, 5]]));\n[ 5, 9, 4, 1, 6, 10, 2, 3, 7, 8 ]\n", count := [ 7, 3, 16, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 16, 5 ], mathmode := "Text", name := "Example", next := 364636, root := ~, start := 364392, stop := 364635 ), rec( content := "\n ", count := [ 7, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 16, 3 ], mathmode := "Text", name := "Description", next := 364653, root := ~, start := 364156, stop := 364652 ) ], count := [ 7, 3, 16, 1 ], mathmode := "Text", name := "ManSection", next := 364667, root := ~, start := 364040, stop := 364666 ), rec( content := "\n\n ", count := [ 7, 3, 16, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "ChromaticNumber\ " ), content := 0, count := [ 7, 3, 17, 2 ], mathmode := "Text", name := "Attr", next := 364732, root := ~, start := 364688, stop := 364731 ), rec( attributes := rec( ), content := [ rec( content := " A non-ne\ gative integer.", count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "Returns", next := 364778, root := ~, start := 364735, stop := 364777 ), rec( attributes := rec( ), content := [ rec( content := "\n A " , count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ roper colouring", count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "E", next := 364824, root := ~, start := 364801, stop := 364823 ), rec( content := " of a di\ graph is a labelling of its\n vertices in such a way that adjacent vertices\ have different labels.\n Equivalently, a proper digraph colouring can be d\ efined to be a ", count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Epimorphism" ), content := 0, count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "Ref", next := 365039, root := ~, start := 365001, stop := 365038 ), rec( content := " from a \ digraph onto a complete digraph. ", count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "P", next := 365084, root := ~, start := 365080, stop := 365083 ), rec( content := "\n\n \ If ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "A", next := 365107, root := ~, start := 365093, stop := 365106 ), rec( content := " is a di\ graph without loops (see ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "Ref", next := 365175, root := ~, start := 365140, stop := 365174 ), rec( content := ", then " , count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ hromaticNumber", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 365204, root := ~, start := 365182, stop := 365203 ), rec( content := " returns\ the least\n non-negative integer ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 365256, root := ~, start := 365248, stop := 365255 ), rec( content := " such th\ at there is a proper colouring of\n ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "A", next := 365316, root := ~, start := 365302, stop := 365315 ), rec( content := " with ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 365330, root := ~, start := 365322, stop := 365329 ), rec( content := " colours\ . In other words, for a digraph with at\n least one vertex, ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ hromaticNumber", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 365423, root := ~, start := 365401, stop := 365422 ), rec( content := " returns\ the least number ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 365457, root := ~, start := 365449, stop := 365456 ), rec( content := "\n su\ ch that ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphColouring(", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "di\ graph", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "A", next := 3655\ 06, root := ~, start := 365492, stop := 365505 ), rec( content := ", n)", count := [ 7, 3, 17, 4 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "C", next := 365514, root := ~, start := 365472, stop := 365513 ), rec( content := " does no\ t return\n ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "K", next := 365546, root := ~, start := 365535, stop := 365545 ), rec( content := ". See ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and a number of colours", Oper := "DigraphColouring" ), content := 0, count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "Ref", next := 365634, root := ~, start := 365552, stop := 365633 ), rec( content := ".\n\n \ ", count := [ 7, 3, 17, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ChromaticNumber(NullDigraph(10));\n1\ngap> ChromaticNumber(CompleteDigra\ ph(10));\n10\ngap> ChromaticNumber(CompleteBipartiteDigraph(5, 5));\n2\ngap> C\ hromaticNumber(Digraph([[], [3], [5], [2, 3], [4]]));\n3\ngap> ChromaticNumber\ (NullDigraph(0));\n0\ngap> D := PetersenGraph(IsMutableDigraph);\n\ngap> ChromaticNumber(D);\n3\n", count := [ 7, 3, 17, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 7, 3, 17, 5 ], mathmode := "Text", name := "Example", next := 366033, root := ~, start := 365641, stop := 366032 ), rec( content := "\n ", count := [ 7, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 17, 3 ], mathmode := "Text", name := "Description", next := 366050, root := ~, start := 364781, stop := 366049 ) ], count := [ 7, 3, 17, 1 ], mathmode := "Text", name := "ManSection", next := 366064, root := ~, start := 364673, stop := 366063 ), rec( content := "\n\n ", count := [ 7, 3, 17, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "D", Name := "DigraphCore" ), content := 0, count := [ 7, 3, 18, 2 ], mathmode := "Text", name := "Attr", next := 366127, root := ~, start := 366089, stop := 366126 ), rec( attributes := rec( ), content := [ rec( content := "A list of\ positive integers.", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "Returns", next := 366179, root := ~, start := 366132, stop := 366178 ), rec( attributes := rec( ), content := [ rec( content := "\n I\ f ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "A", next := 366215, root := ~, start := 366207, stop := 366214 ), rec( content := " is a di\ graph, then ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphCore", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "C", next := 366253, root := ~, start := 366235, stop := 366252 ), rec( content := " returns\ a list of vertices\n corresponding to the ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ore", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "C", next := 366319, root := ~, start := 366308, stop := 366318 ), rec( content := " of ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "A", next := 366331, root := ~, start := 366323, stop := 366330 ), rec( content := ". In par\ ticular, the subdigraph\n of ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "A", next := 366380, root := ~, start := 366372, stop := 366379 ), rec( content := " induced\ by this list is isomorphic to the core of ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "A", next := 366439, root := ~, start := 366431, stop := 366438 ), rec( content := ".", count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "P", next := 366444, root := ~, start := 366440, stop := 366443 ), rec( content := "\n\n \ The ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ ore", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "E", next := 366465, root := ~, start := 366454, stop := 366464 ), rec( content := " of a di\ graph ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "C", next := 366487, root := ~, start := 366479, stop := 366486 ), rec( content := " is the \ minimal subdigraph ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "A", next := 366522, root := ~, start := 366514, stop := 366521 ), rec( content := " of\n \ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "C", next := 366538, root := ~, start := 366530, stop := 366537 ), rec( content := " which i\ s a homomorphic image of ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "C", next := 366579, root := ~, start := 366571, stop := 366578 ), rec( content := ". The co\ re of a digraph \n is unique up to isomorphism.\n ", count := [ 7, 3, 18, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := DigraphSymmetricClosure(CycleDigraph(8));\n\ngap> DigraphCore(D);\n[ 1, 2 ]\ngap> D := Pet\ ersenGraph();\n\ngap> DigraphCor\ e(D);\n[ 1 .. 10 ]\ngap> D := Digraph(IsMutableDigraph, [[3], [3], [4], [5], [\ 2]]);\n\ngap> DigraphCore(D);\n[ 2, \ 3, 4, 5 ]\n", count := [ 7, 3, 18, 5 ], name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 5 ], mathmode := "Text", name := "Example", next := 367063, root := ~, start := 366643, stop := 367062 ), rec( content := "\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 18, 3 ], mathmode := "Text", name := "Description", next := 367082, root := ~, start := 366184, stop := 367081 ) ], count := [ 7, 3, 18, 1 ], mathmode := "Text", name := "ManSection", next := 367098, root := ~, start := 366072, stop := 367097 ), rec( content := "\n\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 3, 0, 1 ], mathmode := "Text", name := "Section", next := 367112, root := ~, start := 334918, stop := 367111 ) , rec( content := "\n\n", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 7, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 367124, root := ~, start := 286090, stop := 367123 ), rec( content := "\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " homomorphisms ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 7, 3, 18, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Finding cliques and independent sets" ), content := [ rec( content := "\n ", count := [ 8, 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Finding cliques and indep\ endent sets", count := [ 8, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 367263, root := ~, start := 367208, stop := 367262 ) , rec( content := "\n\n In ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ", a ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "clique", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 367299, root := ~, start := 367286, stop := 367298 ), rec( content := " of a digraph is a set of mutual\ ly adjacent\n vertices of the digraph, and an ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "independent set", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 367401, root := ~, start := 367379, stop := 367400 ), rec( content := " is a set of mutually\n non-a\ djacent vertices of the digraph. A ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal clique", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 367488, root := ~, start := 367467, stop := 367487 ), rec( content := " is a clique\n which is not p\ roperly contained in another clique, and a ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal\n independen\ t set", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 367598, root := ~, start := 367562, stop := 367597 ), rec( content := " is an independent set which is \ not properly contained\n in another independent set. Using this definition\ in ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ", cliques\n and independent s\ ets are both permitted, but not required, to contain vertices\n at which th\ ere is a loop. Another name for a clique is a ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "complete\n subgraph" , count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "E", next := 367906, root := ~, start := 367876, stop := 367905 ), rec( content := ".\n ", count := [ 8, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "P", next := 367916, root := ~, start := 367912, stop := 367915 ), rec( content := "\n\n ", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " provides extensive functionalit\ y for computing cliques and\n independent sets of a digraph, whether maxima\ l or not. The fundamental\n algorithm used in most of the methods in ", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " to calculate cliques\n and i\ ndependent sets is a version of the Bron-Kerbosch algorithm.\n Calculating \ the cliques and independent sets of a digraph is a well-known\n and hard pr\ oblem, and searching for cliques or independent sets in a\n digraph can be \ very lengthy, even for a digraph with a small number of\n vertices. ", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " uses several strategies to incr\ ease the performance\n of these calculations.\n ", count := [ 8, 0, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "P", next := 368554, root := ~, start := 368550, stop := 368553 ), rec( content := "\n\n From the definition of c\ liques and independent sets, it follows that the\n presence of loops and mu\ ltiple edges in a digraph is irrelevant to the\n existence of cliques and i\ ndependent sets in the digraph. See ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "DigraphHasLoops" ), content := 0, count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "Ref", next := 368810, root := ~, start := 368775, stop := 368809 ), rec( content := " and ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsMultiDigraph" ), content := 0, count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "Ref", next := 368843, root := ~, start := 368815, stop := 368842 ), rec( content := " for more\n information about\ these properties. Therefore given a digraph\n ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "A", next := 368938, root := ~, start := 368924, stop := 368937 ), rec( content := ", the cliques and independent se\ ts of ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "A", next := 368990, root := ~, start := 368976, stop := 368989 ), rec( content := " are\n equal to the cliques a\ nd independent sets of the digraph:\n\n ", count := [ 8, 0, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphRe\ moveLoops(DigraphRemoveAllMultipleEdges(", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "C", next := 369146, root := ~, start := 369090, stop := 369145 ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "A", next := 369160, root := ~, start := 369146, stop := 369159 ), rec( attributes := rec( ), content := [ rec( content := "))", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "C", next := 369169, root := ~, start := 369160, stop := 369168 ), rec( content := ".\n ", count := [ 8, 0, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 7 ], name := "Item", next := 369184, root := ~, start := 369075, stop := 369183 ) ], count := [ 8, 0, 0, 6 ], mathmode := "Text", name := "List", next := 369196, root := ~, start := 369062, stop := 369195 ), rec( content := "\n\n See ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphRemoveLoops" ), content := 0, count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "Ref", next := 369238, root := ~, start := 369206, stop := 369237 ), rec( content := " and ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphRemoveAllMultipleEdges" ), content := 0, count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "Ref", next := 369292, root := ~, start := 369243, stop := 369291 ), rec( content := " for more information about thes\ e\n attributes. Furthermore, the cliques of this digraph are equal to the\ \n cliques of the digraph formed by removing any edge ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[u,v]", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "C", next := 369468, root := ~, start := 369456, stop := 369467 ), rec( content := " for which\n the correspondin\ g reverse edge ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[v,u]", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "C", next := 369526, root := ~, start := 369514, stop := 369525 ), rec( content := " is not present. Therefore,\n \ overall, the cliques of ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "A", next := 369597, root := ~, start := 369583, stop := 369596 ), rec( content := " are equal to the cliques of the\ \n symmetric digraph:\n\n ", count := [ 8, 0, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "MaximalSy\ mmetricSubdigraphWithoutLoops(", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "C", next := 369732, root := ~, start := 369686, stop := 369731 ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "A", next := 369746, root := ~, start := 369732, stop := 369745 ), rec( attributes := rec( ), content := [ rec( content := ")", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "C", next := 369754, root := ~, start := 369746, stop := 369753 ), rec( content := ".\n ", count := [ 8, 0, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 11 ], name := "Item", next := 369769, root := ~, start := 369671, stop := 369768 ) ], count := [ 8, 0, 0, 10 ], mathmode := "Text", name := "List", next := 369781, root := ~, start := 369658, stop := 369780 ), rec( content := "\n\n See ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "MaximalSymmetricSubdigraphWitho\ utLoops" ), content := 0, count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "Ref", next := 369843, root := ~, start := 369791, stop := 369842 ), rec( content := " for more\n information about\ this. The ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "AutomorphismGroup", Label := "for a digraph" ), content := 0, count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "Ref", next := 369945, root := ~, start := 369885, stop := 369944 ), rec( content := " of this symmetric digraph conta\ ins the\n automorphism group of ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "A", next := 370025, root := ~, start := 370011, stop := 370024 ), rec( content := " as a subgroup. By performing t\ he\n search for maximal cliques with the help of this larger automorphism g\ roup\n to reduce the search space, the computation time may be reduced. Th\ e\n functions and attributes which return representatives of cliques of\n \ ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "A", next := 370303, root := ~, start := 370289, stop := 370302 ), rec( content := " will return orbit representativ\ es of cliques under the\n action of the automorphism group of the ", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal symmetric subdigr\ aph\n without loops", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "E", next := 370458, root := ~, start := 370403, stop := 370457 ), rec( content := " on sets of vertices.", count := [ 8, 0, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "P", next := 370483, root := ~, start := 370479, stop := 370482 ), rec( content := "\n\n The independent sets of \ a digraph are equal to the independent sets of the\n ", count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphSymmetricClosure" ), content := 0, count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "Ref", next := 370605, root := ~, start := 370568, stop := 370604 ), rec( content := ". Therefore, overall, the indep\ endent\n sets of ", count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "A", next := 370670, root := ~, start := 370656, stop := 370669 ), rec( content := " are equal to the independent se\ ts of the symmetric\n digraph:\n\n ", count := [ 8, 0, 0, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphSy\ mmetricClosure(DigraphRemoveLoops(DigraphRemoveAllMultipleEdges(\n ", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "C", next := 370857, root := ~, start := 370768, stop := 370856 ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "A", next := 370871, root := ~, start := 370857, stop := 370870 ), rec( attributes := rec( ), content := [ rec( content := ")))", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "C", next := 370881, root := ~, start := 370871, stop := 370880 ), rec( content := ".\n ", count := [ 8, 0, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 16 ], name := "Item", next := 370896, root := ~, start := 370753, stop := 370895 ) ], count := [ 8, 0, 0, 15 ], mathmode := "Text", name := "List", next := 370908, root := ~, start := 370740, stop := 370907 ), rec( content := "\n\n Again, the automorphism \ group of this symmetric digraph contains the\n automorphism group of ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "A", next := 371023, root := ~, start := 371009, stop := 371022 ), rec( content := ". Since a search for independent\ sets\n is equal to a search for cliques in the ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphDual" ), content := 0, count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "Ref", next := 371130, root := ~, start := 371105, stop := 371129 ), rec( content := ", the\n methods used in ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " usually transform a search for \ independent sets\n into a search for cliques, as described above. The func\ tions and\n attributes which return representatives of independent sets of\ \n ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "A", next := 371370, root := ~, start := 371356, stop := 371369 ), rec( content := " will return orbit representativ\ es of independent sets under\n the action of the automorphism group of the \ ", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "symmetric closure", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "E", next := 371503, root := ~, start := 371479, stop := 371502 ), rec( content := " of the\n digraph formed by r\ emoving loops and multiple edges.", count := [ 8, 0, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "P", next := 371571, root := ~, start := 371567, stop := 371570 ), rec( content := "\n\n Please note that in ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := ", cliques and independent sets a\ re not\n required to be maximal. Some authors use the word clique to mean\n\ ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "E", next := 371733, root := ~, start := 371719, stop := 371732 ), rec( content := " clique, and some authors use th\ e phrase independent set to\n mean ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "maximal", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "E", next := 371816, root := ~, start := 371802, stop := 371815 ), rec( content := " independent set, but please not\ e that ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " does\n not use this definiti\ on.\n ", count := [ 8, 0, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 0, 0, 20 ], mathmode := "Text", name := "P", next := 371908, root := ~, start := 371904, stop := 371907 ), rec( content := "\n\n ", count := [ 8, 0, 0, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Finding cliques", count := [ 8, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 371955, root := ~, start := 371921, stop := 371954 ), rec( content := "\n ", count := [ 8, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, l", Name := "IsClique" ), content := 0, count := [ 8, 1, 1, 2 ], mathmode := "Text", name := "Oper", next := 372013, root := ~, start := 371973, stop := 372012 ), rec( attributes := rec( Arg := "digraph, l", Name := "IsMaximalClique\ " ), content := 0, count := [ 8, 1, 1, 4 ], mathmode := "Text", name := "Oper", next := 372061, root := ~, start := 372014, stop := 372060 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 372082, root := ~, start := 372071, stop := 372081 ), rec( content := " or ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 372098, root := ~, start := 372086, stop := 372097 ), rec( content := ".", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "Returns", next := 372109, root := ~, start := 372062, stop := 372108 ), rec( attributes := rec( ), content := [ rec( content := "\n If ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 372143, root := ~, start := 372129, stop := 372142 ), rec( content := " is a di\ graph and ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 372169, root := ~, start := 372161, stop := 372168 ), rec( content := " is a du\ plicate-free list of\n vertices of ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 372226, root := ~, start := 372212, stop := 372225 ), rec( content := ", then\n\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sClique(", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 372251, root := ~, start := 372235, stop := 372250 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 372265, root := ~, start := 372251, stop := 372264 ), rec( attributes := rec( ), content := [ rec( content := ",\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 372273, root := ~, start := 372265, stop := 372272 ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 372281, root := ~, start := 372273, stop := 372280 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 372289, root := ~, start := 372281, stop := 372288 ), rec( content := " returns\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 372309, root := ~, start := 372298, stop := 372308 ), rec( content := "\n if\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 372325, root := ~, start := 372317, stop := 372324 ), rec( content := " is a ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ lique", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "E", next := 372344, root := ~, start := 372331, stop := 372343 ), rec( content := " of ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 372362, root := ~, start := 372348, stop := 372361 ), rec( content := " and ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 372379, root := ~, start := 372367, stop := 372378 ), rec( content := " if it i\ s\n not. Similarly,\n ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximalClique(", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 372437, root := ~, start := 372414, stop := 372436 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 372451, root := ~, start := 372437, stop := 372450 ), rec( attributes := rec( ), content := [ rec( content := ",\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 372459, root := ~, start := 372451, stop := 372458 ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 372467, root := ~, start := 372459, stop := 372466 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "C", next := 372475, root := ~, start := 372467, stop := 372474 ), rec( content := " returns\ \n ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 372499, root := ~, start := 372488, stop := 372498 ), rec( content := " if ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 372511, root := ~, start := 372503, stop := 372510 ), rec( content := " is a ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal clique", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "E", next := 372538, root := ~, start := 372517, stop := 372537 ), rec( content := " of ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "A", next := 372556, root := ~, start := 372542, stop := 372555 ), rec( content := " and\n \ ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "K", next := 372577, root := ~, start := 372565, stop := 372576 ), rec( content := " if it i\ s not. ", count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "P", next := 372597, root := ~, start := 372593, stop := 372596 ), rec( content := "\n\n \ A ", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ lique", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "E", next := 372618, root := ~, start := 372605, stop := 372617 ), rec( content := " of a di\ graph is a set of mutually adjacent vertices of the\n digraph. A ", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal clique", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "E", next := 372714, root := ~, start := 372693, stop := 372713 ), rec( content := " is a cl\ ique that is not properly\n contained in another clique. A clique is permit\ ted, but not required, to\n contain vertices at which there is a loop.\n \ ", count := [ 8, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(4);;\ngap> IsClique(D, [1, 3, 2]);\ntrue\ngap> IsMa\ ximalClique(D, [1, 3, 2]);\nfalse\ngap> IsMaximalClique(D, DigraphVertices(D))\ ;\ntrue\ngap> D := Digraph([[1, 2, 4, 4], [1, 3, 4], [2, 1], [1, 2]]);\n\ngap> IsClique(D, [2, 3, 4]);\nfa\ lse\ngap> IsMaximalClique(D, [1, 2, 4]);\ntrue\ngap> D := CompleteDigraph(IsMu\ tableDigraph, 4);;\ngap> IsClique(D, [1, 3, 2]);\ntrue", count := [ 8, 1, 1, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 1, 7 ], mathmode := "Text", name := "Example", next := 373336, root := ~, start := 372876, stop := 373335 ), rec( content := "\n ", count := [ 8, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 1, 5 ], mathmode := "Text", name := "Description", next := 373353, root := ~, start := 372110, stop := 373352 ) ], count := [ 8, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 373367, root := ~, start := 371960, stop := 373366 ), rec( content := "\n\n ", count := [ 8, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, hook, u\ ser_param, limit, include,\n exclude, max, size, reps", Name := "CliquesFinder" ), content := 0, count := [ 8, 1, 2, 2 ], mathmode := "Text", name := "Func", next := 373494, root := ~, start := 373388, stop := 373493 ), rec( attributes := rec( ), content := [ rec( content := "The argum\ ent ", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "u\ ser_param", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "A", next := 373536, root := ~, start := 373519, stop := 373535 ), rec( content := ".", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "Returns", next := 373547, root := ~, start := 373497, stop := 373546 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function finds cliques of the digraph ", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "A", next := 373625, root := ~, start := 373611, stop := 373624 ), rec( content := " subject\ to the\n conditions imposed by the other arguments as described below. Not\ e\n that a clique is represented by the immutable list of the vertices that\ \n it contains.\n ", count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "P", next := 373813, root := ~, start := 373809, stop := 373812 ), rec( content := "\n\n \ Let ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "C", next := 373831, root := ~, start := 373823, stop := 373830 ), rec( content := " denote \ the automorphism group of the maximal symmetric\n subdigraph of ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "A", next := 373919, root := ~, start := 373905, stop := 373918 ), rec( content := " without\ loops (see ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 8, 1, 2, 4 ] , mathmode := "Text", name := "Ref", next := 373998, root := ~, start := 373939, stop := 373997 ), rec( content := " and ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraphWithoutLoops" ), content := 0, count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "Ref", next := 374061, root := ~, start := 374003, stop := 374060 ), rec( content := ").\n\n \ ", count := [ 8, 1, 2, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( attributes := rec( ), content := [ rec( content :\ = "hook", count := [ 8, 1, 2, 5 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 8, 1, 2, 5 ], mathmode := "Text", name := "A", next := 37\ 4099, root := ~, start := 374088, stop := 374098 ) ], count := [ 8, 1, 2, 5 ], name := "Mar\ k", next := 374106, root := ~, start := 374082, stop := 374105 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be a f\ unction or ", count := [ 8, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 8, 1, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 6 ], mathmode := "Text", name := "K", next := 374177, root := \ ~, start := 374166, stop := 374176 ), rec( content := ".", count := [ 8, 1, 2,\ 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := 0, count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "P", next\ := 374182, root := ~, start := 374178, stop := 374181 ), rec( content := "\n\ \n If ", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "hook", coun\ t := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 7 ], mathmode := "Text", name := "A", next := 374206, root := \ ~, start := 374195, stop := 374205 ), rec( content := " is a function, then it\ should have two arguments\n ", count := [ 8, 1, 2, 7 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "user_param", count := [ 8, 1, 2, 7 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name\ := "A", next := 374281, root := ~, start := 374264, stop := 374280 ), rec( cont\ ent := " (see below) and a clique ", count := [ 8, 1, 2, 7 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "c", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "C", next\ := 374315, root := ~, start := 374307, stop := 374314 ), rec( content := ". T\ he function\n ", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes\ := rec( ), content := [ rec( content := "hook", count := [ 8, 1, 2, 7 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 7 ], mat\ hmode := "Text", name := "A", next := 374352, root := ~, start := 374341, stop\ := 374351 ), rec( content := "(", count := [ 8, 1, 2, 7 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec(\ content := "user_param", count := [ 8, 1, 2, 7 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name :=\ "A", next := 374370, root := ~, start := 374353, stop := 374369 ), rec( conte\ nt := ", c)", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "C", next :\ = 374378, root := ~, start := 374338, stop := 374377 ), rec( content := " is c\ alled every time a new\n clique ", count := [ 8, 1, 2, 7 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "c", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "\ C", next := 374429, root := ~, start := 374421, stop := 374428 ), rec( content\ := " is found by ", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ liquesFinder", count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 7 ], mathmode := "Text", name := "C", next \ := 374462, root := ~, start := 374442, stop := 374461 ), rec( content := ".", coun\ t := [ 8, 1, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 8, 1, 2, 8 ], mathmode := "Text", name\ := "P", next := 374467, root := ~, start := 374463, stop := 374466 ), rec( cont\ ent := "\n\n If ", count := [ 8, 1, 2, 8 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "hook", count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "A", next := 3\ 74491, root := ~, start := 374480, stop := 374490 ), rec( content := " is ", coun\ t := [ 8, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 1, 2, 8 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 8 ] , math\ mode := "Text", name := "K", next := 374506, root := ~, start := 374495, stop \ := 374505 ), rec( content := ", then a default function is used that\n \ simply adds every new clique found by ", count := [ 8, 1, 2, 8 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "CliquesFinder", count := [ 8, 1, 2, 8 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 8 ], mathmode := "Text", name\ := "C", next := 374611, root := ~, start := 374591, stop := 374610 ), rec( cont\ ent := " to\n ", count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "user_param", count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 8 ], mathmode := "Text", name := "A", next \ := 374640, root := ~, start := 374623, stop := 374639 ), rec( content := ", wh\ ich must be a list in this case.\n ", count := [ 8, 1, 2, 8 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 6 ], name := "Ite\ m", next := 374690, root := ~, start := 374113, stop := 374689 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "user_param", count := [ 8, 1, 2, 9 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 8, 1, 2, 9 ], mathmode := "Text", name := "A", next\ := 374721, root := ~, start := 374704, stop := 374720 ) ], count := [ 8, 1, 2, 9 ], name := "Mar\ k", next := 374728, root := ~, start := 374698, stop := 374727 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If ", count := [ 8, 1, 2, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "hook", count := [ 8, 1, 2, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode\ := "Text", name := "A", next := 374764, root := ~, start := 374753, stop := 3\ 74763 ), rec( content := " is a function, then ", count := [ 8, 1, 2, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "user_param", count := [ 8, 1, 2, 10 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := \ "Text", name := "A", next := 374802, root := ~, start := 374785, stop := 37480\ 1 ), rec( content := " can be any ", count := [ 8, 1, 2, 10 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "GAP", count := [ 8, 1, 2, 10 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "Text", name := "\ Package", next := 23, root := ~, start := 1, stop := 22 ), rec( content := "\n\ object. The object ", count := [ 8, 1, 2, 10 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "user_param", count := [ 8, 1, 2, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "Text", name := "A" , next\ := 374864, root := ~, start := 374847, stop := 374863 ), rec( content := " is\ used as the first argument for\n the function ", count := [ 8, 1, 2, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "hook", count := [ 8, 1, 2, 10 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode\ := "Text", name := "A", next := 374931, root := ~, start := 374920, stop := 3\ 74930 ), rec( content := ". For example, ", count := [ 8, 1, 2, 10 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "user_param", count := [ 8, 1, 2, 10 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "Te\ xt", name := "A", next := 374963, root := ~, start := 374946, stop := 374962 ) , rec( cont\ ent := " might be a\n list, and ", count := [ 8, 1, 2, 10 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( attributes := rec( ), content := [ rec( content := "hook", count := \ [ 8, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :=\ [ 8, 1, 2, 10 ], mathmode := "Text", name := "A", next := 375007, root := ~, \ start := 374996, stop := 375006 ), rec( content := "(", count := [ 8, 1, 2, 10\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "user_param", count := [ 8, 1, 2, 10 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mat\ hmode := "Text", name := "A", next := 375025, root := ~, start := 375008, stop\ := 375024 ), rec( content := ", c)", count := [ 8, 1, 2, 10 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "\ Text", name := "C", next := 375033, root := ~, start := 374993, stop := 375032\ ), rec( content := " might add the size\n of the clique ", count := [ \ 8, 1, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "c", count := [ 8, 1, 2, 10 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], math\ mode := "Text", name := "C", next := 375083, root := ~, start := 375075, stop \ := 375082 ), rec( content := " to the list ", count := [ 8, 1, 2, 10 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "user_param", count := [ 8, 1, 2, 10 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], mathmode := "\ Text", name := "A", next := 375113, root := ~, start := 375096, stop := 375112\ ), rec( content := ". ", count := [ 8, 1, 2, 10 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ \ 8, 1, 2, 11 ], mathmode := "Text", name := "P", next := 375119, root := ~, star\ t := 375115, stop := 375118 ), rec( content := "\n\n If the value of ", coun\ t := [ 8, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "hook", count := [ 8, 1, 2, 11\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 1\ 1 ], mathmode := "Text", name := "A", next := 375156, root := ~, start := 3751\ 45, stop := 375155 ), rec( content := " is ", count := [ 8, 1, 2, 11 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "fail", count := [ 8, 1, 2, 11 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 11 ], mathmode := "Text", name\ := "K", next := 375171, root := ~, start := 375160, stop := 375170 ), rec( cont\ ent := ", then the value of\n ", count := [ 8, 1, 2, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "user_param", count := [ 8, 1, 2, 11 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 11 ], mathmode := "Text", name\ := "A", next := 375216, root := ~, start := 375199, stop := 375215 ), rec( cont\ ent := " must be a list.\n ", count := [ 8, 1, 2, 11 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 10 ], name := "Item", next := 3752\ 46, root := ~, start := 374735, stop := 375245 ), rec( attributes := rec( ), content := [ rec( attr\ ibutes := rec( ), content := [ rec( content := "limit", count := [ 8, 1, 2, 1\ 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, \ 12 ], mathmode := "Text", name := "A", next := 375272, root := ~, start := 375\ 260, stop := 375271 ) ], count := [ 8, 1, 2, 12 ], name := "Mark", next := 3752\ 79, root := ~, start := 375254, stop := 375278 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n This argument should be a positive integer or ", count := [ \ 8, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "infinity", count := [ 8, 1, 2, 1\ 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, \ 13 ], mathmode := "Text", name := "K", next := 375362, root := ~, start := 375\ 347, stop := 375361 ), rec( content := ".\n ", count := [ 8, 1, 2, 13 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "CliquesFinder", count := [ 8, 1, 2, 13 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 13 ], mathmode \ := "Text", name := "C", next := 375392, root := ~, start := 375372, stop := 37\ 5391 ), rec( content := " will return after it has found\n ", count := [ \ 8, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "limit", count := [ 8, 1, 2, 13 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 13 ] , math\ mode := "Text", name := "A", next := 375444, root := ~, start := 375432, stop \ := 375443 ), rec( content := " cliques or the search is complete.\n ", coun\ t := [ 8, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 13 ], name := "Ite\ m", next := 375493, root := ~, start := 375286, stop := 375492 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "include", count := [ 8, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 14 ], mathmode := "Text", name := "A", next\ := 375521, root := ~, start := 375507, stop := 375520 ), rec( content := " an\ d ", count := [ 8, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "exclude", count \ := [ 8, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 8, 1, 2, 14 ], mathmode := "Text", name := "A", next := 375540, root := \ ~, start := 375526, stop := 375539 ) ], count := [ 8, 1, 2, 14 ], name := "Mar\ k", next := 375547, root := ~, start := 375501, stop := 375546 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n These arguments should each\ be a (possibly empty) duplicate-free list\n of vertices of ", count :=\ [ 8, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "digraph", count := [ 8, 1, 2,\ 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2\ , 15 ], mathmode := "Text", name := "A", next := 375676, root := ~, start := 3\ 75662, stop := 375675 ), rec( content := " (i.e. positive integers less than t\ he\n number of vertices of ", count := [ 8, 1, 2, 15 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "digraph", count := [ 8, 1, 2, 15 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 15 ], mathmode := "Text", name \ := "A", next := 375759, root := ~, start := 375745, stop := 375758 ), rec( cont\ ent := "). ", count := [ 8, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 2, 16 ], math\ mode := "Text", name := "P", next := 375766, root := ~, start := 375762, stop \ := 375765 ), rec( content := "\n\n ", count := [ 8, 1, 2, 16 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "CliquesFinder", count := [ 8, 1, 2, 16 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 16 ], mathmode :=\ "Text", name := "C", next := 375796, root := ~, start := 375776, stop := 3757\ 95 ), rec( content := " will only look for cliques containing all of the\n \ vertices in ", count := [ 8, 1, 2, 16 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "in\ clude", count := [ 8, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root :\ = ~ ) ], count := [ 8, 1, 2, 16 ], mathmode := "Text", name := "A", next := 37\ 5880, root := ~, start := 375866, stop := 375879 ), rec( content := " and cont\ aining none of the vertices in\n ", count := [ 8, 1, 2, 16 ], mathmode \ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content \ := [ rec( content := "exclude", count := [ 8, 1, 2, 16 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 16 ], mathmode := "Text", name\ := "A", next := 375942, root := ~, start := 375928, stop := 375941 ), rec( cont\ ent := ". ", count := [ 8, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 2, 17 ], math\ mode := "Text", name := "P", next := 375948, root := ~, start := 375944, stop \ := 375947 ), rec( content := "\n\n Note that the search may be much mor\ e efficient if each of these lists\n is invariant under the action of " , coun\ t := [ 8, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G", count := [ 8, 1, 2, 17 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 17 ] , math\ mode := "Text", name := "C", next := 376078, root := ~, start := 376070, stop \ := 376077 ), rec( content := " on sets of vertices.\n ", count := [ 8, 1,\ 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 15 ], name := "Ite\ m", next := 376113, root := ~, start := 375554, stop := 376112 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "max", count := [ 8, 1, 2, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 1, 2, 18 ], mathmode := "Text", name := "A", next := \ 376137, root := ~, start := 376127, stop := 376136 ) ], count := [ 8, 1, 2, 18 ], name := "Mar\ k", next := 376144, root := ~, start := 376121, stop := 376143 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be ", coun\ t := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "true", count := [ 8, 1, 2, 19\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 1\ 9 ], mathmode := "Text", name := "K", next := 376201, root := ~, start := 3761\ 90, stop := 376200 ), rec( content := " or ", count := [ 8, 1, 2, 19 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "false", count := [ 8, 1, 2, 19 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 19 ], mathmode := "Text" , name\ := "K", next := 376217, root := ~, start := 376205, stop := 376216 ), rec( cont\ ent := ". If ", count := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "max"\ , count := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 8, 1, 2, 19 ], mathmode := "Text", name := "A", next := 376233, root :=\ ~, start := 376223, stop := 376232 ), rec( content := " is\n true then\ ", count := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "CliquesFinder", count := [ 8,\ 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8\ , 1, 2, 19 ], mathmode := "Text", name := "C", next := 376275, root := ~, star\ t := 376255, stop := 376274 ), rec( content := " will only search for ", count\ := [ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "maximal", count := [ 8, 1, 2,\ 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2\ , 19 ], mathmode := "Text", name := "E", next := 376311, root := ~, start := 3\ 76297, stop := 376310 ), rec( content := "\n cliques. If ", count := [ \ 8, 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "max", count := [ 8, 1, 2, 19 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 19 ], math\ mode := "Text", name := "K", next := 376342, root := ~, start := 376332, stop \ := 376341 ), rec( content := " is ", count := [ 8, 1, 2, 19 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "false", count := [ 8, 1, 2, 19 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 19 ], mathmode := "Text", name :=\ "K", next := 376358, root := ~, start := 376346, stop := 376357 ), rec( conte\ nt := " then non-maximal cliques may be\n found.\n ", count := [ 8\ , 1, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 19 ], name := "Ite\ m", next := 376419, root := ~, start := 376151, stop := 376418 ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( content \ := "size", count := [ 8, 1, 2, 20 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 8, 1, 2, 20 ], mathmode := "Text", name := "A", next :=\ 376444, root := ~, start := 376433, stop := 376443 ) ], count := [ 8, 1, 2, 20 ], name := "Mar\ k", next := 376451, root := ~, start := 376427, stop := 376450 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n This argument should be ", coun\ t := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 1, 2, 21\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 2\ 1 ], mathmode := "Text", name := "K", next := 376508, root := ~, start := 3764\ 97, stop := 376507 ), rec( content := " or a positive integer.\n If ", coun\ t := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "size", count := [ 8, 1, 2, 21\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 2\ 1 ], mathmode := "Text", name := "A", next := 376554, root := ~, start := 3765\ 43, stop := 376553 ), rec( content := " is a positive integer then ", count :=\ [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "CliquesFinder", count := [ 8,\ 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8\ , 1, 2, 21 ], mathmode := "Text", name := "C", next := 376602, root := ~, star\ t := 376582, stop := 376601 ), rec( content := " will\n only search for\ cliques that contain precisely ", count := [ 8, 1, 2, 21 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "size", count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "A\ ", next := 376674, root := ~, start := 376663, stop := 376673 ), rec( content \ := " vertices.\n If ", count := [ 8, 1, 2, 21 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "size", count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "A", next\ := 376707, root := ~, start := 376696, stop := 376706 ), rec( content := " is\ ", count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 1, 2, 21\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 2\ 1 ], mathmode := "Text", name := "K", next := 376722, root := ~, start := 3767\ 11, stop := 376721 ), rec( content := " then cliques of any size may be found.\ \n ", count := [ 8, 1, 2, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 1, 2, 21 ], name := "Item", next := 376775, root := ~, start := 376\ 458, stop := 376774 ), rec( attributes := rec( ), content := [ rec( attribute\ s := rec( ), content := [ rec( content := "reps", count := [ 8, 1, 2, 22 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 22 ], math\ mode := "Text", name := "A", next := 376800, root := ~, start := 376789, stop \ := 376799 ) ], count := [ 8, 1, 2, 22 ], name := "Mark", next := 376807, root := ~, start := 376\ 783, stop := 376806 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n This argument should be ", count := [ 8, 1, 2, 23 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "true", count := [ 8, 1, 2, 23 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 23 ], mathmode := "Text", name :=\ "K", next := 376864, root := ~, start := 376853, stop := 376863 ), rec( conte\ nt := " or ", count := [ 8, 1, 2, 23 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "false", cou\ nt := [ 8, 1, 2, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 23 ], mathmode := "Text", name := "K", next := 376880, root :=\ ~, start := 376868, stop := 376879 ), rec( content := ".", count := [ 8, 1, 2\ , 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := 0, count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "P" , next\ := 376885, root := ~, start := 376881, stop := 376884 ), rec( content := "\n\ \n If ", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "reps", coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "A", next := 376909, root :=\ ~, start := 376898, stop := 376908 ), rec( content := " is ", count := [ 8, 1\ , 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "true", count := [ 8, 1, 2, 24 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 24 ], math\ mode := "Text", name := "K", next := 376924, root := ~, start := 376913, stop \ := 376923 ), rec( content := " then the arguments ", count := [ 8, 1, 2, 24 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "include", count := [ 8, 1, 2, 24 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 24 ], mathmode := "Te\ xt", name := "A", next := 376958, root := ~, start := 376944, stop := 376957 ) , rec( cont\ ent := " and\n ", count := [ 8, 1, 2, 24 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "exclude", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "A", next\ := 376985, root := ~, start := 376971, stop := 376984 ), rec( content := " ar\ e each required to be invariant under the action of\n ", count := [ 8, \ 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "G", count := [ 8, 1, 2, 24 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 24 ], math\ mode := "Text", name := "C", next := 377056, root := ~, start := 377048, stop \ := 377055 ), rec( content := " on sets of vertices. In this case, ", count :=\ [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "CliquesFinder", count := [ 8,\ 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8\ , 1, 2, 24 ], mathmode := "Text", name := "C", next := 377113, root := ~, star\ t := 377093, stop := 377112 ), rec( content := " will\n find representa\ tives of the orbits of the desired cliques under the\n action of ", coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G", count := [ 8, 1, 2, 24 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 24 ] , math\ mode := "Text", name := "C", next := 377221, root := ~, start := 377213, stop \ := 377220 ), rec( content := ", ", count := [ 8, 1, 2, 24 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "although representatives may be returned that\n are in \ the same orbit", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "E", next\ := 377307, root := ~, start := 377223, stop := 377306 ), rec( content := ".\n\ If ", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "reps", coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "A", next := 377331, root :=\ ~, start := 377320, stop := 377330 ), rec( content := " is false then ", coun\ t := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "CliquesFinder", count := [ 8,\ 1, 2, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8\ , 1, 2, 24 ], mathmode := "Text", name := "C", next := 377366, root := ~, star\ t := 377346, stop := 377365 ), rec( content := " will not take this into\n \ consideration.", count := [ 8, 1, 2, 24 ], mathmode := "Text", name := "PC\ DATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1,\ 2, 25 ], mathmode := "Text", name := "P", next := 377417, root := ~, start :=\ 377413, stop := 377416 ), rec( content := "\n\n For a digraph such tha\ t ", count := [ 8, 1, 2, 25 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := [ rec( content := "G", count := [ 8\ , 1, 2, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 8, 1, 2, 25 ], mathmode := "Text", name := "C", next := 377459, root := ~, star\ t := 377451, stop := 377458 ), rec( content := " is non-trivial, the search fo\ r\n clique representatives can be much more efficient than the search f\ or\n all cliques.\n ", count := [ 8, 1, 2, 25 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 23 ], name := "Item", next := 3776\ 03, root := ~, start := 376814, stop := 377602 ) ], count := [ 8, 1, 2, 5 ], mathmode := "Text", name := "List", next := 377615, root := ~, start := 374069, stop := 377614 ), rec( content := "\n\n \ This function uses a version of the Bron-Kerbosch algorithm.\n ", count := [ 8, 1, 2, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(5);\n\n\ gap> user_param := [];;\ngap> f := function(a, b) # Calculate size of clique\ \n> AddSet(user_param, Size(b));\n> end;;\ngap> CliquesFinder(D, f, user_par\ am, infinity, [], [], false, fail,\n> true);\n[ 1, 2, 3, 4, 5\ ]\ngap> CliquesFinder(D, fail, [], 5, [2, 4], [3], false, fail, false);\n[ [ \ 2, 4 ], [ 1, 2, 4 ], [ 2, 4, 5 ], [ 1, 2, 4, 5 ] ]\ngap> CliquesFinder(D, fail\ , [], 2, [2, 4], [3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ] ]\ngap> Cl\ iquesFinder(D, fail, [], infinity, [], [], true, 5, false);\n[ [ 1, 2, 3, 4, 5\ ] ]\ngap> CliquesFinder(D, fail, [], infinity, [1, 3], [], false, 3, false);\ \n[ [ 1, 2, 3 ], [ 1, 3, 4 ], [ 1, 3, 5 ] ]\ngap> CliquesFinder(D, fail, [], i\ nfinity, [1, 3], [], true, 3, false);\n[ ]\ngap> D := CompleteDigraph(IsMutab\ leDigraph, 5);\n\ngap> user_param :\ = [];;\ngap> f := function(a, b) # Calculate size of clique\n> AddSet(user_\ param, Size(b));\n> end;;\ngap> CliquesFinder(D, f, user_param, infinity, [], \ [], false, fail,\n> true);\n[ 1, 2, 3, 4, 5 ]\ngap> CliquesFi\ nder(D, fail, [], 5, [2, 4], [3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 \ ], [ 2, 4, 5 ], [ 1, 2, 4, 5 ] ]\ngap> CliquesFinder(D, fail, [], 2, [2, 4], [\ 3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ] ]\ngap> CliquesFinder(D, fai\ l, [], infinity, [], [], true, 5, false);\n[ [ 1, 2, 3, 4, 5 ] ]\ngap> Cliques\ Finder(D, fail, [], infinity, [1, 3], [], false, 3, false);\n[ [ 1, 2, 3 ], [ \ 1, 3, 4 ], [ 1, 3, 5 ] ]\ngap> CliquesFinder(D, fail, [], infinity, [1, 3], []\ , true, 3, false);\n[ ]", count := [ 8, 1, 2, 28 ], name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 28 ], mathmode := "Text", name := "Example", next := 379340, root := ~, start := 377686, stop := 379339 ), rec( content := "\n ", count := [ 8, 1, 2, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 2, 3 ], mathmode := "Text", name := "Description", next := 379357, root := ~, start := 373550, stop := 379356 ) ], count := [ 8, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 379371, root := ~, start := 373373, stop := 379370 ), rec( content := "\n\n ", count := [ 8, 1, 2, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[, size]]]", Name := "DigraphClique" ), content := 0, count := [ 8, 1, 3, 2 ], mathmode := "Text", name := "Func", next := 379464, root := ~, start := 379392, stop := 379463 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n size]]]", Name := "DigraphMaximalClique" ), content := 0, count := [ 8, 1, 3, 4 ], mathmode := "Text", name := "Func", next := 379550, root := ~, start := 379467, stop := 379549 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of positive integers, or ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "K", next := 379616, root := ~, start := 379605, stop := 379615 ), rec( content := ".", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Returns", next := 379627, root := ~, start := 379553, stop := 379626 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "A", next := 379665, root := ~, start := 379651, stop := 379664 ), rec( content := " is a di\ graph, then these functions returns a clique\n of ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "A", next := 379739, root := ~, start := 379725, stop := 379738 ), rec( content := " if one \ exists that satisfies the arguments, else it\n returns ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "K", next := 379815, root := ~, start := 379804, stop := 379814 ), rec( content := ". A cli\ que is defined by the set of vertices that it\n contains; see ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsCliqu\ e" ), content := 0, count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Ref", next := 379909, root := ~, start := 379887, stop := 379908 ), rec( content := " and ", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alClique" ), content := 0, count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Ref", next := 379943, root := ~, start := 379914, stop := 379942 ), rec( content := ".", count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "P", next := 379948, root := ~, start := 379944, stop := 379947 ), rec( content := "\n\n \ The optional arguments ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 379991, root := ~, start := 379977, stop := 379990 ), rec( content := " and ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 380010, root := ~, start := 379996, stop := 380009 ), rec( content := " must ea\ ch be a\n (possibly empty) duplicate-free list of vertices of ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 380096, root := ~, start := 380082, stop := 380095 ), rec( content := ", and\n \ the optional argument ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 380139, root := ~, start := 380128, stop := 380138 ), rec( content := " must be\ a positive integer. By default,\n ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 380199, root := ~, start := 380185, stop := 380198 ), rec( content := " and ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 380218, root := ~, start := 380204, stop := 380217 ), rec( content := " are emp\ ty. These functions will search for\n a clique of ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 380292, root := ~, start := 380278, stop := 380291 ), rec( content := " that in\ cludes the vertices of ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 380337, root := ~, start := 380323, stop := 380336 ), rec( content := "\n bu\ t does not include any vertices in ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 380393, root := ~, start := 380379, stop := 380392 ), rec( content := "; if the\ argument\n ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 380426, root := ~, start := 380415, stop := 380425 ), rec( content := " is supp\ lied, then additionally the clique will be required to\n contain precisely \ ", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "A", next := 380522, root := ~, start := 380511, stop := 380521 ), rec( content := " vertice\ s.", count := [ 8, 1, 3, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "P", next := 380536, root := ~, start := 380532, stop := 380535 ), rec( content := "\n\n \ If ", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "A", next := 380559, root := ~, start := 380545, stop := 380558 ), rec( content := " is not \ a clique, then these functions return ", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "K", next := 380616, root := ~, start := 380605, stop := 380615 ), rec( content := ".\n O\ therwise, the functions behave in the following way, depending on the\n num\ ber of arguments:\n\n ", count := [ 8, 1, 3, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "One or two arguments", count := [ 8, 1\ , 3, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 8 ], name := "Mar\ k", next := 380769, root := ~, start := 380736, stop := 380768 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If one or two arguments are\ supplied, then ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Dig\ raphClique", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "C", next :=\ 380854, root := ~, start := 380834, stop := 380853 ), rec( content := " and\n\ ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphMaxi\ malClique", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "C", next := \ 380894, root := ~, start := 380867, stop := 380893 ), rec( content := " greedi\ ly enlarge the clique ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := \ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :\ = "include", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "A", next :=\ 380937, root := ~, start := 380923, stop := 380936 ), rec( content := "\n \ until it can not continue. The result is guaranteed to be a maximal\n \ clique. This may or may not return an answer more quickly than using\n \ ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( Func := "DigraphMaximalCliques" ), content := 0, coun\ t := [ 8, 1, 3, 9 ], mathmode := "Text", name := "Ref", next := 381135, root :\ = ~, start := 381100, stop := 381134 ), rec( content := ". with a limit of 1.\ \n ", count := [ 8, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 1, 3, 9 ], name := "Item", next := 381170, root := ~, start := 380\ 776, stop := 381169 ), rec( attributes := rec( ), content := [ rec( content :\ = "Three arguments", count := [ 8, 1, 3, 10 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 8, 1, 3, 10 ], name := "Mark", next := 381205 , root := ~, start := 381\ 177, stop := 381204 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n If three arguments are supplied, then ", count := [ 8, 1, 3, 11 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphClique", count := [ 8, 1, 3, 11 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 11 ], mathmode \ := "Text", name := "C", next := 381285, root := ~, start := 381265, stop := 38\ 1284 ), rec( content := " greedily\n enlarges the clique ", count := [ \ 8, 1, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "include", count := [ 8, 1, 3, 11\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 1\ 1 ], mathmode := "Text", name := "A", next := 381337, root := ~, start := 3813\ 23, stop := 381336 ), rec( content := " until it can not continue, although\n \ this clique may not be maximal.", count := [ 8, 1, 3, 11 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ 0, count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "P", next := 381417, root\ := ~, start := 381413, stop := 381416 ), rec( content := "\n\n Given t\ hree arguments, ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Di\ graphMaximalClique", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "C\ ", next := 381477, root := ~, start := 381450, stop := 381476 ), rec( content \ := " returns the maximal\n clique returned by ", count := [ 8, 1, 3, 12\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ ), content := [ rec( content := "DigraphMaximalCliques(", count := [ 8, 1, 3\ , 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, \ 3, 12 ], mathmode := "Text", name := "C", next := 381554, root := ~, start := \ 381525, stop := 381553 ), rec( attributes := rec( ), content := [ rec( conten\ t := "digraph", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA"\ , root := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "A", next\ := 381568, root := ~, start := 381554, stop := 381567 ), rec( attributes := re\ c( ), content := [ rec( content := ",\n ", count := [ 8, 1, 3, 12 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 12 ], math\ mode := "Text", name := "C", next := 381585, root := ~, start := 381568, stop \ := 381584 ), rec( attributes := rec( ), content := [ rec( content := "include\ ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "A", next := 381599, root\ := ~, start := 381585, stop := 381598 ), rec( attributes := rec( ), content \ := [ rec( content := ", ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name\ := "C", next := 381608, root := ~, start := 381599, stop := 381607 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 1, 3,\ 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3\ , 12 ], mathmode := "Text", name := "A", next := 381622, root := ~, start := 3\ 81608, stop := 381621 ), rec( attributes := rec( ), content := [ rec( content\ := ", 1)", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "C", next :\ = 381633, root := ~, start := 381622, stop := 381632 ), rec( content := " if o\ ne exists,\n else ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "fail", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "K", next\ := 381673, root := ~, start := 381662, stop := 381672 ), rec( content := ".\n\ ", count := [ 8, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root \ := ~ ) ], count := [ 8, 1, 3, 11 ], name := "Item", next := 381688, root := ~, start := 381\ 212, stop := 381687 ), rec( attributes := rec( ), content := [ rec( content :\ = "Four arguments", count := [ 8, 1, 3, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 3, 13 ], name := "Mark", next := 381722, root := ~, start := 381\ 695, stop := 381721 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n If four arguments are supplied, then ", count := [ 8, 1, 3, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphClique", count := [ 8, 1, 3, 14 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ], mathmode \ := "Text", name := "C", next := 381801, root := ~, start := 381781, stop := 38\ 1800 ), rec( content := " returns the\n clique returned by ", count := [ \ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribu\ tes := rec( ), content := [ rec( content := "DigraphCliques(", count := [ 8, \ 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8,\ 1, 3, 14 ], mathmode := "Text", name := "C", next := 381863, root := ~, start\ := 381841, stop := 381862 ), rec( attributes := rec( ), content := [ rec( co\ ntent := "digraph", count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "A" , next\ := 381877, root := ~, start := 381863, stop := 381876 ), rec( attributes := re\ c( ), content := [ rec( content := ",\n ", count := [ 8, 1, 3, 14 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ] , math\ mode := "Text", name := "C", next := 381896, root := ~, start := 381877, stop \ := 381895 ), rec( attributes := rec( ), content := [ rec( content := "include\ ", count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "A", next := 381910, root\ := ~, start := 381896, stop := 381909 ), rec( attributes := rec( ), content \ := [ rec( content := ", ", count := [ 8, 1, 3, 14 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ], mathmode := "Text", name\ := "C", next := 381919, root := ~, start := 381910, stop := 381918 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 1, 3,\ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3\ , 14 ], mathmode := "Text", name := "A", next := 381933, root := ~, start := 3\ 81919, stop := 381932 ), rec( attributes := rec( ), content := [ rec( content\ := ", 1,\n ", count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "\ C", next := 381953, root := ~, start := 381933, stop := 381952 ), rec( attribu\ tes := rec( ), content := [ rec( content := "size", count := [ 8, 1, 3, 14 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 14 ] , math\ mode := "Text", name := "A", next := 381964, root := ~, start := 381953, stop \ := 381963 ), rec( attributes := rec( ), content := [ rec( content := ")", cou\ nt := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 3, 14 ], mathmode := "Text", name := "C", next := 381972, root :=\ ~, start := 381964, stop := 381971 ), rec( content := " if one exists, else " , coun\ t := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 1, 3, 14\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 1\ 4 ], mathmode := "Text", name := "K", next := 382004, root := ~, start := 3819\ 93, stop := 382003 ), rec( content := ". This clique may\n not be maxim\ al.", count := [ 8, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 3, 15 ], mathm\ ode := "Text", name := "P", next := 382049, root := ~, start := 382045, stop :\ = 382048 ), rec( content := " Given four arguments, ", count := [ 8, 1, 3, 15 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphMaximalClique", count := [ 8, 1, 3, 15 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], math\ mode := "Text", name := "C", next := 382099, root := ~, start := 382072, stop \ := 382098 ), rec( content := "\n returns the maximal clique returned by\ \n ", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphMaxi\ malCliques(", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "C", next\ := 382184, root := ~, start := 382155, stop := 382183 ), rec( attributes := re\ c( ), content := [ rec( content := "digraph", count := [ 8, 1, 3, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], math\ mode := "Text", name := "A", next := 382198, root := ~, start := 382184, stop \ := 382197 ), rec( attributes := rec( ), content := [ rec( content := ", ", co\ unt := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 3, 15 ], mathmode := "Text", name := "C", next := 382207, root :=\ ~, start := 382198, stop := 382206 ), rec( attributes := rec( ), content := [ \ rec( content := "include", count := [ 8, 1, 3, 15 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name\ := "A", next := 382221, root := ~, start := 382207, stop := 382220 ), rec( attr\ ibutes := rec( ), content := [ rec( content := ",\n ", count := [ 8, 1\ , 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, \ 1, 3, 15 ], mathmode := "Text", name := "C", next := 382238, root := ~, start \ := 382221, stop := 382237 ), rec( attributes := rec( ), content := [ rec( con\ tent := "exclude", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "A", next\ := 382252, root := ~, start := 382238, stop := 382251 ), rec( attributes := re\ c( ), content := [ rec( content := ", 1, ", count := [ 8, 1, 3, 15 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmo\ de := "Text", name := "C", next := 382264, root := ~, start := 382252, stop :=\ 382263 ), rec( attributes := rec( ), content := [ rec( content := "size", co\ unt := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 1, 3, 15 ], mathmode := "Text", name := "A", next := 382275, root :=\ ~, start := 382264, stop := 382274 ), rec( attributes := rec( ), content := [ \ rec( content := ")", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "C\ ", next := 382283, root := ~, start := 382275, stop := 382282 ), rec( content \ := " if one exists, else\n ", count := [ 8, 1, 3, 15 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "fail", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 8, 1, 3, 15 ], mathmode := "Text", name := \ "K", next := 382323, root := ~, start := 382312, stop := 382322 ), rec( conten\ t := ".\n ", count := [ 8, 1, 3, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 1, 3, 14 ], name := "Item", next := 382338, root := ~, start := 381\ 729, stop := 382337 ) ], count := [ 8, 1, 3, 8 ], mathmode := "Text", name := "List", next := 382350, root := ~, start := 380723, stop := 382349 ), rec( content := "\n\n \ ", count := [ 8, 1, 3, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([[2, 3, 4], [1, 3], [1, 2], [1, 5], []]);\n\ngap> IsSymmetricDigraph(D);\nfalse\ngap> Digraph\ Clique(D);\n[ 5 ]\ngap> DigraphMaximalClique(D);\n[ 5 ]\ngap> DigraphClique(D,\ [1, 2]);\n[ 1, 2, 3 ]\ngap> DigraphMaximalClique(D, [1, 3]);\n[ 1, 3, 2 ]\nga\ p> DigraphClique(D, [1], [2]);\n[ 1, 4 ]\ngap> DigraphMaximalClique(D, [1], [3\ , 4]);\nfail\ngap> DigraphClique(D, [1, 5]);\nfail\ngap> DigraphClique(D, [], \ [], 2);\n[ 1, 2 ]\ngap> D := Digraph(IsMutableDigraph,\n> [[2,\ 3, 4], [1, 3], [1, 2], [1, 5], []]);\n\ngap> IsSymmetricDigraph(D);\nfalse\ngap> DigraphClique(D);\n[ 5 ]", count := [ 8, 1, 3, 18 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 3, 18 ], mathmode := "Text", name := "Example", next := 383055, root := ~, start := 382356, stop := 383054 ), rec( content := "\n ", count := [ 8, 1, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 3, 5 ], mathmode := "Text", name := "Description", next := 383072, root := ~, start := 379630, stop := 383071 ) ], count := [ 8, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 383086, root := ~, start := 379377, stop := 383085 ), rec( content := "\n\n ", count := [ 8, 1, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[, limit[, size]]]]", Name := "DigraphMaximalCliques" ), content := 0, count := [ 8, 1, 4, 2 ], mathmode := "Text", name := "Func", next := 383196, root := ~, start := 383107, stop := 383195 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n limit[, size]]]]", Name := "DigraphMaximalCliquesReps" ), content := 0, count := [ 8, 1, 4, 4 ], mathmode := "Text", name := "Func", next := 383296, root := ~, start := 383199, stop := 383295 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n limit[, size]]]]", Name := "DigraphCliques" ), content := 0, count := [ 8, 1, 4, 6 ], mathmode := "Text", name := "Func", next := 383385, root := ~, start := 383299, stop := 383384 ), rec( attributes := rec( Arg := "digraph[, includ\ e[,\n exclude[, limit[, size]]]]", Name := "DigraphCliquesReps" ), content := 0, count := [ 8, 1, 4, 8 ], mathmode := "Text", name := "Func", next := 383478, root := ~, start := 383388, stop := 383477 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalC\ liquesAttr" ), content := 0, count := [ 8, 1, 4, 10 ], mathmode := "Text", name := "Attr", next := 383535, root := ~, start := 383481, stop := 383534 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalC\ liquesRepsAttr" ), content := 0, count := [ 8, 1, 4, 12 ], mathmode := "Text", name := "Attr", next := 383596, root := ~, start := 383538, stop := 383595 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of lists of positive integers.", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Returns", next := 383666, root := ~, start := 383599, stop := 383665 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "A", next := 383704, root := ~, start := 383690, stop := 383703 ), rec( content := " is digr\ aph, then these functions and attributes use ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Cliques\ Finder" ), content := 0, count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Ref", next := 383790, root := ~, start := 383757, stop := 383789 ), rec( content := " to retu\ rn cliques of ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "A", next := 383826, root := ~, start := 383812, stop := 383825 ), rec( content := ". A\n \ clique is defined by the set of vertices that it contains; see ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsCliqu\ e" ), content := 0, count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Ref", next := 383925, root := ~, start := 383897, stop := 383924 ), rec( content := " and ", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alClique" ), content := 0, count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Ref", next := 383959, root := ~, start := 383930, stop := 383958 ), rec( content := ".", count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "P", next := 383964, root := ~, start := 383960, stop := 383963 ), rec( content := "\n\n \ The optional arguments ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 384007, root := ~, start := 383993, stop := 384006 ), rec( content := " and ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 384026, root := ~, start := 384012, stop := 384025 ), rec( content := " must ea\ ch be a\n (possibly empty) list of vertices of ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 384097, root := ~, start := 384083, stop := 384096 ), rec( content := ", the op\ tional argument\n ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 384137, root := ~, start := 384125, stop := 384136 ), rec( content := " must be\ either a positive integer or ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "C", next := 384190, root := ~, start := 384175, stop := 384189 ), rec( content := ", and th\ e\n optional argument ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 384233, root := ~, start := 384222, stop := 384232 ), rec( content := " must be\ a positive integer. If not\n specified, then ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 384304, root := ~, start := 384290, stop := 384303 ), rec( content := " and ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 384323, root := ~, start := 384309, stop := 384322 ), rec( content := " are cho\ sen to be empty\n lists, and ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "A", next := 384374, root := ~, start := 384362, stop := 384373 ), rec( content := " is set \ to ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "C", next := 384400, root := ~, start := 384385, stop := 384399 ), rec( content := ". ", count := [ 8, 1, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "P", next := 384406, root := ~, start := 384402, stop := 384405 ), rec( content := "\n\n \ The functions will return as many suitable cliques as possible, up to the\n \ number ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 384509, root := ~, start := 384497, stop := 384508 ), rec( content := ". These\ functions will find cliques that contain all\n of the vertices of ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 384600, root := ~, start := 384586, stop := 384599 ), rec( content := " but do \ not contain any of the\n vertices of ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 384661, root := ~, start := 384647, stop := 384660 ), rec( content := ". The a\ rgument ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 384688, root := ~, start := 384677, stop := 384687 ), rec( content := " restric\ ts the search\n to those cliques that contain precisely ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "A", next := 384765, root := ~, start := 384754, stop := 384764 ), rec( content := " vertice\ s.\n If the function or attribute has ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximal", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "C", next := 384827, root := ~, start := 384813, stop := 384826 ), rec( content := " in its \ name, then only\n maximal cliques will be returned; otherwise non-maximal c\ liques may be\n returned. ", count := [ 8, 1, 4, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "P", next := 384944, root := ~, start := 384940, stop := 384943 ), rec( content := "\n\n \ Let ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "C", next := 384962, root := ~, start := 384954, stop := 384961 ), rec( content := " denote \ the automorphism group of maximal symmetric subdigraph\n of ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "A", next := 385046, root := ~, start := 385032, stop := 385045 ), rec( content := " without\ loops (see ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "Ref", next := 385125, root := ~, start := 385066, stop := 385124 ), rec( content := " and ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Maximal\ SymmetricSubdigraphWithoutLoops" ), content := 0, count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "Ref", next := 385188, root := ~, start := 385130, stop := 385187 ), rec( content := ").\n\n \ ", count := [ 8, 1, 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Distinct cliques", count := [ 8, 1, 4,\ 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 17 ], name := "Mar\ k", next := 385238, root := ~, start := 385209, stop := 385237 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 8, 1, 4, 18 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphMaximalCliques", count := [ 8, 1, 4, 18 ], ma\ thmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ], math\ mode := "Text", name := "C", next := 385288, root := ~, start := 385260, stop \ := 385287 ), rec( content := " and ", count := [ 8, 1, 4, 18 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "DigraphCliques", count := [ 8, 1, 4, 18 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ], mathmode := "Text\ ", name := "C", next := 385314, root := ~, start := 385293, stop := 385313 ), rec( cont\ ent := " each return a\n duplicate-free list of at most ", count := [ 8\ , 1, 4, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "limit", count := [ 8, 1, 4, 18 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ] , math\ mode := "Text", name := "A", next := 385380, root := ~, start := 385368, stop \ := 385379 ), rec( content := " cliques of ", count := [ 8, 1, 4, 18 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conten\ t := [ rec( content := "digraph", count := [ 8, 1, 4, 18 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ], mathmode := "Text\ ", name := "A", next := 385406, root := ~, start := 385392, stop := 385405 ), rec( cont\ ent := "\n that satisfy the arguments.", count := [ 8, 1, 4, 18 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := 0, count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "P", next := 3\ 85446, root := ~, start := 385442, stop := 385445 ), rec( content := "\n\n \ The computation may be significantly faster if ", count := [ 8, 1, 4, 19 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "include", count := [ 8, 1, 4, 19 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 19 ], mathmode := "Te\ xt", name := "A", next := 385517, root := ~, start := 385503, stop := 385516 ) , rec( cont\ ent := " and\n ", count := [ 8, 1, 4, 19 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "exclude", count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "A", next\ := 385544, root := ~, start := 385530, stop := 385543 ), rec( content := " ar\ e invariant under the action of ", count := [ 8, 1, 4, 19 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "G", count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 1, 4, 19 ], mathmode := "Text", name := "C", next\ := 385587, root := ~, start := 385579, stop := 385586 ), rec( content := "\n \ on sets of vertices.\n ", count := [ 8, 1, 4, 19 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 18 ], name := "Item\ ", next := 385630, root := ~, start := 385245, stop := 385629 ), rec( attributes :\ = rec( ), content := [ rec( content := "Orbit representatives of cliques", coun\ t := [ 8, 1, 4, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 20 ], name := "Mar\ k", next := 385683, root := ~, start := 385638, stop := 385682 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n To use ", count := [ 8, 1, \ 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "DigraphMaximalCliquesReps", count := [\ 8, 1, 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 8, 1, 4, 21 ], mathmode := "Text", name := "C", next := 385744, root := ~, star\ t := 385712, stop := 385743 ), rec( content := " or ", count := [ 8, 1, 4, 21 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphCliquesReps", count := [ 8, 1, 4, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], math\ mode := "Text", name := "C", next := 385773, root := ~, start := 385748, stop \ := 385772 ), rec( content := ",\n the arguments ", count := [ 8, 1, 4, \ 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "include", count := [ 8, 1, 4, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], math\ mode := "Text", name := "A", next := 385811, root := ~, start := 385797, stop \ := 385810 ), rec( content := " and ", count := [ 8, 1, 4, 21 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ \ rec( content := "exclude", count := [ 8, 1, 4, 21 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], mathmode := "Text", name\ := "A", next := 385830, root := ~, start := 385816, stop := 385829 ), rec( cont\ ent := " must each be invariant\n under the action of ", count := [ 8, \ 1, 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes\ := rec( ), content := [ rec( content := "G", count := [ 8, 1, 4, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], math\ mode := "Text", name := "C", next := 385890, root := ~, start := 385882, stop \ := 385889 ), rec( content := " on sets of vertices.", count := [ 8, 1, 4, 21 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := 0, count := [ 8, 1, 4, 22 ], mathmode := "Text", name := "P", next := 3\ 85915, root := ~, start := 385911, stop := 385914 ), rec( content := "\n\n \ If this is the case, then ", count := [ 8, 1, 4, 22 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "DigraphMaximalCliquesReps", count := [ 8, 1, 4, 22 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 22 ], mathmode := "Tex\ t", name := "C", next := 385983, root := ~, start := 385951, stop := 385982 ), rec( cont\ ent := " and\n ", count := [ 8, 1, 4, 22 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "DigraphCliquesReps", count := [ 8, 1, 4, 22 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 22 ], mathmode := "Text", name :\ = "C", next := 386021, root := ~, start := 385996, stop := 386020 ), rec( cont\ ent := " each return a duplicate-free list of at most\n ", count := [ 8\ , 1, 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "limit", count := [ 8, 1, 4, 22 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 22 ] , math\ mode := "Text", name := "A", next := 386087, root := ~, start := 386075, stop \ := 386086 ), rec( content := " orbits representatives (under the action of ", coun\ t := [ 8, 1, 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G", count := [ 8, 1, 4, 22 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 22 ] , math\ mode := "Text", name := "C", next := 386140, root := ~, start := 386132, stop \ := 386139 ), rec( content := " on\n sets vertices) of cliques of ", coun\ t := [ 8, 1, 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "digraph", count := [ 8, 1, 4,\ 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4\ , 22 ], mathmode := "Text", name := "A", next := 386195, root := ~, start := 3\ 86181, stop := 386194 ), rec( content := " that satisfy the\n arguments\ .", count := [ 8, 1, 4, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 8, 1, 4, 23 ], mathmode := "Text", name\ := "P", next := 386235, root := ~, start := 386231, stop := 386234 ), rec( cont\ ent := "\n\n The representatives are not guaranteed to be in distinct o\ rbits.\n However, if fewer than ", count := [ 8, 1, 4, 23 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "lim", count := [ 8, 1, 4, 23 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 23 ], mathmode := "Text", name\ := "A", next := 386351, root := ~, start := 386341, stop := 386350 ), rec( cont\ ent := " results are returned, then there\n will be at least one repres\ entative from each orbit of maximal cliques.\n ", count := [ 8, 1, 4, 23 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 21 ], name := "Ite\ m", next := 386478, root := ~, start := 385690, stop := 386477 ) ], count := [ 8, 1, 4, 17 ], mathmode := "Text", name := "List", next := 386490, root := ~, start := 385196, stop := 386489 ), rec( content := "\n\n \ ", count := [ 8, 1, 4, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := Digraph([\n> [2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]])\ ;\n\ngap> IsSymmetricDigraph(D);\ \ntrue\ngap> G := AutomorphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4)\ ])\ngap> DigraphMaximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]\ng\ ap> DigraphMaximalCliquesReps(D);\n[ [ 1, 2, 3 ], [ 3, 4 ] ]\ngap> Orbit(G, [1\ , 2, 3], OnSets);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ] ]\ngap> Orbit(G, [3, 4], OnSets)\ ;\n[ [ 3, 4 ] ]\ngap> DigraphMaximalCliquesReps(D, [3, 4], [], 1);\n[ [ 3, 4 ]\ ]\ngap> DigraphMaximalCliques(D, [1, 2], [5, 6], 1, 2);\n[ ]\ngap> DigraphCl\ iques(D, [1], [5, 6], infinity, 2);\n[ [ 1, 2 ], [ 1, 3 ] ]\ngap> D := Digraph\ (IsMutableDigraph, [\n> [2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]])\ ;\n\ngap> IsSymmetricDigraph(D);\nt\ rue\ngap> G := AutomorphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4) ])\ \ngap> DigraphMaximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]", count := [ 8, 1, 4, 26 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 4, 26 ], mathmode := "Text", name := "Example", next := 387505, root := ~, start := 386496, stop := 387504 ), rec( content := "\n ", count := [ 8, 1, 4, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 4, 13 ], mathmode := "Text", name := "Description", next := 387522, root := ~, start := 383669, stop := 387521 ) ], count := [ 8, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 387536, root := ~, start := 383092, stop := 387535 ), rec( content := "\n\n ", count := [ 8, 1, 4, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "CliqueNumber" ) , content := 0, count := [ 8, 1, 5, 2 ], mathmode := "Text", name := "Attr", next := 387598, root := ~, start := 387557, stop := 387597 ), rec( attributes := rec( ), content := [ rec( content := "A non-neg\ ative integer.", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "Returns", next := 387643, root := ~, start := 387601, stop := 387642 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "A", next := 387681, root := ~, start := 387667, stop := 387680 ), rec( content := " is a di\ graph, then ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ liqueNumber(", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "digraph", count := [ 8, 1, 5, 3 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := \ "Text", name := "A", next := 387731, root := ~, start := 387717, stop := 38773\ 0 ), rec( content := ")", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "C", next := 387736, root := ~, start := 387701, stop := 387735 ), rec( content := "\n re\ turns the largest integer ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "C", next := 387777, root := ~, start := 387769, stop := 387776 ), rec( content := " such th\ at ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "A", next := 387802, root := ~, start := 387788, stop := 387801 ), rec( content := " contain\ s a\n clique with ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "C", next := 387838, root := ~, start := 387830, stop := 387837 ), rec( content := " vertice\ s as an induced subdigraph.\n ", count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "P", next := 387882, root := ~, start := 387878, stop := 387881 ), rec( content := "\n\n \ A ", count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ lique", count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "E", next := 387903, root := ~, start := 387890, stop := 387902 ), rec( content := " of a di\ graph is a set of mutually adjacent vertices of the\n digraph. Loops and mu\ ltiple edges are ignored for the purpose of\n determining the clique number\ of a digraph. \n ", count := [ 8, 1, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CompleteDigraph(4);;\ngap> CliqueNumber(D);\n4\ngap> D := Digraph([\ [1, 2, 4, 4], [1, 3, 4], [2, 1], [1, 2]]);\n\ngap> CliqueNumber(D);\n3\ngap> D := CompleteDigraph(IsMutabl\ eDigraph, 4);;\ngap> CliqueNumber(D);\n4", count := [ 8, 1, 5, 5 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 1, 5, 5 ], mathmode := "Text", name := "Example", next := 388381, root := ~, start := 388085, stop := 388380 ), rec( content := "\n", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 5, 3 ], mathmode := "Text", name := "Description", next := 388396, root := ~, start := 387646, stop := 388395 ) ], count := [ 8, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 388410, root := ~, start := 387542, stop := 388409 ), rec( content := "\n\n ", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 388424, root := ~, start := 371912, stop := 388423 ) , rec( content := "\n\n ", count := [ 8, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Finding independe\ nt sets", count := [ 8, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 388480, root := ~, start := 388437, stop := 388479 ), rec( content := "\n ", count := [ 8, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, l", Name := "IsIndependentSe\ t" ), content := 0, count := [ 8, 2, 1, 2 ], mathmode := "Text", name := "Oper", next := 388548, root := ~, start := 388500, stop := 388547 ), rec( attributes := rec( Arg := "digraph, l", Name := "IsMaximalIndepe\ ndentSet" ), content := 0, count := [ 8, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 388606, root := ~, start := 388551, stop := 388605 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ) , content := [ rec( content := "t\ rue", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 388629, root := ~, start := 388618, stop := 388628 ), rec( content := " or ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 388645, root := ~, start := 388633, stop := 388644 ), rec( content := ".", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 388656, root := ~, start := 388609, stop := 388655 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 388694, root := ~, start := 388680, stop := 388693 ), rec( content := " is a di\ graph and ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 388720, root := ~, start := 388712, stop := 388719 ), rec( content := " is a du\ plicate-free list of\n vertices of ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 388779, root := ~, start := 388765, stop := 388778 ), rec( content := ", then\n\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sIndependentSet(", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 388814, root := ~, start := 388790, stop := 388813 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 388828, root := ~, start := 388814, stop := 388827 ), rec( attributes := rec( ), content := [ rec( content := ",\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 388836, root := ~, start := 388828, stop := 388835 ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 388844, root := ~, start := 388836, stop := 388843 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 388852, root := ~, start := 388844, stop := 388851 ), rec( content := " returns\ \n ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 388876, root := ~, start := 388865, stop := 388875 ), rec( content := " if ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 388888, root := ~, start := 388880, stop := 388887 ), rec( content := " is an " , count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ndependent set", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "E", next := 388917, root := ~, start := 388895, stop := 388916 ), rec( content := " of ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 388935, root := ~, start := 388921, stop := 388934 ), rec( content := " and\n \ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 388956, root := ~, start := 388944, stop := 388955 ), rec( content := " if it i\ s not. Similarly,\n ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sMaximalIndependentSet(", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 389018, root := ~, start := 388987, stop := 389017 ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 389032, root := ~, start := 389018, stop := 389031 ), rec( attributes := rec( ), content := [ rec( content := ",\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 389040, root := ~, start := 389032, stop := 389039 ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 389048, root := ~, start := 389040, stop := 389047 ), rec( attributes := rec( ), content := [ rec( content := ")\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "C", next := 389056, root := ~, start := 389048, stop := 389055 ), rec( content := "\n re\ turns ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "t\ rue", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 389080, root := ~, start := 389069, stop := 389079 ), rec( content := " if ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 389092, root := ~, start := 389084, stop := 389091 ), rec( content := " is a ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal independent set", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "E", next := 389128, root := ~, start := 389098, stop := 389127 ), rec( content := " of\n \ ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "A", next := 389150, root := ~, start := 389136, stop := 389149 ), rec( content := " and ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ alse", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "K", next := 389167, root := ~, start := 389155, stop := 389166 ), rec( content := " if it i\ s not. ", count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "P", next := 389187, root := ~, start := 389183, stop := 389186 ), rec( content := "\n\n \ An ", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ ndependent set", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "E", next := 389218, root := ~, start := 389196, stop := 389217 ), rec( content := " of a di\ graph is a set of mutually non-adjacent\n vertices of the digraph. A ", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ aximal independent set", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "E", next := 389327, root := ~, start := 389297, stop := 389326 ), rec( content := " is an i\ ndependent\n set that is not properly contained in another independent set.\ An\n independent set is permitted, but not required, to contain vertices a\ t\n which there is a loop.\n ", count := [ 8, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(4);;\ngap> IsIndependentSet(D, [1]);\ntrue\ngap> IsMax\ imalIndependentSet(D, [1]);\nfalse\ngap> IsIndependentSet(D, [1, 4, 3]);\nfals\ e\ngap> IsIndependentSet(D, [2, 4]);\ntrue\ngap> IsMaximalIndependentSet(D, [2\ , 4]);\ntrue\ngap> D := CycleDigraph(IsMutableDigraph, 4);;\ngap> IsIndependen\ tSet(D, [1]);\ntrue", count := [ 8, 2, 1, 7 ], name := "PCDATA", root := ~ ) ] , count := [ 8, 2, 1, 7 ], mathmode := "Text", name := "Example", next := 389871, root := ~, start := 389522, stop := 389870 ), rec( content := "\n ", count := [ 8, 2, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 389888, root := ~, start := 388659, stop := 389887 ) ], count := [ 8, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 389902, root := ~, start := 388485, stop := 389901 ), rec( content := "\n\n ", count := [ 8, 2, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[, size]]]", Name := "DigraphIndependentSet" ), content := 0, count := [ 8, 2, 2, 2 ], mathmode := "Text", name := "Func", next := 390007, root := ~, start := 389923, stop := 390006 ), rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n size]]]", Name := "DigraphMaximalIndependentSet" ), content := 0, count := [ 8, 2, 2, 4 ], mathmode := "Text", name := "Func", next := 390105, root := ~, start := 390010, stop := 390104 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of positive integers, or ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "K", next := 390171, root := ~, start := 390160, stop := 390170 ), rec( content := ".", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Returns", next := 390182, root := ~, start := 390108, stop := 390181 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "A", next := 390220, root := ~, start := 390206, stop := 390219 ), rec( content := " is a di\ graph, then these functions returns an independent\n set of ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "A", next := 390304, root := ~, start := 390290, stop := 390303 ), rec( content := " if one \ exists that satisfies the arguments, else it\n returns ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "K", next := 390380, root := ~, start := 390369, stop := 390379 ), rec( content := ". An in\ dependent set is defined by the set of vertices\n that it contains; see ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsIndep\ endentSet" ), content := 0, count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Ref", next := 390492, root := ~, start := 390462, stop := 390491 ), rec( content := " and ", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alIndependentSet" ), content := 0, count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Ref", next := 390540, root := ~, start := 390497, stop := 390539 ), rec( content := ".", count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "P", next := 390545, root := ~, start := 390541, stop := 390544 ), rec( content := "\n\n \ The optional arguments ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 390588, root := ~, start := 390574, stop := 390587 ), rec( content := " and ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 390607, root := ~, start := 390593, stop := 390606 ), rec( content := " must ea\ ch be a\n (possibly empty) duplicate-free list of vertices of ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 390693, root := ~, start := 390679, stop := 390692 ), rec( content := ", and th\ e\n optional argument ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 390736, root := ~, start := 390725, stop := 390735 ), rec( content := " must be\ a positive integer. By default,\n ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 390796, root := ~, start := 390782, stop := 390795 ), rec( content := " and ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 390815, root := ~, start := 390801, stop := 390814 ), rec( content := " are emp\ ty. These functions will search\n for an independent set of ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 390899, root := ~, start := 390885, stop := 390898 ), rec( content := " that in\ cludes the vertices of\n ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 390948, root := ~, start := 390934, stop := 390947 ), rec( content := " but doe\ s not include any vertices in ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 391000, root := ~, start := 390986, stop := 390999 ), rec( content := ";\n i\ f the argument ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 391033, root := ~, start := 391022, stop := 391032 ), rec( content := " is supp\ lied, then additionally the independent\n set will be required to contain p\ recisely ", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "A", next := 391138, root := ~, start := 391127, stop := 391137 ), rec( content := " vertice\ s.", count := [ 8, 2, 2, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "P", next := 391152, root := ~, start := 391148, stop := 391151 ), rec( content := "\n\n \ If ", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "A", next := 391175, root := ~, start := 391161, stop := 391174 ), rec( content := " is not \ an independent set, then these functions return\n ", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ail", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "K", next := 391246, root := ~, start := 391235, stop := 391245 ), rec( content := ". Other\ wise, the functions behave in the following way,\n depending on the number \ of arguments:\n\n ", count := [ 8, 2, 2, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "One or two arguments", count := [ 8, 2\ , 2, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 8 ], name := "Mar\ k", next := 391396, root := ~, start := 391363, stop := 391395 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If one or two arguments are\ supplied, then ", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Dig\ raphIndependentSet", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "C", next\ := 391489, root := ~, start := 391461, stop := 391488 ), rec( content := "\n \ and ", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphMaxi\ malIndependentSet", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "C", next\ := 391537, root := ~, start := 391502, stop := 391536 ), rec( content := " gr\ eedily enlarge the\n independent set ", count := [ 8, 2, 2, 9 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "include", count := [ 8, 2, 2, 9 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 9 ], mathmode := "Text" , name\ := "A", next := 391597, root := ~, start := 391583, stop := 391596 ), rec( cont\ ent := " until it can not continue. The result\n is guaranteed to be a\ maximal independent set. This may or may not\n return an answer more q\ uickly than using ", count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Func := "DigraphMaximalIndependentS\ ets" ), content := 0, count := [ 8, 2, 2, 9 ], mathmode := "Text", name := "Re\ f", next := 391814, root := ~, start := 391761, stop := 391813 ), rec( content\ := ". with a limit of 1.\n ", count := [ 8, 2, 2, 9 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 9 ], name := "Item", next := 3918\ 49, root := ~, start := 391403, stop := 391848 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "Three arguments", count := [ 8, 2, 2, 10 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 10 ], name := "Mark", next := 3918\ 84, root := ~, start := 391856, stop := 391883 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n If three arguments are supplied, then ", count := [ 8, 2, 2,\ 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "DigraphIndependentSet", count := [ 8, 2, \ 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2,\ 2, 11 ], mathmode := "Text", name := "C", next := 391972, root := ~, start :=\ 391944, stop := 391971 ), rec( content := "\n greedily enlarges the in\ dependent set ", count := [ 8, 2, 2, 11 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "incl\ ude", count := [ 8, 2, 2, 11 ], mathmode := "Text", name := "PCDATA", root := \ ~ ) ], count := [ 8, 2, 2, 11 ], mathmode := "Text", name := "A", next := 3920\ 33, root := ~, start := 392019, stop := 392032 ), rec( content := " until it c\ an not\n continue, although this independent set may not be maximal.", coun\ t := [ 8, 2, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 8, 2, 2, 12 ], mathmode := "Text", name\ := "P", next := 392122, root := ~, start := 392118, stop := 392121 ), rec( cont\ ent := "\n\n Given three arguments, ", count := [ 8, 2, 2, 12 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), conte\ nt := [ rec( content := "DigraphMaximalIndependentSet", count := [ 8, 2, 2, 12\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 1\ 2 ], mathmode := "Text", name := "C", next := 392190, root := ~, start := 3921\ 55, stop := 392189 ), rec( content := " returns the\n maximal independe\ nt set returned by\n ", count := [ 8, 2, 2, 12 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "DigraphMaximalIndependentSets(", count := [ 8, 2, 2, 12 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 12 ], mathmode :=\ "Text", name := "C", next := 392292, root := ~, start := 392255, stop := 3922\ 91 ), rec( attributes := rec( ), content := [ rec( content := "digraph", coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "A", next := 392306, root :=\ ~, start := 392292, stop := 392305 ), rec( attributes := rec( ), content := [ \ rec( content := ",\n ", count := [ 8, 2, 2, 12 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 12 ], mathmode := "Text", name\ := "C", next := 392323, root := ~, start := 392306, stop := 392322 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "include", count := [ 8, 2, 2,\ 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2\ , 12 ], mathmode := "Text", name := "A", next := 392337, root := ~, start := 3\ 92323, stop := 392336 ), rec( attributes := rec( ), content := [ rec( content\ := ", ", count := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 8, 2, 2, 12 ], mathmode := "Text", name := "C", next := \ 392346, root := ~, start := 392337, stop := 392345 ), rec( attributes := rec( \ ), content := [ rec( content := "exclude", count := [ 8, 2, 2, 12 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 12 ], mathmod\ e := "Text", name := "A", next := 392360, root := ~, start := 392346, stop := \ 392359 ), rec( attributes := rec( ), content := [ rec( content := ", 1)", cou\ nt := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "C", next := 392371, root :=\ ~, start := 392360, stop := 392370 ), rec( content := " if one exists,\n \ else ", count := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 12 ], mathmode := "Text", name := "K", next := 392411, root :=\ ~, start := 392400, stop := 392410 ), rec( content := ".\n ", count := [ \ 8, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 11 ], name := "Ite\ m", next := 392426, root := ~, start := 391891, stop := 392425 ), rec( attributes :\ = rec( ), content := [ rec( content := "Four arguments", count := [ 8, 2, 2, \ 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 13 ], name := "Mar\ k", next := 392460, root := ~, start := 392433, stop := 392459 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n If four arguments are suppl\ ied, then ", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "DigraphInde\ pendentSet", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "C", next \ := 392547, root := ~, start := 392519, stop := 392546 ), rec( content := "\n \ returns the independent set returned by\n ", count := [ 8, 2, 2, \ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "DigraphIndependentSets(", count := [ 8, 2\ , 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, \ 2, 2, 14 ], mathmode := "Text", name := "C", next := 392634, root := ~, start \ := 392604, stop := 392633 ), rec( attributes := rec( ), content := [ rec( con\ tent := "digraph", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "A", next\ := 392648, root := ~, start := 392634, stop := 392647 ), rec( attributes := re\ c( ), content := [ rec( content := ", ", count := [ 8, 2, 2, 14 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode \ := "Text", name := "C", next := 392657, root := ~, start := 392648, stop := 39\ 2656 ), rec( attributes := rec( ), content := [ rec( content := "include", co\ unt := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 14 ], mathmode := "Text", name := "A", next := 392671, root :=\ ~, start := 392657, stop := 392670 ), rec( attributes := rec( ), content := [ \ rec( content := ",\n ", count := [ 8, 2, 2, 14 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode := "Text", name\ := "C", next := 392688, root := ~, start := 392671, stop := 392687 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 2, 2,\ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2\ , 14 ], mathmode := "Text", name := "A", next := 392702, root := ~, start := 3\ 92688, stop := 392701 ), rec( attributes := rec( ), content := [ rec( content\ := ", 1, ", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "C", next \ := 392714, root := ~, start := 392702, stop := 392713 ), rec( attributes := re\ c( ), content := [ rec( content := "size", count := [ 8, 2, 2, 14 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], mathmod\ e := "Text", name := "A", next := 392725, root := ~, start := 392714, stop := \ 392724 ), rec( attributes := rec( ), content := [ rec( content := ")", count \ := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 8, 2, 2, 14 ], mathmode := "Text", name := "C", next := 392733, root := \ ~, start := 392725, stop := 392732 ), rec( content := " if one exists, else\n \ ", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "fail", coun\ t := [ 8, 2, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 14 ], mathmode := "Text", name := "K", next := 392773, root :=\ ~, start := 392762, stop := 392772 ), rec( content := ". This independent set\ may not be maximal.", count := [ 8, 2, 2, 14 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, \ 2, 2, 15 ], mathmode := "Text", name := "P", next := 392819, root := ~, start \ := 392815, stop := 392818 ), rec( content := " Given four\n arguments, \ ", count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "DigraphMaximalIndependentSet"\ , count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 8, 2, 2, 15 ], mathmode := "Text", name := "C", next := 392885, root :=\ ~, start := 392850, stop := 392884 ), rec( content := " returns the maximal\n\ independent set returned by\n ", count := [ 8, 2, 2, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphMaximalIndependentSets(", count := [ 8, 2, 2,\ 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2\ , 15 ], mathmode := "Text", name := "C", next := 392987, root := ~, start := 3\ 92950, stop := 392986 ), rec( attributes := rec( ), content := [ rec( content\ := "digraph", count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "A", next\ := 393001, root := ~, start := 392987, stop := 393000 ), rec( attributes := re\ c( ), content := [ rec( content := ",\n ", count := [ 8, 2, 2, 15 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 15 ] , math\ mode := "Text", name := "C", next := 393020, root := ~, start := 393001, stop \ := 393019 ), rec( attributes := rec( ), content := [ rec( content := "include\ ", count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "A", next := 393034, root\ := ~, start := 393020, stop := 393033 ), rec( attributes := rec( ), content \ := [ rec( content := ", ", count := [ 8, 2, 2, 15 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 15 ], mathmode := "Text", name\ := "C", next := 393043, root := ~, start := 393034, stop := 393042 ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 2, 2,\ 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2\ , 15 ], mathmode := "Text", name := "A", next := 393057, root := ~, start := 3\ 93043, stop := 393056 ), rec( attributes := rec( ), content := [ rec( content\ := ", 1,\n ", count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 8, 2, 2, 15 ], mathmode := "Text", name := "\ C", next := 393077, root := ~, start := 393057, stop := 393076 ), rec( attribu\ tes := rec( ), content := [ rec( content := "size", count := [ 8, 2, 2, 15 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 15 ] , math\ mode := "Text", name := "A", next := 393088, root := ~, start := 393077, stop \ := 393087 ), rec( attributes := rec( ), content := [ rec( content := ")", cou\ nt := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 2, 15 ], mathmode := "Text", name := "C", next := 393096, root :=\ ~, start := 393088, stop := 393095 ), rec( content := " if one exists, else " , coun\ t := [ 8, 2, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "fail", count := [ 8, 2, 2, 15\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 1\ 5 ], mathmode := "Text", name := "K", next := 393128, root := ~, start := 3931\ 17, stop := 393127 ), rec( content := ".\n ", count := [ 8, 2, 2, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 14 ], name := "Ite\ m", next := 393143, root := ~, start := 392467, stop := 393142 ) ], count := [ 8, 2, 2, 8 ], mathmode := "Text", name := "List", next := 393155, root := ~, start := 391350, stop := 393154 ), rec( content := "\n ", count := [ 8, 2, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := ChainDigraph(6);\n\ngap> D\ igraphIndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphMaximalIndependentSet(D);\n\ [ 6, 4, 2 ]\ngap> DigraphIndependentSet(D, [2, 4]);\n[ 2, 4, 6 ]\ngap> Digraph\ MaximalIndependentSet(D, [1, 3]);\n[ 1, 3, 6 ]\ngap> DigraphIndependentSet(D, \ [2, 4], [6]);\n[ 2, 4 ]\ngap> DigraphMaximalIndependentSet(D, [2, 4], [6]);\nf\ ail\ngap> DigraphIndependentSet(D, [1], [], 2);\n[ 1, 3 ]\ngap> DigraphMaximal\ IndependentSet(D, [1], [], 2);\nfail\ngap> DigraphMaximalIndependentSet(D, [1]\ , [], 3);\n[ 1, 3, 5 ]\ngap> D := ChainDigraph(IsMutableDigraph, 6);\n\ngap> DigraphIndependentSet(D);\n[ 6, 4, 2 \ ]\ngap> DigraphMaximalIndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphIndependent\ Set(D, [2, 4]);\n[ 2, 4, 6 ]\ngap> DigraphMaximalIndependentSet(D, [1, 3]);\n[\ 1, 3, 6 ]\ngap> DigraphIndependentSet(D, [2, 4], [6]);\n[ 2, 4 ]\ngap> Digrap\ hMaximalIndependentSet(D, [2, 4], [6]);\nfail\ngap> DigraphIndependentSet(D, [\ 1], [], 2);\n[ 1, 3 ]\ngap> DigraphMaximalIndependentSet(D, [1], [], 2);\nfail\ \ngap> DigraphMaximalIndependentSet(D, [1], [], 3);\n[ 1, 3, 5 ]", count := [ 8, 2, 2, 18 ], name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 2, 18 ], mathmode := "Text", name := "Example", next := 394308, root := ~, start := 393160, stop := 394307 ), rec( content := "\n ", count := [ 8, 2, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 2, 5 ], mathmode := "Text", name := "Description", next := 394325, root := ~, start := 390185, stop := 394324 ) ], count := [ 8, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 394339, root := ~, start := 389908, stop := 394338 ), rec( content := "\n\n ", count := [ 8, 2, 2, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph[, includ\ e[, exclude[,\n limit[, size]]]]", Name := "DigraphMaximalIndependentSets" ), content := 0, count := [ 8, 2, 3, 2 ], mathmode := "Text", name := "Func", next := 394461, root := ~, start := 394360, stop := 394460 ), rec( attributes := rec( Arg := "digraph[, includ\ e[,\n exclude[, limit[, size]]]]", Name := "DigraphMaximalIndependentSetsRe\ ps" ), content := 0, count := [ 8, 2, 3, 4 ], mathmode := "Text", name := "Func", next := 394569, root := ~, start := 394464, stop := 394568 ), rec( attributes := rec( Arg := "digraph[, includ\ e[,\n exclude[, limit[, size]]]]", Name := "DigraphIndependentSets" ), content := 0, count := [ 8, 2, 3, 6 ], mathmode := "Text", name := "Func", next := 394666, root := ~, start := 394572, stop := 394665 ), rec( attributes := rec( Arg := "digraph[, includ\ e[,\n exclude[, limit[, size]]]]", Name := "DigraphIndependentSetsReps" ), content := 0, count := [ 8, 2, 3, 8 ], mathmode := "Text", name := "Func", next := 394767, root := ~, start := 394669, stop := 394766 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalI\ ndependentSetsAttr" ), content := 0, count := [ 8, 2, 3, 10 ], mathmode := "Text", name := "Attr", next := 394832, root := ~, start := 394770, stop := 394831 ), rec( attributes := rec( Arg := "digraph", Name := "DigraphMaximalI\ ndependentSetsRepsAttr" ), content := 0, count := [ 8, 2, 3, 12 ], mathmode := "Text", name := "Attr", next := 394901, root := ~, start := 394835, stop := 394900 ), rec( attributes := rec( ), content := [ rec( content := "An immuta\ ble list of lists of positive integers.", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Returns", next := 394971, root := ~, start := 394904, stop := 394970 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "A", next := 395009, root := ~, start := 394995, stop := 395008 ), rec( content := " is digr\ aph, then these functions and attributes use ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Cliques\ Finder" ), content := 0, count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Ref", next := 395095, root := ~, start := 395062, stop := 395094 ), rec( content := " to retu\ rn independent sets of ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "A", next := 395140, root := ~, start := 395126, stop := 395139 ), rec( content := ". An\n \ independent set is defined by the set of vertices that it contains; see\n \ ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsMaxim\ alIndependentSet" ), content := 0, count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Ref", next := 395262, root := ~, start := 395225, stop := 395261 ), rec( content := " and ", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "IsIndep\ endentSet" ), content := 0, count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Ref", next := 395303, root := ~, start := 395267, stop := 395302 ), rec( content := ".", count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "P", next := 395308, root := ~, start := 395304, stop := 395307 ), rec( content := "\n\n \ The optional arguments ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 395351, root := ~, start := 395337, stop := 395350 ), rec( content := " and ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 395370, root := ~, start := 395356, stop := 395369 ), rec( content := " must ea\ ch be a\n (possibly empty) list of vertices of ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 395441, root := ~, start := 395427, stop := 395440 ), rec( content := ", the op\ tional argument\n ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 395481, root := ~, start := 395469, stop := 395480 ), rec( content := " must be\ either a positive integer or ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "C", next := 395534, root := ~, start := 395519, stop := 395533 ), rec( content := ", and th\ e\n optional argument ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 395577, root := ~, start := 395566, stop := 395576 ), rec( content := " must be\ a positive integer. If not\n specified, then ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 395648, root := ~, start := 395634, stop := 395647 ), rec( content := " and ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 395667, root := ~, start := 395653, stop := 395666 ), rec( content := " are cho\ sen to be empty\n lists, and ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "A", next := 395718, root := ~, start := 395706, stop := 395717 ), rec( content := " is set \ to ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nfinity", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "C", next := 395744, root := ~, start := 395729, stop := 395743 ), rec( content := ". ", count := [ 8, 2, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "P", next := 395750, root := ~, start := 395746, stop := 395749 ), rec( content := "\n\n \ The functions will return as many suitable independent sets as possible, up\n \ to the number ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "l\ imit", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 395862, root := ~, start := 395850, stop := 395861 ), rec( content := ". These\ functions will find independent sets\n that contain all of the vertices of\ ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ nclude", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 395962, root := ~, start := 395948, stop := 395961 ), rec( content := " but do \ not\n contain any of the vertices of ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ xclude", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 396023, root := ~, start := 396009, stop := 396022 ), rec( content := " The arg\ ument ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 396048, root := ~, start := 396037, stop := 396047 ), rec( content := "\n re\ stricts the search to those cliques that contain precisely ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ ize", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "A", next := 396125, root := ~, start := 396114, stop := 396124 ), rec( content := "\n ve\ rtices. If the function or attribute has ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "M\ aximal", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "C", next := 396188, root := ~, start := 396174, stop := 396187 ), rec( content := " in its \ name,\n then only maximal independent sets will be returned; otherwise non-\ maximal\n independent sets may be returned.\n ", count := [ 8, 2, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "P", next := 396327, root := ~, start := 396323, stop := 396326 ), rec( content := "\n\n \ Let ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "C", next := 396345, root := ~, start := 396337, stop := 396344 ), rec( content := " denote \ the ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "Ref", next := 396410, root := ~, start := 396357, stop := 396409 ), rec( content := "\n of\ the ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ SymmetricClosure" ), content := 0, count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "Ref", next := 396459, root := ~, start := 396422, stop := 396458 ), rec( content := " of the \ digraph formed from\n ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "A", next := 396505, root := ~, start := 396491, stop := 396504 ), rec( content := " by remo\ ving loops and ignoring the multiplicity of edges.\n\n ", count := [ 8, 2, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Distinct independent sets", count := [ \ 8, 2, 3, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 17 ], name := "Mar\ k", next := 396620, root := ~, start := 396582, stop := 396619 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n ", count := [ 8, 2, 3, 18 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphMaximalIndependentSets", count := [ 8, 2, 3, \ 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3,\ 18 ], mathmode := "Text", name := "C", next := 396678, root := ~, start := 39\ 6642, stop := 396677 ), rec( content := " and ", count := [ 8, 2, 3, 18 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "DigraphIndependentSets", count := [ 8, 2, 3, 18 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 18 ], math\ mode := "Text", name := "C", next := 396712, root := ~, start := 396683, stop \ := 396711 ), rec( content := "\n each return a duplicate-free list of a\ t most ", count := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "limit", cou\ nt := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 3, 18 ], mathmode := "Text", name := "A", next := 396778, root :=\ ~, start := 396766, stop := 396777 ), rec( content := " independent\n \ sets of ", count := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "digraph", c\ ount := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 3, 18 ], mathmode := "Text", name := "A", next := 396821, root :=\ ~, start := 396807, stop := 396820 ), rec( content := " that satisfy the argu\ ments.", count := [ 8, 2, 3, 18 ], mathmode := "Text", name := "PCDATA", root \ := ~ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 19 ], math\ mode := "Text", name := "P", next := 396853, root := ~, start := 396849, stop \ := 396852 ), rec( content := "\n\n The computation may be significantly\ faster if ", count := [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "include", c\ ount := [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 8, 2, 3, 19 ], mathmode := "Text", name := "A", next := 396924, root :=\ ~, start := 396910, stop := 396923 ), rec( content := " and\n ", count\ := [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "exclude", count := [ 8, 2, 3,\ 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3\ , 19 ], mathmode := "Text", name := "A", next := 396951, root := ~, start := 3\ 96937, stop := 396950 ), rec( content := " are invariant under the action of " , coun\ t := [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "G", count := [ 8, 2, 3, 19 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 19 ] , math\ mode := "Text", name := "C", next := 396994, root := ~, start := 396986, stop \ := 396993 ), rec( content := " on sets of\n vertices.\n ", count :\ = [ 8, 2, 3, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 18 ], name := "Ite\ m", next := 397037, root := ~, start := 396627, stop := 397036 ), rec( attributes :\ = rec( ), content := [ rec( content := "Representatives of distinct orbits of\ independent sets", count := [ 8, 2, 3, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 8, 2, 3, 20 ], name := "Mark", next := 397112, root := ~, start := 397\ 045, stop := 397111 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n To use ", count := [ 8, 2, 3, 21 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content :=\ "DigraphMaximalIndependentSetsReps", count := [ 8, 2, 3, 21 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmode := "\ Text", name := "C", next := 397181, root := ~, start := 397141, stop := 397180\ ), rec( content := " or\n ", count := [ 8, 2, 3, 21 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ r\ ec( content := "DigraphIndependentSetsReps", count := [ 8, 2, 3, 21 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmo\ de := "Text", name := "C", next := 397226, root := ~, start := 397193, stop :=\ 397225 ), rec( content := ", the arguments ", count := [ 8, 2, 3, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "include", count := [ 8, 2, 3, 21 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmode := "Te\ xt", name := "A", next := 397256, root := ~, start := 397242, stop := 397255 ) , rec( cont\ ent := " and\n ", count := [ 8, 2, 3, 21 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "exclude", count := [ 8, 2, 3, 21 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmode := "Text", name := "A", next\ := 397283, root := ~, start := 397269, stop := 397282 ), rec( content := " mu\ st each be invariant under the action of ", count := [ 8, 2, 3, 21 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "G", count := [ 8, 2, 3, 21 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], mathmode := "Text", name\ := "C", next := 397335, root := ~, start := 397327, stop := 397334 ), rec( cont\ ent := " on\n sets of vertices.", count := [ 8, 2, 3, 21 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ 0, count := [ 8, 2, 3, 22 ], mathmode := "Text", name := "P", next := 397368, root\ := ~, start := 397364, stop := 397367 ), rec( content := "\n\n If this\ is the case, then ", count := [ 8, 2, 3, 22 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "DigraphMaximalIndependentSetsReps", count := [ 8, 2, 3, 22 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 22 ], mathmode := "T\ ext", name := "C", next := 397444, root := ~, start := 397404, stop := 397443 ), rec( cont\ ent := " and\n ", count := [ 8, 2, 3, 22 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "DigraphIndependentSetsReps", count := [ 8, 2, 3, 22 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 22 ], mathmode := "Text" , name\ := "C", next := 397490, root := ~, start := 397457, stop := 397489 ), rec( cont\ ent := " each return a list of\n at most ", count := [ 8, 2, 3, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "limit", count := [ 8, 2, 3, 22 ], mathmode := "Text"\ , name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 22 ], mathmode := "Text\ ", name := "A", next := 397541, root := ~, start := 397529, stop := 397540 ), rec( cont\ ent := " orbits representatives (under the action of\n ", count := [ 8,\ 2, 3, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribute\ s := rec( ), content := [ rec( content := "G", count := [ 8, 2, 3, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 22 ], math\ mode := "Text", name := "C", next := 397602, root := ~, start := 397594, stop \ := 397601 ), rec( content := " on sets of vertices) of independent sets of ", coun\ t := [ 8, 2, 3, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "digraph", count := [ 8, 2, 3,\ 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3\ , 22 ], mathmode := "Text", name := "A", next := 397661, root := ~, start := 3\ 97647, stop := 397660 ), rec( content := "\n that satisfy the arguments\ . ", count := [ 8, 2, 3, 22 ], mathmode := "Text", name := "PCDATA", root := ~\ ), rec( attributes := rec( ), content := 0, count := [ 8, 2, 3, 23 ], mathmo\ de := "Text", name := "P", next := 397702, root := ~, start := 397698, stop :=\ 397701 ), rec( content := "\n\n The representatives are not guaranteed\ to be in distinct orbits.\n However, if ", count := [ 8, 2, 3, 23 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "lim", count := [ 8, 2, 3, 23 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 23 ], mathmode := "Text", name\ := "A", next := 397807, root := ~, start := 397797, stop := 397806 ), rec( cont\ ent := " is not specified, or fewer than ", count := [ 8, 2, 3, 23 ], mathmode\ := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content\ := [ rec( content := "lim", count := [ 8, 2, 3, 23 ], mathmode := "Text", nam\ e := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 23 ], mathmode := "Text", name\ := "A", next := 397850, root := ~, start := 397840, stop := 397849 ), rec( cont\ ent := "\n results are returned, then there will be at least one repres\ entative\n from each orbit of maximal independent sets.\n ", count\ := [ 8, 2, 3, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 21 ], name := "Ite\ m", next := 397994, root := ~, start := 397119, stop := 397993 ) ], count := [ 8, 2, 3, 17 ], mathmode := "Text", name := "List", next := 398006, root := ~, start := 396569, stop := 398005 ), rec( content := "\n\n \ ", count := [ 8, 2, 3, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(5);\n\ngap> D\ igraphMaximalIndependentSetsReps(D);\n[ [ 1, 3 ] ]\ngap> DigraphIndependentSet\ sReps(D);\n[ [ 1 ], [ 1, 3 ] ]\ngap> Set(DigraphMaximalIndependentSets(D));\n[\ [ 1, 3 ], [ 1, 4 ], [ 2, 4 ], [ 2, 5 ], [ 3, 5 ] ]\ngap> DigraphMaximalIndepe\ ndentSets(D, [1]);\n[ [ 1, 3 ], [ 1, 4 ] ]\ngap> DigraphIndependentSets(D, [],\ [4, 5]);\n[ [ 1 ], [ 2 ], [ 3 ], [ 1, 3 ] ]\ngap> DigraphIndependentSets(D, [\ ], [4, 5], 1, 2);\n[ [ 1, 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\ \n\ngap> DigraphMaximalIndependentSe\ tsReps(D);\n[ [ 1, 3 ] ]\ngap> DigraphIndependentSetsReps(D);\n[ [ 1 ], [ 1, 3\ ] ]\ngap> Set(DigraphMaximalIndependentSets(D));\n[ [ 1, 3 ], [ 1, 4 ], [ 2, \ 4 ], [ 2, 5 ], [ 3, 5 ] ]\ngap> DigraphMaximalIndependentSets(D, [1]);\n[ [ 1,\ 3 ], [ 1, 4 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5]);\n[ [ 1 ], [ 2 ],\ [ 3 ], [ 1, 3 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5], 1, 2);\n[ [ 1, \ 3 ] ]", count := [ 8, 2, 3, 26 ], name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 26 ], mathmode := "Text", name := "Example", next := 399034, root := ~, start := 398012, stop := 399033 ), rec( content := "\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 3, 13 ], mathmode := "Text", name := "Description", next := 399051, root := ~, start := 394974, stop := 399050 ) ], count := [ 8, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 399065, root := ~, start := 394345, stop := 399064 ), rec( content := "\n\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 399079, root := ~, start := 388428, stop := 399078 ) , rec( content := "\n", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 8, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 399090, root := ~, start := 367151, stop := 399089 ), rec( content := "\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " cliques ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n ", count := [ 8, 2, 3, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "Visualising and IO" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Visualising and IO", count := [ 9, 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 399184, root := ~, start := 399147, stop := 399183 ) , rec( content := "\n\n ", count := [ 9, 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Visualising a dig\ raph", count := [ 9, 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 399237, root := ~, start := 399197, stop := 399236 ), rec( content := "\n ", count := [ 9, 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str, [opts]", Name := "Splash" ), content := 0, count := [ 9, 1, 1, 2 ], mathmode := "Text", name := "Func", next := 399300, root := ~, start := 399261, stop := 399299 ), rec( attributes := rec( ), content := [ rec( content := "Nothing." , count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "Returns", next := 399332, root := ~, start := 399305, stop := 399331 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function attempts to convert the string ", count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "A", next := 399412, root := ~, start := 399402, stop := 399411 ), rec( content := " into a \ pdf\n document and open this document, i.e. to splash it all over your mo\ nitor.", count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "P", next := 399507, root := ~, start := 399503, stop := 399506 ), rec( content := "\n\n \ The string ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "A", next := 399536, root := ~, start := 399526, stop := 399535 ), rec( content := " must co\ rrespond to a valid ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "C", next := 399574, root := ~, start := 399564, stop := 399573 ), rec( content := " or\n \ ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ aTeX", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "C", next := 399596, root := ~, start := 399584, stop := 399595 ), rec( content := " text fi\ le and you must have have ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "C", next := 399645, root := ~, start := 399630, stop := 399644 ), rec( content := " and\n \ ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "p\ dflatex", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "C", next := 399671, root := ~, start := 399656, stop := 399670 ), rec( content := " install\ ed on your computer. For details about these file\n formats, see ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.latex-project.org", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "URL", next := 399788, root := ~, start := 399749, stop := 399787 ), rec( content := " and\n \ ", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "URL", next := 399833, root := ~, start := 399799, stop := 399832 ), rec( content := ".", count := [ 9, 1, 1, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1, 5 ], mathmode := "Text", name := "P", next := 399838, root := ~, start := 399834, stop := 399837 ), rec( content := "\n\n \ This function is provided to allow convenient, immediate viewing of the\n \ pictures produced by the function ", count := [ 9, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DotDigr\ aph" ), content := 0, count := [ 9, 1, 1, 5 ], mathmode := "Text", name := "Ref", next := 399982, root := ~, start := 399958, stop := 399981 ), rec( content := ".", count := [ 9, 1, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "P", next := 399987, root := ~, start := 399983, stop := 399986 ), rec( content := "\n\n \ The optional second argument ", count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ pts", count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "A", next := 400035, root := ~, start := 400024, stop := 400034 ), rec( content := " should \ be a record with\n components corresponding to various options, given bel\ ow.\n\n ", count := [ 9, 1, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "path", count := [ 9, 1, 1, 7 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 7 ], name := "Mar\ k", next := 400163, root := ~, start := 400146, stop := 400162 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this should be a string r\ epresenting the path to the directory where\n you want ", count := [ \ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "Splash", count := [ 9, 1, 1, 8 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 8 ], math\ mode := "Text", name := "C", next := 400290, root := ~, start := 400277, stop \ := 400289 ), rec( content := " to do its work. The default value of this\n \ option is ", count := [ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"~\ /\"", count := [ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ 9, 1, 1, 8 ], mathmode := "Text", name := "C", next := 400364 , root\ := ~, start := 400353, stop := 400363 ), rec( content := ".\n ", count\ := [ 9, 1, 1, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 8 ], name := "Ite\ m", next := 400381, root := ~, start := 400172, stop := 400380 ), rec( attributes :\ = rec( ), content := [ rec( content := "directory", count := [ 9, 1, 1, 9 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 9 ], name := "Mar\ k", next := 400413, root := ~, start := 400391, stop := 400412 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this should be a string r\ epresenting the name of the directory in\n ", count := [ 9, 1, 1, 10 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "path", count := [ 9, 1, 1, 10 ], mathmode := "Text",\ name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 10 ], mathmode := "Text" , name\ := "C", next := 400526, root := ~, start := 400515, stop := 400525 ), rec( cont\ ent := " where you want ", count := [ 9, 1, 1, 10 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conten\ t := "Splash", count := [ 9, 1, 1, 10 ], mathmode := "Text", name := "PCDATA",\ root := ~ ) ], count := [ 9, 1, 1, 10 ], mathmode := "Text", name := "C", next\ := 400555, root := ~, start := 400542, stop := 400554 ), rec( content := " to\ do its work. This function\n will create this directory if does not \ already exist. ", count := [ 9, 1, 1, 10 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1,\ 11 ], mathmode := "Text", name := "P", next := 400654, root := ~, start := 40\ 0650, stop := 400653 ), rec( content := "\n\n The default value of th\ is option is ", count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"tmp.viz\"\ ", count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ )\ ], count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "C", next := 400718, root\ := ~, start := 400702, stop := 400717 ), rec( content := " if the option\n \ ", count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "path", coun\ t := [ 9, 1, 1, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 11 ], mathmode := "Text", name := "C", next := 400754, root :=\ ~, start := 400743, stop := 400753 ), rec( content := " is present, and the r\ esult of\n ", count := [ 9, 1, 1, 11 ], mathmode := "Text", name := "\ PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref", Func := "Dire\ ctoryTemporary" ), content := 0, count := [ 9, 1, 1, 11 ], mathmode := "Text", name\ := "Ref", next := 400842, root := ~, start := 400795, stop := 400841 ), rec( cont\ ent := " is used otherwise.\n ", count := [ 9, 1, 1, 11 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 10 ], name := "Ite\ m", next := 400877, root := ~, start := 400422, stop := 400876 ), rec( attributes :\ = rec( ), content := [ rec( content := "filename", count := [ 9, 1, 1, 12 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 12 ], name := "Mar\ k", next := 400908, root := ~, start := 400887, stop := 400907 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this should be a string r\ epresenting the name of the file where\n ", count := [ 9, 1, 1, 13 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "str", count := [ 9, 1, 1, 13 ], mathmode := "Text", \ name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 13 ], mathmode := "Text", name\ := "A", next := 401018, root := ~, start := 401008, stop := 401017 ), rec( cont\ ent := " will be written. The default value of this option is\n ", coun\ t := [ 9, 1, 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "\"vizpicture\"", count := [ 9\ , 1, 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 9, 1, 1, 13 ], mathmode := "Text", name := "C", next := 401102, root := ~, star\ t := 401083, stop := 401101 ), rec( content := ".\n ", count := [ 9, 1,\ 1, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 13 ], name := "Ite\ m", next := 401119, root := ~, start := 400917, stop := 401118 ), rec( attributes :\ = rec( ), content := [ rec( content := "viewer", count := [ 9, 1, 1, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 14 ], name := "Mar\ k", next := 401148, root := ~, start := 401129, stop := 401147 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this should be a string r\ epresenting the name of the program which\n should open the files pro\ duced by ", count := [ 9, 1, 1, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "GraphViz", \ count := [ 9, 1, 1, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 15 ], mathmode := "Text", name := "C", next := 401300, root :=\ ~, start := 401285, stop := 401299 ), rec( content := " or ", count := [ 9, 1\ , 1, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes \ := rec( ), content := [ rec( content := "pdflatex", count := [ 9, 1, 1, 15 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 15 ] , math\ mode := "Text", name := "C", next := 401319, root := ~, start := 401304, stop \ := 401318 ), rec( content := ".\n ", count := [ 9, 1, 1, 15 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 15 ], name := "Ite\ m", next := 401336, root := ~, start := 401157, stop := 401335 ), rec( attributes :\ = rec( ), content := [ rec( content := "type", count := [ 9, 1, 1, 16 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 16 ], name := "Mar\ k", next := 401363, root := ~, start := 401346, stop := 401362 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n this option can be used t\ o specify that the string ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "str", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", r\ oot := ~ ) ], count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "A", next \ := 401450, root := ~, start := 401440, stop := 401449 ), rec( content := " con\ tains\n a ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( Only := "LaTeX" ), content := [ r\ ec( content := "{\\LaTeX}", count := [ 9, 1, 1, 17 ], name := "PCDATA", root :\ = ~ ) ], count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "Alt", next := \ 33, root := ~, start := 1, stop := 32 ), rec( attributes := rec( Not := "LaTeX\ " ), content := [ rec( content := "LaTeX", count := [ 9, 1, 1, 17 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], name := \ "Alt", next := 61, root := ~, start := 33, stop := 60 ), rec( content := " or \ ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := [ rec( content := "dot", count := [ 9, 1, 1, 17 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 17\ ], mathmode := "Text", name := "C", next := 401493, root := ~, start := 40148\ 3, stop := 401492 ), rec( content := " document. You can specify this option i\ n\n ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "str", count\ := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 17 ], mathmode := "Text", name := "A", next := 401555, root :=\ ~, start := 401545, stop := 401554 ), rec( content := " directly by making th\ e first line ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"%latex\""\ , count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 9, 1, 1, 17 ], mathmode := "Text", name := "C", next := 401605, root :=\ ~, start := 401590, stop := 401604 ), rec( content := " or\n ", coun\ t := [ 9, 1, 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "\"//dot\"", count := [ 9, 1, \ 1, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1,\ 1, 17 ], mathmode := "Text", name := "C", next := 401633, root := ~, start :=\ 401619, stop := 401632 ), rec( content := ". There is no default value for t\ his option, this\n option must be specified in ", count := [ 9, 1, 1,\ 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "str", count := [ 9, 1, 1, 17 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], mathmode\ := "Text", name := "A", next := 401732, root := ~, start := 401722, stop := 4\ 01731 ), rec( content := " or in ", count := [ 9, 1, 1, 17 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ re\ c( content := "opt.type", count := [ 9, 1, 1, 17 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], mathmode := "Text", name \ := "A", next := 401754, root := ~, start := 401739, stop := 401753 ), rec( cont\ ent := ".\n ", count := [ 9, 1, 1, 17 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 9, 1, 1, 17 ], name := "Item", next := 40177\ 1, root := ~, start := 401372, stop := 401770 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "engine", count := [ 9, 1, 1, 18 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 1, 1, 18 ], name := "Mark", next := 401800, root := ~, start := 401\ 781, stop := 401799 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n this option can be used to specify the GraphViz engine to use\n\ to render a ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content \ := "dot", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next := \ 401920, root := ~, start := 401910, stop := 401919 ), rec( content := " docume\ nt. The valid choices are ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "\"dot\"", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next\ := 401965, root := ~, start := 401953, stop := 401964 ), rec( content := ",\n\ ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"neato\"",\ count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ] , coun\ t := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next := 401991, root :=\ ~, start := 401977, stop := 401990 ), rec( content := ", ", count := [ 9, 1, \ 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :=\ rec( ), content := [ rec( content := "\"circo\"", count := [ 9, 1, 1, 19 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 19 ], math\ mode := "Text", name := "C", next := 402007, root := ~, start := 401993, stop \ := 402006 ), rec( content := ", ", count := [ 9, 1, 1, 19 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec\ ( content := "\"twopi\"", count := [ 9, 1, 1, 19 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode := "Text", name \ := "C", next := 402023, root := ~, start := 402009, stop := 402022 ), rec( cont\ ent := ", ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "\"fdp\"", c\ ount := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next := 402037, root :=\ ~, start := 402025, stop := 402036 ), rec( content := ",\n ", count \ := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "\"sfdp\"", count := [ 9, 1, 1\ , 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, \ 1, 19 ], mathmode := "Text", name := "C", next := 402062, root := ~, start := \ 402049, stop := 402061 ), rec( content := ", and ", count := [ 9, 1, 1, 19 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "\"patchwork\"", count := [ 9, 1, 1, 19 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode \ := "Text", name := "C", next := 402086, root := ~, start := 402068, stop := 40\ 2085 ), rec( content := ". Please refer to the\n GraphViz documentati\ on for details on these engines.\n The default value for this option \ is ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := [ rec( content := "\"dot\"", count\ := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 19 ], mathmode := "Text", name := "C", next := 402230, root :=\ ~, start := 402218, stop := 402229 ), rec( content := ", and it\n mu\ st be specified in ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "P\ CDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := \ "opt.engine", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", \ root := ~ ) ], count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "A", next\ := 402287, root := ~, start := 402270, stop := 402286 ), rec( content := ".\n\ ", count := [ 9, 1, 1, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 1, 1, 19 ], name := "Item", next := 402304, root := ~, start := 401\ 809, stop := 402303 ), rec( attributes := rec( ), content := [ rec( content :\ = "filetype", count := [ 9, 1, 1, 20 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 1, 1, 20 ], name := "Mark", next := 402335, root := ~, start := 402\ 314, stop := 402334 ), rec( attributes := rec( ), content := [ rec( content :\ = "\n this should be a string representing the type of file which\n \ ", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "Splash", co\ unt := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], coun\ t := [ 9, 1, 1, 21 ], mathmode := "Text", name := "C", next := 402444, root :=\ ~, start := 402431, stop := 402443 ), rec( content := " should produce. For " , coun\ t := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( Only := "LaTeX" ), content := [ rec( content := "{\\LaTeX}", co\ unt := [ 9, 1, 1, 21 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 2\ 1 ], mathmode := "Text", name := "Alt", next := 33, root := ~, start := 1, stop\ := 32 ), rec( attributes := rec( Not := "LaTeX" ), content := [ rec( content \ := "LaTeX", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA", ro\ ot := ~ ) ], count := [ 9, 1, 1, 21 ], name := "Alt", next := 61, root := ~, star\ t := 33, stop := 60 ), rec( content := " files, this option is\n igno\ red and the default value ", count := [ 9, 1, 1, 21 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "\"pdf\"", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "C", next\ := 402547, root := ~, start := 402535, stop := 402546 ), rec( content := " is\ used.\n ", count := [ 9, 1, 1, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 1, 21 ], name := "Item", next := 402572, root := ~, start := 402\ 344, stop := 402571 ) ], count := [ 9, 1, 1, 7 ], mathmode := "Text", name := "List", next := 402586, root := ~, start := 400131, stop := 402585 ), rec( content := "\n\n \ This function was written by Attila Egri-Nagy and Manuel Delgado with some\n\ minor changes by J. D. Mitchell.", count := [ 9, 1, 1, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 1, 24 ], mathmode := "Text", name := "P", next := 402711, root := ~, start := 402707, stop := 402710 ), rec( content := "\n \ ", count := [ 9, 1, 1, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> Splash(DotDigraph(RandomDigraph(4)));\n", count := [ 9, 1, 1, 25 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 1, 25 ], mathmode := "Text", name := "Log", next := 402785, root := ~, start := 402718, stop := 402784 ), rec( content := "\n ", count := [ 9, 1, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 1, 3 ], mathmode := "Text", name := "Description", next := 402804, root := ~, start := 399337, stop := 402803 ) ], count := [ 9, 1, 1, 1 ], mathmode := "Text", name := "ManSection", next := 402820, root := ~, start := 399244, stop := 402819 ), rec( content := "\n\n ", count := [ 9, 1, 1, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DotDigraph" ), content := 0, count := [ 9, 1, 2, 2 ], mathmode := "Text", name := "Attr", next := 402884, root := ~, start := 402845, stop := 402883 ), rec( attributes := rec( Arg := "digraph, vert, e\ dge", Name := "DotColoredDigraph" ), content := 0, count := [ 9, 1, 2, 4 ], mathmode := "Text", name := "Oper", next := 402947, root := ~, start := 402889, stop := 402946 ), rec( attributes := rec( Arg := "digraph", Name := "DotVertexLabell\ edDigraph" ), content := 0, count := [ 9, 1, 2, 6 ], mathmode := "Text", name := "Oper", next := 403005, root := ~, start := 402952, stop := 403004 ), rec( attributes := rec( Arg := "digraph, vert", Name := "DotVertexColore\ dDigraph" ), content := 0, count := [ 9, 1, 2, 8 ], mathmode := "Text", name := "Oper", next := 403068, root := ~, start := 403010, stop := 403067 ), rec( attributes := rec( Arg := "digraph, edge", Name := "DotEdgeColoredD\ igraph" ), content := 0, count := [ 9, 1, 2, 10 ], mathmode := "Text", name := "Oper", next := 403129, root := ~, start := 403073, stop := 403128 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "Returns", next := 403162, root := ~, start := 403134, stop := 403161 ), rec( attributes := rec( ), content := [ rec( content := "\n " , count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "C", next := 403204, root := ~, start := 403187, stop := 403203 ), rec( content := " produce\ s a graphical representation of the digraph\n ", count := [ 9, 1, 2, 11 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "A", next := 403276, root := ~, start := 403262, stop := 403275 ), rec( content := ". Vertic\ es are displayed as circles, numbered consistently\n with ", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "A", next := 403360, root := ~, start := 403346, stop := 403359 ), rec( content := ". Edges \ are displayed as arrowed lines between\n vertices, with the arrowhead of \ each line pointing towards the range\n of the edge.", count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "P", next := 403504, root := ~, start := 403500, stop := 403503 ), rec( content := "\n\n \ ", count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otColoredDigraph", count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "C", next := 403536, root := ~, start := 403512, stop := 403535 ), rec( content := " differs\ from ", count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "C", next := 403567, root := ~, start := 403550, stop := 403566 ), rec( content := " only in\ \n that the values in given in the two lists are used to color the vertic\ es and\n edges of the graph when displayed. The list for vertex colours s\ hould be\n a list of length equal to the number of vertices, containing s\ trings that\n are accepted by the graphviz software, which is the one use\ d for graph\n representation. The list for edge colours should be a list \ of lists\n with the same shape of the outneighbours of the digraph that c\ ontains strings \n that correspond to colours accepted by the graphviz so\ ftware. If the lists \n are not the appropriate size, or have holes then \ the function will return\n an error.", count := [ 9, 1, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "P", next := 404235, root := ~, start := 404231, stop := 404234 ), rec( content := "\n\n \ ", count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otVertexColoredDigraph", count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "C", next := 404273, root := ~, start := 404243, stop := 404272 ), rec( content := " differs\ from ", count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "C", next := 404304, root := ~, start := 404287, stop := 404303 ), rec( content := " only in\ \n that the values in given in the list are used to color the vertices\n \ of the graph when displayed. The list for vertex colours should be\n \ a list of length equal to the number of vertices, containing strings that\n \ are accepted by the graphviz software, which is the one used for graph\n \ representation. If the list is not the appropriate size, or has holes \n \ then the function will return an error.", count := [ 9, 1, 2, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "P", next := 404743, root := ~, start := 404739, stop := 404742 ), rec( content := "\n\n \ ", count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otEdgeColoredDigraph", count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "C", next := 404779, root := ~, start := 404751, stop := 404778 ), rec( content := " differs\ from ", count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "C", next := 404810, root := ~, start := 404793, stop := 404809 ), rec( content := " only in\ \n that the values in given in the list are used to color the vertices an\ d\n edges of the graph when displayed. The list for edge colours should b\ e a list\n of lists with the same shape of the outneighbours of the digra\ ph that contains\n strings that correspond to colours accepted by the gra\ phviz software. If the \n list is not the appropriate size, or has holes \ then the function will return\n an error.", count := [ 9, 1, 2, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "P", next := 405252, root := ~, start := 405248, stop := 405251 ), rec( content := "\n\n \ ", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otVertexLabelledDigraph", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "C", next := 405291, root := ~, start := 405260, stop := 405290 ), rec( content := " differs\ from ", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "C", next := 405322, root := ~, start := 405305, stop := 405321 ), rec( content := " only in\ \n that the values in ", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "Ref", next := 405389, root := ~, start := 405356, stop := 405388 ), rec( content := " are use\ d to label\n the vertices in the produced picture rather than the numbers\ ", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "C", next := 405483, root := ~, start := 405475, stop := 405482 ), rec( content := " to\n \ the number of vertices of the digraph. ", count := [ 9, 1, 2, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "P", next := 405536, root := ~, start := 405532, stop := 405535 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "C", next := 405571, root := ~, start := 405561, stop := 405570 ), rec( content := " format \ (also known as ", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "C", next := 405609, root := ~, start := 405594, stop := 405608 ), rec( content := ")\n \ format. For details about this file format, and information about how to\n \ display or edit this format see ", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "URL", next := 405762, root := ~, start := 405728, stop := 405761 ), rec( content := ". ", count := [ 9, 1, 2, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "P", next := 405768, root := ~, start := 405764, stop := 405767 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "C", next := 405816, root := ~, start := 405799, stop := 405815 ), rec( content := " or\n \ ", count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otVertexLabelledDigraph", count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "C", next := 405857, root := ~, start := 405826, stop := 405856 ), rec( content := " can be \ written to a file using\n the command ", count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "Ref", next := 405949, root := ~, start := 405907, stop := 405948 ), rec( content := ".", count := [ 9, 1, 2, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 2, 18 ], mathmode := "Text", name := "P", next := 405954, root := ~, start := 405950, stop := 405953 ), rec( content := "\n \ ", count := [ 9, 1, 2, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> adj := List([1 .. 4], x -> [1, 1, 1, 1]);\n[ [ 1, 1, 1, 1 ], [ 1, 1, 1, \ 1 ], [ 1, 1, 1, 1 ], [ 1, 1, 1, 1 ] ]\ngap> adj[1][3] := 0;\n0\ngap> gr := Dig\ raphByAdjacencyMatrix(adj);\n\nga\ p> D := CompleteDigraph(4);\n\ngap\ > vertcolors := [];;\ngap> vertcolors[1] := \"blue\";; vertcolors[2] := \"red\ \";; \ngap> vertcolors[3] := \"green\";; vertcolors[4] := \"yellow\";;\ngap> e\ dgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];;\ngap> edge\ colors[3] := [];; edgecolors[4] := [];; \ngap> edgecolors[1][2] := \"lightblue\ \";;\ngap> edgecolors[1][3] := \"pink\";;\ngap> edgecolors[1][4] := \"purple\"\ ;;\ngap> edgecolors[2][1] := \"lightblue\";;\ngap> edgecolors[2][3] := \"pink\ \";; \ngap> edgecolors[2][4] := \"purple\";; \ngap> edgecolors[3][1] := \"ligh\ tblue\";; \ngap> edgecolors[3][2] := \"pink\";; \ngap> edgecolors[3][4] := \"p\ urple\";;\ngap> edgecolors[4][1] := \"lightblue\";; \ngap> edgecolors[4][2] :=\ \"pink\";;\ngap> edgecolors[4][3] := \"purple\";;\ngap> Print(DotColoredDigra\ ph(D, vertcolors, edgecolors));\n//dot\ndigraph hgn{\nnode [shape=circle]\n1[c\ olor=blue, style=filled]\n2[color=red, style=filled]\n3[color=green, style=fil\ led]\n4[color=yellow, style=filled]\n1 -> 2[color=lightblue]\n1 -> 3[color=pin\ k]\n1 -> 4[color=purple]\n2 -> 1[color=lightblue]\n2 -> 3[color=pink]\n2 -> 4[\ color=purple]\n3 -> 1[color=lightblue]\n3 -> 2[color=pink]\n3 -> 4[color=purpl\ e]\n4 -> 1[color=lightblue]\n4 -> 2[color=pink]\n4 -> 3[color=purple]\n}\ngap>\ D := EmptyDigraph(3);\n\ngap> vertco\ lors := [];;\ngap> vertcolors[1] := \"blue\";; vertcolors[2] := \"red\";;\ngap\ > vertcolors[3] := \"green\";;\ngap> edgecolors := [];;\ngap> edgecolors[1] :=\ [];; edgecolors[2] := [];; \ngap> edgecolors[3] := [];;\ngap> Print(DotColore\ dDigraph(D, vertcolors, edgecolors));\n//dot\ndigraph hgn{\nnode [shape=circle\ ]\n1[color=blue, style=filled]\n2[color=red, style=filled]\n3[color=green, sty\ le=filled]\n}\ngap> D := Digraph([[2], [1, 3], [2]]);\n\ngap> vertcolors := [];;\ngap> vertcolors[1] := \"blue\"\ ;;\ngap> vertcolors[2] := \"pink\";;\ngap> vertcolors[3] := \"purple\";;\ngap>\ edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];;\ngap> ed\ gecolors[3] := [];;\ngap> edgecolors[1][2] := \"green\";; edgecolors[2][1] := \ \"green\";;\ngap> edgecolors[2][3] := \"red\";; edgecolors[3][2] := \"red\";;\ \ngap> Print(DotSymmetricColoredDigraph(D, vertcolors, edgecolors));\n//dot\ng\ raph hgn{\nnode [shape=circle]\n\n1[color=blue, style=filled]\n2[color=pink, s\ tyle=filled]\n3[color=purple, style=filled]\n1 -- 2[color=green]\n2 -- 3[color\ =red]\n}\ngap> D := Digraph([[2, 3], [1, 3], [1]]);\n\ngap> vertcolors := [];;\ngap> vertcolors[1] := \"blue\";;\ vertcolors[2] := \"red\";;\ngap> vertcolors[3] := \"green\";;\ngap> edgecolor\ s := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];;\ngap> edgecolors[3\ ] := [];;\ngap> edgecolors[1][2] := \"orange\";; edgecolors[1][3] := \"yellow\ \";;\ngap> edgecolors[2][1] := \"orange\";; edgecolors[2][3] := \"pink\";;\nga\ p> edgecolors[3][1] := \"yellow\";;\ngap> Print(DotColoredDigraph(D, vertcolor\ s, edgecolors));;\n//dot\ndigraph hgn{\nnode [shape=circle]\n1[color=blue, sty\ le=filled]\n2[color=red, style=filled]\n3[color=green, style=filled]\n1 -> 2[c\ olor=orange]\n1 -> 3[color=yellow]\n2 -> 1[color=orange]\n2 -> 3[color=pink]\n\ 3 -> 1[color=yellow]\n}\ngap> D := Digraph(IsMutableDigraph, [[2, 3], [1, 3], \ [1]]);\n\ngap> vertcolors := [];;\ng\ ap> vertcolors[1] := \"blue\";; vertcolors[2] := \"red\";;\ngap> vertcolors[3]\ := \"green\";;\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolor\ s[2] := [];;\ngap> edgecolors[3] := [];;\ngap> edgecolors[1][2] := \"orange\";\ ; edgecolors[1][3] := \"yellow\";;\ngap> edgecolors[2][1] := \"orange\";; edge\ colors[2][3] := \"pink\";;\ngap> edgecolors[3][1] := \"yellow\";;\ngap> Print(\ DotColoredDigraph(D, vertcolors, edgecolors));;\n//dot\ndigraph hgn{\nnode [sh\ ape=circle]\n1[color=blue, style=filled]\n2[color=red, style=filled]\n3[color=\ green, style=filled]\n1 -> 2[color=orange]\n1 -> 3[color=yellow]\n2 -> 1[color\ =orange]\n2 -> 3[color=pink]\n3 -> 1[color=yellow]\n}\ngap> D;\n\ngap> DotSymmetricDigraph(gr2){[12 .. 70]};\n\" h\ gn{\\nnode [shape=circle]\\n\\n1\\n2\\n3\\n4\\n1 -- 2\\n2 -- 3\\n3 -- 3\\n3 -\ \"\ngap> DotSymmetricDigraph(gr1);\nError, the argument must be a symmetri\ c digraph,\ngap> D := CompleteDigraph(4);\n\ngap> vertcolors := [];;\ngap> vertcolors[1] := \"blue\";; vertcolor\ s[2] := \"red\";; \ngap> vertcolors[3] := \"green\";; vertcolors[4] := \"yello\ w\";;\ngap> Print(DotVertexColoredDigraph(D, vertcolors));\n//dot\ndigraph hgn\ {\nnode [shape=circle]\n1[color=blue, style=filled]\n2[color=red, style=filled\ ]\n3[color=green, style=filled]\n4[color=yellow, style=filled]\n1 -> 2\n1 -> 3\ \n1 -> 4\n2 -> 1\n2 -> 3\n2 -> 4\n3 -> 1\n3 -> 2\n3 -> 4\n4 -> 1\n4 -> 2\n4 ->\ 3\n}\ngap> D := CompleteDigraph(4);\n\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];\ ;\ngap> edgecolors[3] := [];; edgecolors[4] := [];; \ngap> edgecolors[1][2] :=\ \"lightblue\";;\ngap> edgecolors[1][3] := \"pink\";;\ngap> edgecolors[1][4] :\ = \"purple\";;\ngap> edgecolors[2][1] := \"lightblue\";;\ngap> edgecolors[2][3\ ] := \"pink\";; \ngap> edgecolors[2][4] := \"purple\";; \ngap> edgecolors[3][1\ ] := \"lightblue\";; \ngap> edgecolors[3][2] := \"pink\";; \ngap> edgecolors[3\ ][4] := \"purple\";;\ngap> edgecolors[4][1] := \"lightblue\";; \ngap> edgecolo\ rs[4][2] := \"pink\";;\ngap> edgecolors[4][3] := \"purple\";;\ngap> Print(DotE\ dgeColoredDigraph(D, edgecolors));\n//dot\ndigraph hgn{\nnode [shape=circle]\n\ 1\n2\n3\n4\n1 -> 2[color=lightblue]\n1 -> 3[color=pink]\n1 -> 4[color=purple]\ \n2 -> 1[color=lightblue]\n2 -> 3[color=pink]\n2 -> 4[color=purple]\n3 -> 1[co\ lor=lightblue]\n3 -> 2[color=pink]\n3 -> 4[color=purple]\n4 -> 1[color=lightbl\ ue]\n4 -> 2[color=pink]\n4 -> 3[color=purple]\n}\ngap> FileString(\"dot/k4.dot\ \", DotDigraph(gr));\n154", count := [ 9, 1, 2, 19 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 19 ], mathmode := "Text", name := "Log", next := 411909, root := ~, start := 405961, stop := 411908 ), rec( content := "\n ", count := [ 9, 1, 2, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 2, 11 ], mathmode := "Text", name := "Description", next := 411928, root := ~, start := 403167, stop := 411927 ) ], count := [ 9, 1, 2, 1 ], mathmode := "Text", name := "ManSection", next := 411944, root := ~, start := 402828, stop := 411943 ), rec( content := "\n\n ", count := [ 9, 1, 2, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DotSymmetricDig\ raph" ), content := 0, count := [ 9, 1, 3, 2 ], mathmode := "Text", name := "Attr", next := 412017, root := ~, start := 411969, stop := 412016 ), rec( attributes := rec( Arg := "digraph, vert, e\ dge", Name := "DotSymmetricColoredDigraph" ), content := 0, count := [ 9, 1, 3, 4 ], mathmode := "Text", name := "Oper", next := 412089, root := ~, start := 412022, stop := 412088 ), rec( attributes := rec( Arg := "digraph, vert", Name := "DotSymmetricVer\ texColoredDigraph" ), content := 0, count := [ 9, 1, 3, 6 ], mathmode := "Text", name := "Oper", next := 412161, root := ~, start := 412094, stop := 412160 ), rec( attributes := rec( Arg := "digraph, edge", Name := "DotSymmetricEdg\ eColoredDigraph" ), content := 0, count := [ 9, 1, 3, 8 ], mathmode := "Text", name := "Oper", next := 412231, root := ~, start := 412166, stop := 412230 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "Returns", next := 412264, root := ~, start := 412236, stop := 412263 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function produces a graphical representation of the symmetric\n digra\ ph ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "A", next := 412384, root := ~, start := 412370, stop := 412383 ), rec( content := ". ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricDigraph", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "C", next := 412412, root := ~, start := 412386, stop := 412411 ), rec( content := " will re\ turn an\n error if ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "A", next := 412457, root := ~, start := 412443, stop := 412456 ), rec( content := " is not \ a symmetric digraph. See\n ", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "Ref", next := 412528, root := ~, start := 412496, stop := 412527 ), rec( content := ".", count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "P", next := 412533, root := ~, start := 412529, stop := 412532 ), rec( content := "\n\n \ The function ", count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricColoredDigraph", count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "C", next := 412587, root := ~, start := 412554, stop := 412586 ), rec( content := " differs\ from ", count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "C", next := 412618, root := ~, start := 412601, stop := 412617 ), rec( content := "\n \ only in that the values given in the two lists are used to color the vertices\ \n and edges of the graph when displayed. The list for vertex colours sho\ uld be\n a list of length equal to the number of vertices, containing str\ ings that\n are accepted by the graphviz software, which is the one used \ for graph\n representation. The list for edge colours should be a list o\ f lists\n with the same shape of the outneighbours of the digraph that co\ ntains strings \n that correspond to colours accepted by the graphviz sof\ tware.\n If the list is not the appropriate size, or has holes then the f\ unction\n will return an error. ", count := [ 9, 1, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "P", next := 413282, root := ~, start := 413278, stop := 413281 ), rec( content := "\n\n \ The function ", count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricVertexColoredDigraph", count := [ 9, 1, 3, 11 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "C", next := 413342, root := ~, start := 413303, stop := 413341 ), rec( content := " differs\ from ", count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "C", next := 413373, root := ~, start := 413356, stop := 413372 ), rec( content := "\n \ only in that the values in given in the list is used to color the vertices\n \ of the graph when displayed. The list for vertex colours should be\n \ a list of length equal to the number of vertices, containing strings that\n \ are accepted by the graphviz software, which is the one used for graph\n \ representation. If the list is not the appropriate size, or has holes \n \ then the function will return an error. ", count := [ 9, 1, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "P", next := 413813, root := ~, start := 413809, stop := 413812 ), rec( content := "\n\n \ The function ", count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricEdgeColoredDigraph", count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "C", next := 413871, root := ~, start := 413834, stop := 413870 ), rec( content := " differs\ from ", count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otDigraph", count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "C", next := 413902, root := ~, start := 413885, stop := 413901 ), rec( content := "\n \ only in that the values given in the list are used to color the edges\n o\ f the graph when displayed. The list for edge colours should be\n a list \ of lists with the same shape of the outneighbours, containing strings that\n \ are accepted by the graphviz software, which is the one used for graph\n \ representation. If the list is not the appropriate size, or has holes \n \ then the function will return an error. ", count := [ 9, 1, 3, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "P", next := 414343, root := ~, start := 414339, stop := 414342 ), rec( content := "\n\n \ Vertices are displayed as circles, numbered consistently with\n ", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "A", next := 414433, root := ~, start := 414419, stop := 414432 ), rec( content := ". Since \ ", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "A", next := 414455, root := ~, start := 414441, stop := 414454 ), rec( content := " is symm\ etric, for every non-loop\n edge there is a complementary edge with oppos\ ite source and range.\n ", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricDigraph", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "C", next := 414594, root := ~, start := 414568, stop := 414593 ), rec( content := " display\ s each pair of complementary edges\n as a single line between the relevan\ t vertices, with no arrowhead.", count := [ 9, 1, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "P", next := 414713, root := ~, start := 414709, stop := 414712 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "C", next := 414748, root := ~, start := 414738, stop := 414747 ), rec( content := " format \ (also known as ", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "C", next := 414786, root := ~, start := 414771, stop := 414785 ), rec( content := ")\n \ format. For details about this file format, and information about how\n \ to display or edit this format see\n ", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "URL", next := 414945, root := ~, start := 414911, stop := 414944 ), rec( content := ". ", count := [ 9, 1, 3, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "P", next := 414951, root := ~, start := 414947, stop := 414950 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otSymmetricDigraph", count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "C", next := 415008, root := ~, start := 414982, stop := 415007 ), rec( content := " can be \ written to a\n file using the command ", count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "Ref", next := 415100, root := ~, start := 415058, stop := 415099 ), rec( content := ".", count := [ 9, 1, 3, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 3, 16 ], mathmode := "Text", name := "P", next := 415105, root := ~, start := 415101, stop := 415104 ), rec( content := "\n\n \ ", count := [ 9, 1, 3, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> star := Digraph([[2, 2, 3, 4], [1, 1], [1], [1, 4]]);\n\ngap> IsSymmetricDigraph(star);\ntrue\ngap> F\ ileString(\"dot/star.dot\", DotSymmetricDigraph(gr));\ngap> D := Digraph([[2],\ [1, 3], [2]]);\n\ngap> vertcolors\ := [];;\ngap> vertcolors[1] := \"blue\";;\ngap> vertcolors[2] := \"pink\";;\n\ gap> vertcolors[3] := \"purple\";;\ngap> edgecolors := [];;\ngap> edgecolors[1\ ] := [];; edgecolors[2] := [];;\ngap> edgecolors[3] := [];;\ngap> edgecolors[1\ ][2] := \"green\";; edgecolors[2][1] := \"green\";;\ngap> edgecolors[2][3] := \ \"red\";; edgecolors[3][2] := \"red\";;\ngap> Print(DotSymmetricColoredDigraph\ (D, vertcolors, edgecolors));\n//dot\ngraph hgn{\nnode [shape=circle]\n\n1[col\ or=blue, style=filled]\n2[color=pink, style=filled]\n3[color=purple, style=fil\ led]\n1 -- 2[color=green]\n2 -- 3[color=red]\n}\ngap> D := Digraph([[2], [1, 3\ ], [2]]);\n\ngap> vertcolors := []\ ;;\ngap> vertcolors[1] := \"blue\";;\ngap> vertcolors[2] := \"pink\";;\ngap> v\ ertcolors[3] := \"purple\";;\ngap> Print(DotSymmetricVertexColoredDigraph(D, v\ ertcolors));\n//dot\ngraph hgn{\nnode [shape=circle]\n\n1[color=blue, style=fi\ lled]\n2[color=pink, style=filled]\n3[color=purple, style=filled]\n1 -- 2\n2 -\ - 3\n}\ngap> D := Digraph([[2], [1, 3], [2]]);\n\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors\ [2] := [];;\ngap> edgecolors[3] := [];;\ngap> edgecolors[1][2] := \"green\";; \ edgecolors[2][1] := \"green\";;\ngap> edgecolors[2][3] := \"red\";; edgecolors\ [3][2] := \"red\";;\ngap> Print(DotSymmetricEdgeColoredDigraph(D, edgecolors))\ ;\n//dot\ngraph hgn{\nnode [shape=circle]\n\n1\n2\n3\n1 -- 2[color=green]\n2 -\ - 3[color=red]\n}\n83", count := [ 9, 1, 3, 17 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 17 ], mathmode := "Text", name := "Log", next := 416859, root := ~, start := 415113, stop := 416858 ), rec( content := "\n ", count := [ 9, 1, 3, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 3, 9 ], mathmode := "Text", name := "Description", next := 416878, root := ~, start := 412269, stop := 416877 ) ], count := [ 9, 1, 3, 1 ], mathmode := "Text", name := "ManSection", next := 416894, root := ~, start := 411952, stop := 416893 ), rec( content := "\n\n ", count := [ 9, 1, 3, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DotPartialOrder\ Digraph" ), content := 0, count := [ 9, 1, 4, 2 ], mathmode := "Text", name := "Attr", next := 416970, root := ~, start := 416919, stop := 416969 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "Returns", next := 417003, root := ~, start := 416975, stop := 417002 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function produces a graphical representation of a partial order\n dig\ raph ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "A", next := 417125, root := ~, start := 417111, stop := 417124 ), rec( content := ". ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPartialOrderDigraph", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "C", next := 417156, root := ~, start := 417127, stop := 417155 ), rec( content := " will re\ turn an error\n if ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "A", next := 417201, root := ~, start := 417187, stop := 417200 ), rec( content := " is not \ a partial order digraph. See ", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsParti\ alOrderDigraph" ), content := 0, count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "Ref", next := 417281, root := ~, start := 417238, stop := 417280 ), rec( content := ".", count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "P", next := 417286, root := ~, start := 417282, stop := 417285 ), rec( content := "\n\n \ Since ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "A", next := 417314, root := ~, start := 417300, stop := 417313 ), rec( content := " is a pa\ rtial order, it is both reflexive and\n transitive. The output of ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPartialOrderDigraph", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "C", next := 417421, root := ~, start := 417392, stop := 417420 ), rec( content := " is the\\ n ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DotDigr\ aph" ), content := 0, count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "Ref", next := 417459, root := ~, start := 417435, stop := 417458 ), rec( content := " of the\\ n ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ReflexiveTransitiveReduction" ), content := 0, count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "Ref", next := 417522, root := ~, start := 417473, stop := 417521 ), rec( content := " of ", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "A", next := 417540, root := ~, start := 417526, stop := 417539 ), rec( content := ".", count := [ 9, 1, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "P", next := 417545, root := ~, start := 417541, stop := 417544 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "C", next := 417580, root := ~, start := 417570, stop := 417579 ), rec( content := " format \ (also known as ", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "C", next := 417618, root := ~, start := 417603, stop := 417617 ), rec( content := ")\n \ format. For details about this file format, and information about how\n \ to display or edit this format see\n ", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "URL", next := 417777, root := ~, start := 417743, stop := 417776 ), rec( content := ". ", count := [ 9, 1, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "P", next := 417783, root := ~, start := 417779, stop := 417782 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPartialOrderDigraph", count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "C", next := 417843, root := ~, start := 417814, stop := 417842 ), rec( content := " can be \ written to a\n file using the command ", count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "Ref", next := 417935, root := ~, start := 417893, stop := 417934 ), rec( content := ".", count := [ 9, 1, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 4, 7 ], mathmode := "Text", name := "P", next := 417940, root := ~, start := 417936, stop := 417939 ), rec( content := "\n\n \ ", count := [ 9, 1, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> poset := Digraph([[1, 4], [2], [2, 3, 4], [4]);\ngap> IsPartialOrderDigr\ aph(gr);\ntrue\ngap> FileString(\"dot/poset.dot\", DotPartialOrderDigraph(gr))\ ;\n83", count := [ 9, 1, 4, 8 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 8 ], mathmode := "Text", name := "Log", next := 418126, root := ~, start := 417948, stop := 418125 ), rec( content := "\n ", count := [ 9, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 4, 3 ], mathmode := "Text", name := "Description", next := 418145, root := ~, start := 417008, stop := 418144 ) ], count := [ 9, 1, 4, 1 ], mathmode := "Text", name := "ManSection", next := 418161, root := ~, start := 416902, stop := 418160 ), rec( content := "\n\n ", count := [ 9, 1, 4, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "DotPreorderDigr\ aph" ), content := 0, count := [ 9, 1, 5, 2 ], mathmode := "Text", name := "Attr", next := 418233, root := ~, start := 418186, stop := 418232 ), rec( attributes := rec( Arg := "digraph", Name := "DotQuasiorderDi\ graph" ), content := 0, count := [ 9, 1, 5, 4 ], mathmode := "Text", name := "Attr", next := 418287, root := ~, start := 418238, stop := 418286 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "Returns", next := 418320, root := ~, start := 418292, stop := 418319 ), rec( attributes := rec( ), content := [ rec( content := "\n T\ his function produces a graphical representation of a preorder\n digraph \ ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "A", next := 418437, root := ~, start := 418423, stop := 418436 ), rec( content := ". ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPreorderDigraph", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "C", next := 418464, root := ~, start := 418439, stop := 418463 ), rec( content := " will re\ turn an error\n if ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "A", next := 418509, root := ~, start := 418495, stop := 418508 ), rec( content := " is not \ a preorder digraph. See ", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsPreor\ derDigraph" ), content := 0, count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "Ref", next := 418580, root := ~, start := 418541, stop := 418579 ), rec( content := ".", count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "P", next := 418585, root := ~, start := 418581, stop := 418584 ), rec( content := "\n\n \ A preorder digraph is reflexive and transitive but in general it is\n n\ ot anti-symmetric and may have strongly connected components\n containing\ more than one vertex. The ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Quotien\ tDigraph" ), content := 0, count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "Ref", next := 418801, root := ~, start := 418772, stop := 418800 ), rec( content := "\n \ ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Q\ ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 418816, root := ~, start := 418808, stop := 418815 ), rec( content := " obtaine\ d by forming the quotient of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 418867, root := ~, start := 418853, stop := 418866 ), rec( content := " by the\\ n partition of its vertices into the strongly connected components\n \ satisfies ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsParti\ alOrderDigraph" ), content := 0, count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "Ref", next := 418997, root := ~, start := 418962, stop := 418996 ), rec( content := ". Thus e\ very vertex of\n ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Q\ ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 419034, root := ~, start := 419026, stop := 419033 ), rec( content := " corresp\ onds to a strongly connected component of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 419098, root := ~, start := 419084, stop := 419097 ), rec( content := ".\n \ The output of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPreorderDigraph", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "C", next := 419145, root := ~, start := 419120, stop := 419144 ), rec( content := " display\ s the \n ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ReflexiveTransitiveReduction" ), content := 0, count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "Ref", next := 419215, root := ~, start := 419166, stop := 419214 ), rec( content := " of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Q\ ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 419227, root := ~, start := 419219, stop := 419226 ), rec( content := " with\n \ vertices displayed as rounded rectangles labelled by all of the vertices\ \n of ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "A", next := 419335, root := ~, start := 419321, stop := 419334 ), rec( content := " in the \ corresponding strongly connected component. ", count := [ 9, 1, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "P", next := 419391, root := ~, start := 419387, stop := 419390 ), rec( content := "\n\n \ The output is in ", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ot", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "C", next := 419426, root := ~, start := 419416, stop := 419425 ), rec( content := " format \ (also known as ", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphViz", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "C", next := 419464, root := ~, start := 419449, stop := 419463 ), rec( content := ")\n \ format. For details about this file format, and information about how\n \ to display or edit this format see\n ", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://www.graphviz.org", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "URL", next := 419623, root := ~, start := 419589, stop := 419622 ), rec( content := ". ", count := [ 9, 1, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "P", next := 419629, root := ~, start := 419625, stop := 419628 ), rec( content := "\n\n \ The string returned by ", count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otPreorderDigraph", count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "C", next := 419685, root := ~, start := 419660, stop := 419684 ), rec( content := " can be \ written to a\n file using the command ", count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "GAP\ Doc", Func := "FileString" ), content := 0, count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "Ref", next := 419777, root := ~, start := 419735, stop := 419776 ), rec( content := ".", count := [ 9, 1, 5, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 5, 9 ], mathmode := "Text", name := "P", next := 419782, root := ~, start := 419778, stop := 419781 ), rec( content := "\n\n \ ", count := [ 9, 1, 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> preset := Digraph([[1, 2, 4, 5], [1, 2, 4, 5], [3, 4], [4], [1, 2, 4, 5]\ );\ngap> IsPreorderDigraph(gr);\ntrue\ngap> FileString(\"dot/preset.dot\", Dot\ ProrderDigraph(gr));\n83", count := [ 9, 1, 5, 10 ], name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 10 ], mathmode := "Text", name := "Log", next := 419987, root := ~, start := 419790, stop := 419986 ), rec( content := "\n ", count := [ 9, 1, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 5, 5 ], mathmode := "Text", name := "Description", next := 420006, root := ~, start := 418325, stop := 420005 ) ], count := [ 9, 1, 5, 1 ], mathmode := "Text", name := "ManSection", next := 420022, root := ~, start := 418169, stop := 420021 ), rec( content := "\n\n ", count := [ 9, 1, 5, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph, verts [\ , colour1, colour2]", Name := "DotHighlightedDigraph" ), content := 0, count := [ 9, 1, 6, 2 ], mathmode := "Text", name := "Oper", next := 420125, root := ~, start := 420047, stop := 420124 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "Returns", next := 420158, root := ~, start := 420130, stop := 420157 ), rec( attributes := rec( ), content := [ rec( content := "\n " , count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otHighlightedDigraph", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "C", next := 420211, root := ~, start := 420183, stop := 420210 ), rec( content := " produce\ s a graphical represenation of the\n digraph ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 420282, root := ~, start := 420268, stop := 420281 ), rec( content := ", where \ the vertices in the list ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ erts", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 420327, root := ~, start := 420315, stop := 420326 ), rec( content := ", and\n \ edges between them, are drawn with colour ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour1", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 420395, root := ~, start := 420381, stop := 420394 ), rec( content := " and all\ other\n vertices and edges in ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 420452, root := ~, start := 420438, stop := 420451 ), rec( content := " are dra\ wn with colour ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour2", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 420489, root := ~, start := 420475, stop := 420488 ), rec( content := ".\n \ If ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour1", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 420514, root := ~, start := 420500, stop := 420513 ), rec( content := " and ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour2", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "A", next := 420533, root := ~, start := 420519, stop := 420532 ), rec( content := " are not\ given then\n ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otHighlightedDigraph", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "C", next := 420587, root := ~, start := 420559, stop := 420586 ), rec( content := " uses bl\ ack and grey respectively. ", count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "P", next := 420626, root := ~, start := 420622, stop := 420625 ), rec( content := "\n\n \ Note that ", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ otHighlightedDigraph", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "C", next := 420672, root := ~, start := 420644, stop := 420671 ), rec( content := " does no\ t validate the colours\n ", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour1", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "A", next := 420723, root := ~, start := 420709, stop := 420722 ), rec( content := " and ", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ olour2", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "A", next := 420742, root := ~, start := 420728, stop := 420741 ), rec( content := " - consu\ lt the GraphViz documentation to\n see what is available.\n\n See ", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DotDigr\ aph" ), content := 0, count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "Ref", next := 420847, root := ~, start := 420823, stop := 420846 ), rec( content := " for mor\ e details on the output.", count := [ 9, 1, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 1, 6, 5 ], mathmode := "Text", name := "P", next := 420883, root := ~, start := 420879, stop := 420882 ), rec( content := "\n\n \ ", count := [ 9, 1, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> digraph := Digraph([[2, 3], [2], [1, 3]]);\n\ngap> FileString(\"dot/my_digraph.dot\",\n> DotHighlightedDigraph(digra\ ph, [1, 2], \"red\", \"black\"));\n264", count := [ 9, 1, 6, 6 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 1, 6, 6 ], mathmode := "Text", name := "Log", next := 421098, root := ~, start := 420891, stop := 421097 ), rec( content := "\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 6, 3 ], mathmode := "Text", name := "Description", next := 421117, root := ~, start := 420163, stop := 421116 ) ], count := [ 9, 1, 6, 1 ], mathmode := "Text", name := "ManSection", next := 421133, root := ~, start := 420030, stop := 421132 ), rec( content := "\n\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 1, 0, 1 ], mathmode := "Text", name := "Section", next := 421147, root := ~, start := 399188, stop := 421146 ) , rec( content := "\n\n ", count := [ 9, 1, 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "Reading and writi\ ng digraphs to a file", count := [ 9, 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 421217, root := ~, start := 421160, stop := 421216 ), rec( content := "\n This section descr\ ibes different ways to store and read graphs\n from a file in the ", count := [ 9, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ 9, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " package.\n ", count := [ 9, 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := " Graph6 " , count := [ 9, 2, 0, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 4 ], name := "Mark", next := 421366, root := ~, start := 421345, stop := 421365 ), rec( attributes := rec( ), content := [ rec( content := "\n\t", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Graph6 ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "E", next := 421396, root := ~, start := 421381, stop := 421395 ), rec( content := " is a gr\ aph data format for storing undirected graphs\n\twith no multiple edges nor lo\ ops of size up to ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ 2^{36} - 1 ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "M", next := 421517, root := ~, start := 421498, stop := 421516 ), rec( content := " in\n\tp\ rintable chracters. The format consists of two parts. The first part\n\tuses o\ ne to eight bytes to store the number of vertices. And the second\n\tpart is t\ he upper half of the adjacency matrix converted into ASCII\n\tcharacters. For \ a more detail description see ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "Graph6"\ ), content := [ rec( content := "\n\thttp://cs.anu.edu.au/~bdm/data/formats.t\ xt", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "URL", next := 421850, root := ~, start := 421781, stop := 421849 ), rec( content := ".\n \ ", count := [ 9, 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 5 ], name := "Item", next := 421865, root := ~, start := 421373, stop := 421864 ), rec( attributes := rec( ), content := [ rec( content := " Sparse6 \ ", count := [ 9, 2, 0, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 6 ], name := "Mark", next := 421895, root := ~, start := 421873, stop := 421894 ), rec( attributes := rec( ), content := [ rec( content := "\n\t", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Sparse6 ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "E", next := 421926, root := ~, start := 421910, stop := 421925 ), rec( content := " is a gr\ aph data format for storing undirected graphs\n\twith possibly multiple edges \ or loops. The maximal number of vertices\n\tallowed is ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ 2^{36} - 1 ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "M", next := 422082, root := ~, start := 422063, stop := 422081 ), rec( content := ". The fo\ rmat consists of two parts. The\n\tfirst part uses one to eight bytes to store\ the number of vertices.\n\tAnd the second part only stores information about \ the edges.\n\tTherefore, the ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Sparse6 ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "E", next := 422285, root := ~, start := 422269, stop := 422284 ), rec( content := " format \ return a more compact encoding\n\tthen ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Graph6 ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "E", next := 422345, root := ~, start := 422330, stop := 422344 ), rec( content := " for spa\ rse graph, i.e. graphs where the number\n\tof edges is much less than the num\ ber of vertices squared. For a more\n\tdetail description see ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Text := "Sparse6\ " ), content := [ rec( content := "\n http://cs.anu.edu.au/~bdm/data/fo\ rmats.txt", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "URL", next := 422567, root := ~, start := 422490, stop := 422566 ), rec( content := ".\n \ ", count := [ 9, 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 7 ], name := "Item", next := 422582, root := ~, start := 421902, stop := 422581 ), rec( attributes := rec( ), content := [ rec( content := " Digraph6\ ", count := [ 9, 2, 0, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 8 ], name := "Mark", next := 422613, root := ~, start := 422590, stop := 422612 ), rec( attributes := rec( ), content := [ rec( content := "\n\t", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraph6", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "E", next := 422643, root := ~, start := 422628, stop := 422642 ), rec( content := " is a ne\ w format based on ", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Graph6 ", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "E", next := 422684, root := ~, start := 422669, stop := 422683 ), rec( content := ", but de\ signed\n\tfor digraphs. The entire adjacency matrix is stored, and\n\ttherefor\ e there is support for directed edges and single-vertex loops.\n\tHowever, mul\ tiple edges are not supported.\n ", count := [ 9, 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 9 ], name := "Item", next := 422886, root := ~, start := 422620, stop := 422885 ), rec( attributes := rec( ), content := [ rec( content := " DiSparse\ 6 ", count := [ 9, 2, 0, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 10 ], name := "Mark", next := 422918, root := ~, start := 422894, stop := 422917 ), rec( attributes := rec( ), content := [ rec( content := "\n\t", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ iSparse6", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "E", next := 422949, root := ~, start := 422933, stop := 422948 ), rec( content := " is a ne\ w format based on ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Sparse6 ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "E", next := 422991, root := ~, start := 422975, stop := 422990 ), rec( content := ", but\n\\ tdesigned for digraphs. In this format the list of edges is\n\tpartitioned int\ o inceasing and decreasing edges, depending whether the\n\tedge has its source\ bigger than the range. Then both sets of edges are\n\twritten separetly in ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := " \ Sparse6 ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "E", next := 423239, root := ~, start := 423223, stop := 423238 ), rec( content := " format \ with a separation symbol\n\tin between.\n ", count := [ 9, 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 11 ], name := "Item", next := 423298, root := ~, start := 422925, stop := 423297 ) ], count := [ 9, 2, 0, 4 ], mathmode := "Text", name := "List", next := 423310, root := ~, start := 421332, stop := 423309 ), rec( content := "\n ", count := [ 9, 2, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "String" ), content := 0, count := [ 9, 2, 1, 2 ], mathmode := "Text", name := "Attr", next := 423365, root := ~, start := 423330, stop := 423364 ), rec( attributes := rec( Arg := "digraph", Name := "PrintString" ), content := 0, count := [ 9, 2, 1, 4 ], mathmode := "Text", name := "Oper", next := 423408, root := ~, start := 423368, stop := 423407 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "Returns", next := 423439, root := ~, start := 423411, stop := 423438 ), rec( attributes := rec( ), content := [ rec( content := "\n Ret\ urns a string ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tring", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "C", next := 423490, root := ~, start := 423477, stop := 423489 ), rec( content := " such th\ at ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ valString(string)", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "C", next := 423526, root := ~, start := 423501, stop := 423525 ), rec( content := "\n is\ equal to ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "A", next := 423557, root := ~, start := 423543, stop := 423556 ), rec( content := ", and ha\ s the same mutability.\n See ", count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "EvalString" ), content := 0, count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "Ref", next := 423636, root := ~, start := 423596, stop := 423635 ), rec( content := ".\n " , count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "P", next := 423646, root := ~, start := 423642, stop := 423645 ), rec( content := "\n\n \ The methods installed for ", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ tring", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "C", next := 423691, root := ~, start := 423678, stop := 423690 ), rec( content := " make so\ me attempts to\n ensure that ", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tring", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "C", next := 423743, root := ~, start := 423730, stop := 423742 ), rec( content := " has as \ short a length as possible, but\n there may exist shorter strings that also\ evaluate to ", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "A", next := 423855, root := ~, start := 423841, stop := 423854 ), rec( content := ".\n " , count := [ 9, 2, 1, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "P", next := 423865, root := ~, start := 423861, stop := 423864 ), rec( content := "\n\n \ It is possible that ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tring", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "C", next := 423904, root := ~, start := 423891, stop := 423903 ), rec( content := " may con\ tain escaped special\n characters. To obtain a representation of ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "A", next := 423993, root := ~, start := 423979, stop := 423992 ), rec( content := " that\n \ can be entered as GAP input, please use\n ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Func := "Print" ), content := 0, count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "Ref", next := 424082, root := ~, start := 424047, stop := 424081 ), rec( content := ".\n N\ ote that ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ rint", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "C", next := 424110, root := ~, start := 424098, stop := 424109 ), rec( content := " for a d\ igraph delegates to\n ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ rintString", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "C", next := 424161, root := ~, start := 424143, stop := 424160 ), rec( content := ", which \ delegates to ", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ tring", count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "C", next := 424195, root := ~, start := 424182, stop := 424194 ), rec( content := ".\n " , count := [ 9, 2, 1, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> D := CycleDigraph(3);\n\ngap> P\ rint(D);\nCycleDigraph(3);\ngap> G := PetersenGraph(IsMutableDigraph);\n\ngap> String(G);\n\"DigraphFromGraph6St\ ring(IsMutableDigraph, \\\"IheA@GUAo\\\");\"\ngap> Print(last);\nDigraphFromGr\ aph6String(IsMutableDigraph, \"IheA@GUAo\");\ngap> DigraphFromGraph6String(IsM\ utableDigraph, \"IheA@GUAo\");\n\n\ ", count := [ 9, 2, 1, 8 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 8 ], mathmode := "Text", name := "Log", next := 424670, root := ~, start := 424201, stop := 424669 ), rec( content := "\n ", count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 1, 5 ], mathmode := "Text", name := "Description", next := 424687, root := ~, start := 423442, stop := 424686 ) ], count := [ 9, 2, 1, 1 ], mathmode := "Text", name := "ManSection", next := 424701, root := ~, start := 423315, stop := 424700 ), rec( content := "\n\n ", count := [ 9, 2, 1, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromGrap\ h6String" ), content := 0, count := [ 9, 2, 2, 2 ], mathmode := "Text", name := "Oper", next := 424781, root := ~, start := 424722, stop := 424780 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromDigr\ aph6String" ), content := 0, count := [ 9, 2, 2, 4 ], mathmode := "Text", name := "Oper", next := 424843, root := ~, start := 424784, stop := 424842 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromSpar\ se6String" ), content := 0, count := [ 9, 2, 2, 6 ], mathmode := "Text", name := "Oper", next := 424905, root := ~, start := 424846, stop := 424904 ), rec( attributes := rec( Arg := "[filt, ]str", Name := "DigraphFromDiSp\ arse6String" ), content := 0, count := [ 9, 2, 2, 8 ], mathmode := "Text", name := "Oper", next := 424967, root := ~, start := 424908, stop := 424966 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "Returns", next := 424999, root := ~, start := 424970, stop := 424998 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "A", next := 425033, root := ~, start := 425023, stop := 425032 ), rec( content := " is a st\ ring encoding a graph in Graph6, Digraph6, Sparse6 or\n DiSparse6 format, t\ hen the corresponding function returns a digraph. In the\n case of either G\ raph6 or Sparse6, formats which do not support directed\n edges, this will \ be a digraph such that for every edge, the edge going in\n the opposite dir\ ection is also present.", count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "P", next := 425376, root := ~, start := 425372, stop := 425375 ), rec( content := "\n\n \ Each of these functions takes an optional first argument ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilt", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "A", next := 425450, root := ~, start := 425439, stop := 425449 ), rec( content := ",\n w\ hich should be either ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "Ref", next := 425509, root := ~, start := 425479, stop := 425508 ), rec( content := "\n or\ ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "Ref", next := 425549, root := ~, start := 425517, stop := 425548 ), rec( content := ",\n a\ nd which specifies whether the output digraph shall\n be mutable or immutab\ le.\n If no first argument is provided, then an immutable\n digraph is r\ eturned by default.\n ", count := [ 9, 2, 2, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> DigraphFromGraph6String(\"?\");\n\ngap> DigraphFromGraph6String(\"C]\");\n\ngap> DigraphFromGraph6String(\"H?AAEM{\");\n\ngap> DigraphFromDigraph6String(\"&?\");\n\ngap> DigraphFromDigraph6String(IsMutableDigraph,\ \"&DOOOW?\");\n\ngap> DigraphFromDi\ graph6String(\"&CQFG\");\n\ngap> D\ igraphFromDigraph6String(\"&IM[SrKLc~lhesbU[F_\");\n\ngap> DigraphFromDiSparse6String(\".CaWBGA?b\");\n\n", count := [ 9, 2, 2, 11 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 2, 11 ], mathmode := "Text", name := "Example", next := 426478, root := ~, start := 425733, stop := 426477 ), rec( content := "\n ", count := [ 9, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 2, 9 ], mathmode := "Text", name := "Description", next := 426497, root := ~, start := 425002, stop := 426496 ) ], count := [ 9, 2, 2, 1 ], mathmode := "Text", name := "ManSection", next := 426511, root := ~, start := 424707, stop := 426510 ), rec( content := "\n\n ", count := [ 9, 2, 2, 12 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "Graph6String" ) , content := 0, count := [ 9, 2, 3, 2 ], mathmode := "Text", name := "Oper", next := 426573, root := ~, start := 426532, stop := 426572 ), rec( attributes := rec( Arg := "digraph", Name := "Digraph6String"\ ), content := 0, count := [ 9, 2, 3, 4 ], mathmode := "Text", name := "Oper", next := 426619, root := ~, start := 426576, stop := 426618 ), rec( attributes := rec( Arg := "digraph", Name := "Sparse6String" ), content := 0, count := [ 9, 2, 3, 6 ], mathmode := "Text", name := "Oper", next := 426664, root := ~, start := 426622, stop := 426663 ), rec( attributes := rec( Arg := "digraph", Name := "DiSparse6String\ " ), content := 0, count := [ 9, 2, 3, 8 ], mathmode := "Text", name := "Oper", next := 426711, root := ~, start := 426667, stop := 426710 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "Returns", next := 426742, root := ~, start := 426714, stop := 426741 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se four functions return a highly compressed string fully describing the\n \ digraph ", count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "A", next := 426865, root := ~, start := 426851, stop := 426864 ), rec( content := ". ", count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "P", next := 426871, root := ~, start := 426867, stop := 426870 ), rec( content := "\n\n \ Graph6 and Digraph6 are formats best used on small, dense graphs, if\n appl\ icable. For larger, sparse graphs use ", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ parse6", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "E", next := 427006, root := ~, start := 426992, stop := 427005 ), rec( content := " and\n \ ", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ isparse6", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "E", next := 427031, root := ~, start := 427015, stop := 427030 ), rec( content := " (this l\ atter also preserves multiple edges). ", count := [ 9, 2, 3, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 3, 11 ], mathmode := "Text", name := "P", next := 427081, root := ~, start := 427077, stop := 427080 ), rec( content := "\n\n \ See ", count := [ 9, 2, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 3, 11 ], mathmode := "Text", name := "Ref", next := 427118, root := ~, start := 427091, stop := 427117 ), rec( content := ".\n " , count := [ 9, 2, 3, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2, 3], [1], [1]]);\n\ngap> Sparse6String(gr);\n\":Bc\"\ngap> DiSparse6String(gr);\n\".Bc{f\ \"\n", count := [ 9, 2, 3, 12 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 12 ], mathmode := "Text", name := "Example", next := 427305, root := ~, start := 427124, stop := 427304 ), rec( content := "\n ", count := [ 9, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 3, 9 ], mathmode := "Text", name := "Description", next := 427322, root := ~, start := 426745, stop := 427321 ) ], count := [ 9, 2, 3, 1 ], mathmode := "Text", name := "ManSection", next := 427336, root := ~, start := 426517, stop := 427335 ), rec( content := "\n\n ", count := [ 9, 2, 3, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename [, code\ r][, mode]", Name := "DigraphFile" ), content := 0, count := [ 9, 2, 4, 2 ], mathmode := "Text", name := "Func", next := 427416, root := ~, start := 427357, stop := 427415 ), rec( attributes := rec( ), content := [ rec( content := "An IO fil\ e object.", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Returns", next := 427456, root := ~, start := 427419, stop := 427455 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "A", next := 427495, root := ~, start := 427480, stop := 427494 ), rec( content := " is a st\ ring representing the name of a file, then\n ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFile", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "C", next := 427568, root := ~, start := 427550, stop := 427567 ), rec( content := " returns\ an ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 9, 2, 4, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 9, 2, 4, 3 ], name := \ "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], name := "Package", next := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " package\ file object for that file.\n ", count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "P", next := 427628, root := ~, start := 427624, stop := 427627 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ oder", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "A", next := 427671, root := ~, start := 427659, stop := 427670 ), rec( content := " is spec\ ified\n and is a function which either encodes a digraph as a string, or de\ codes a\n string into a digraph, then this function will be used when readi\ ng or\n writing to the returned file object. If the optional argument ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "c\ oder", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "A", next := 427917, root := ~, start := 427905, stop := 427916 ), rec( content := "\n is\ not specified, then the encoding of the digraphs in the returned file\n ob\ ject must be specified in the the file extension. The file extension must\n \ be one of: ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ g6", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 428100, root := ~, start := 428090, stop := 428099 ), rec( content := ", ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ s6", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 428112, root := ~, start := 428102, stop := 428111 ), rec( content := ", ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ d6", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 428124, root := ~, start := 428114, stop := 428123 ), rec( content := ", ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ ds6", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 428137, root := ~, start := 428126, stop := 428136 ), rec( content := ", ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ txt", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 428150, root := ~, start := 428139, stop := 428149 ), rec( content := ",\n " , count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ p", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 428165, root := ~, start := 428156, stop := 428164 ), rec( content := ", or ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ pickle", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "C", next := 428184, root := ~, start := 428170, stop := 428183 ), rec( content := "; more d\ etails of these file formats is given\n below. ", count := [ 9, 2, 4, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "P", next := 428245, root := ~, start := 428241, stop := 428244 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ode", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "A", next := 428287, root := ~, start := 428276, stop := 428286 ), rec( content := " is spec\ ified, then it must be one of:\n ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "w\"", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 428340, root := ~, start := 428330, stop := 428339 ), rec( content := " (for wr\ ite), ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "a\"", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 428364, root := ~, start := 428354, stop := 428363 ), rec( content := " (for ap\ pend), or ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "r\"", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 428392, root := ~, start := 428382, stop := 428391 ), rec( content := " (for re\ ad).\n If ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "m\ ode", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "A", next := 428423, root := ~, start := 428412, stop := 428422 ), rec( content := " is not \ specified, then ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ "r\"", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "C", next := 428457, root := ~, start := 428447, stop := 428456 ), rec( content := " is used\ by default. ", count := [ 9, 2, 4, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "P", next := 428482, root := ~, start := 428478, stop := 428481 ), rec( content := "\n\n \ If ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "A", next := 428506, root := ~, start := 428491, stop := 428505 ), rec( content := " ends in\ one of: ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "C", next := 428533, root := ~, start := 428523, stop := 428532 ), rec( content := ", ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "C", next := 428546, root := ~, start := 428535, stop := 428545 ), rec( content := ", or\n \ ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "C", next := 428565, root := ~, start := 428555, stop := 428564 ), rec( content := ", then t\ he digraphs which are read from, or written to, the\n returned file object \ are decompressed, or compressed, appropriately.\n ", count := [ 9, 2, 4, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "P", next := 428706, root := ~, start := 428702, stop := 428705 ), rec( content := "\n\n \ The file object returned by ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFile", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "C", next := 428758, root := ~, start := 428740, stop := 428757 ), rec( content := " can be \ given as the first\n argument for either of the functions ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "ReadDig\ raphs" ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "Ref", next := 428852, root := ~, start := 428826, stop := 428851 ), rec( content := " or ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "Ref", next := 428889, root := ~, start := 428856, stop := 428888 ), rec( content := ". The pu\ rpose of this is to reduce the overhead of\n recreating the file object ins\ ide the functions ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "ReadDig\ raphs" ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "Ref", next := 429018, root := ~, start := 428992, stop := 429017 ), rec( content := "\n or\ ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "Ref", next := 429053, root := ~, start := 429026, stop := 429052 ), rec( content := " when, f\ or example, reading or writing many\n digraphs in a loop.\n ", count := [ 9, 2, 4, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 4, 8 ], mathmode := "Text", name := "P", next := 429129, root := ~, start := 429125, stop := 429128 ), rec( content := "\n\n \ The currently supported file formats, and associated filename extensions,\n \ are:\n ", count := [ 9, 2, 4, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "graph6 (.g6)", count := [ 9, 2, 4, 9 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 9 ], name := "Mar\ k", next := 429260, root := ~, start := 429235, stop := 429259 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tA standard and widely-used format\ for undirected graphs, with no support\n\tfor loops or multiple edges. Only \ symmetric graphs are allowed -- each\n\tedge is combined with its converse edg\ e to produce a single undirected\n\tedge. This format is best used for \"dens\ e\" graphs -- those with many\n\tedges per vertex.\n ", count := [ 9, 2, \ 4, 10 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 10 ], name := "Ite\ m", next := 429596, root := ~, start := 429267, stop := 429595 ), rec( attributes :\ = rec( ), content := [ rec( content := "sparse6 (.s6)", count := [ 9, 2, 4, 1\ 1 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 11 ], name := "Mar\ k", next := 429629, root := ~, start := 429603, stop := 429628 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tUnlike graph6, sparse6 has suppor\ t for loops and multiple edges.\n\tHowever, its use is still limited to symmet\ ric graphs. This format is\n\tbetter-suited to \"sparse\" graphs -- those wit\ h few edges per vertex.\n ", count := [ 9, 2, 4, 12 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 12 ], name := "Item", next := 4298\ 64, root := ~, start := 429636, stop := 429863 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph6 (.d6)", count := [ 9, 2, 4, 13 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 13 ], name := "Mark", next := 4298\ 98, root := ~, start := 429871, stop := 429897 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis format is based on graph6, but stores direction information -\ \n\ttherefore is not limited to symmetric graphs. Loops are allowed, but\n\tm\ ultiple edges are not. Best compression with \"dense\" graphs.\n ", coun\ t := [ 9, 2, 4, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 14 ], name := "Ite\ m", next := 430128, root := ~, start := 429905, stop := 430127 ), rec( attributes :\ = rec( ), content := [ rec( content := "disparse6 (.ds6)", count := [ 9, 2, 4\ , 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 15 ], name := "Mar\ k", next := 430164, root := ~, start := 430135, stop := 430163 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tAny type of digraph can be encode\ d in disparse6: directions, loops, and\n\tmultiple edges are all allowed. Sim\ ilar to sparse6, this has the best\n\tcompression rate with \"sparse\" graphs.\ \n ", count := [ 9, 2, 4, 16 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 4, 16 ], name := "Item", next := 430376, root := ~, start := 430\ 171, stop := 430375 ), rec( attributes := rec( ), content := [ rec( content :\ = "plain text (.txt)", count := [ 9, 2, 4, 17 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 9, 2, 4, 17 ], name := "Mark", next := 4304\ 13, root := ~, start := 430383, stop := 430412 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis is a human-readable format which stores graphs in the form\n \ ", count := [ 9, 2, 4, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "0 7 0 8 1\ 7 2 8 3 8 4 8 5 8 6 8", count := [ 9, 2, 4, 18 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 18 ], mathmode := "Text", name\ := "C", next := 430545, root := ~, start := 430500, stop := 430544 ), rec( cont\ ent := " i.e. pairs of vertices\n describing edges in a graph. More spe\ cifically, the vertices making up\n one edge must be separated by a sin\ gle space, and pairs of vertices must\n be separated by two spaces. ", coun\ t := [ 9, 2, 4, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 4, 19 ], mathmode := "Text", name\ := "P", next := 430769, root := ~, start := 430765, stop := 430768 ), rec( cont\ ent := "\n\n See ", count := [ 9, 2, 4, 19 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ReadPlainTextDigrap\ h" ), content := 0, count := [ 9, 2, 4, 19 ], mathmode := "Text", name := "Ref\ ", next := 430817, root := ~, start := 430783, stop := 430816 ), rec( content \ := " for a more flexible way to store\n digraphs in a plain text file. \ ", count := [ 9, 2, 4, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 4, 20 ], mathmode := "Text", name\ := "P", next := 430894, root := ~, start := 430890, stop := 430893 ), rec( cont\ ent := "\n ", count := [ 9, 2, 4, 20 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 4, 18 ], name := "Item", next := 430908, root := ~, start := 430\ 420, stop := 430907 ), rec( attributes := rec( ), content := [ rec( content :\ = "pickled (", count := [ 9, 2, 4, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := ".p", count \ := [ 9, 2, 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 9, 2, 4, 21 ], mathmode := "Text", name := "F", next := 430940, root := \ ~, start := 430931, stop := 430939 ), rec( content := " or ", count := [ 9, 2,\ 4, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := ".pickle", count := [ 9, 2, 4, 21 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 21 ], math\ mode := "Text", name := "F", next := 430958, root := ~, start := 430944, stop \ := 430957 ), rec( content := ")", count := [ 9, 2, 4, 21 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 21 ], name := "Mark", next := 4309\ 66, root := ~, start := 430916, stop := 430965 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n Digraphs are pickled using the ", count := [ 9, 2, 4, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := \ "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ re\ c( content := "\n http://gap-packages.github.io/io/\n", count := [ 9, 2, 4\ , 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, \ 4, 22 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17\ , stop := 77 ) ], count := [ 9, 2, 4, 22 ], name := "Alt", next := 84, root :=\ ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ), conten\ t := [ rec( attributes := rec( ), content := [ rec( content := "IO", count :=\ [ 9, 2, 4, 22 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 22 ], n\ ame := "Package", next := 122, root := ~, start := 101, stop := 121 ) ], count\ := [ 9, 2, 4, 22 ], mathmode := "Text", name := "Alt", next := 128, root := ~ , star\ t := 84, stop := 127 ), rec( content := " package. This is particularly good\n\ when the ", count := [ 9, 2, 4, 22 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphGroup" ), content :\ = 0, count := [ 9, 2, 4, 22 ], mathmode := "Text", name := "Ref", next := 4311\ 02, root := ~, start := 431076, stop := 431101 ), rec( content := " is non-tri\ vial.\n ", count := [ 9, 2, 4, 22 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 9, 2, 4, 22 ], name := "Item", next := 431132, root := ~, start := 430\ 973, stop := 431131 ) ], count := [ 9, 2, 4, 9 ], mathmode := "Text", name := "List", next := 431144, root := ~, start := 429222, stop := 431143 ), rec( content := "\n\n \ ", count := [ 9, 2, 4, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ngap\ > file := DigraphFile(filename, \"w\");;\ngap> for i in [1 .. 10] do\n> WriteD\ igraphs(file, Digraph([[1, 3], [2], [1, 2]]));\n> od;\ngap> IO_Close(file);;\n\ gap> file := DigraphFile(filename, \"r\");;\ngap> ReadDigraphs(file, 9);\n\ngap> IO_Close(file);;\n", count := [ 9, 2, 4, 25 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 4, 25 ], mathmode := "Text", name := "Example", next := 431542, root := ~, start := 431150, stop := 431541 ), rec( content := "\n ", count := [ 9, 2, 4, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 4, 3 ], mathmode := "Text", name := "Description", next := 431561, root := ~, start := 427459, stop := 431560 ) ], count := [ 9, 2, 4, 1 ], mathmode := "Text", name := "ManSection", next := 431575, root := ~, start := 427342, stop := 431574 ), rec( content := "\n\n ", count := [ 9, 2, 4, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename[, decod\ er][, n]", Name := "ReadDigraphs" ), content := 0, count := [ 9, 2, 5, 2 ], mathmode := "Text", name := "Func", next := 431654, root := ~, start := 431596, stop := 431653 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ , or a list of digraphs.", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Returns", next := 431709, root := ~, start := 431657, stop := 431708 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "A", next := 431748, root := ~, start := 431733, stop := 431747 ), rec( content := " is a st\ ring containing the name of a file containing\n encoded digraphs or an ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 9, 2, 5, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 9, 2, 5, 3 ], name := \ "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], name := "Package", next := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " file ob\ ject created using ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ File" ), content := 0, count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Ref", next := 431891, root := ~, start := 431860, stop := 431890 ), rec( content := ", then " , count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 431917, root := ~, start := 431898, stop := 431916 ), rec( content := " returns\ the digraphs\n encoded in the file as a list. Note that if ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "A", next := 432003, root := ~, start := 431988, stop := 432002 ), rec( content := " is a\n \ compressed file, which has been compressed appropriately to give a filename\ \n extension of ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 432116, root := ~, start := 432106, stop := 432115 ), rec( content := ", ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 432129, root := ~, start := 432118, stop := 432128 ), rec( content := ", or ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 432144, root := ~, start := 432134, stop := 432143 ), rec( content := ", then\n\ ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "C", next := 432174, root := ~, start := 432155, stop := 432173 ), rec( content := " can rea\ d ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "A", next := 432199, root := ~, start := 432184, stop := 432198 ), rec( content := " without\ it first needing to be\n decompressed.\n ", count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "P", next := 432257, root := ~, start := 432253, stop := 432256 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "A", next := 432302, root := ~, start := 432288, stop := 432301 ), rec( content := " is spec\ ified\n and is a function which decodes a string into a digraph,\n then \ ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "C", next := 432405, root := ~, start := 432386, stop := 432404 ), rec( content := " will us\ e ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "A", next := 432429, root := ~, start := 432415, stop := 432428 ), rec( content := " to deco\ de the digraphs\n contained in ", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "A", next := 432485, root := ~, start := 432470, stop := 432484 ), rec( content := ".", count := [ 9, 2, 5, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "P", next := 432490, root := ~, start := 432486, stop := 432489 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "A", next := 432529, root := ~, start := 432521, stop := 432528 ), rec( content := " is spec\ ified, then ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "C", next := 432568, root := ~, start := 432549, stop := 432567 ), rec( content := "\n re\ turns the ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "A", next := 432593, root := ~, start := 432585, stop := 432592 ), rec( content := "th digra\ ph encoded in the file ", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "A", next := 432639, root := ~, start := 432624, stop := 432638 ), rec( content := ".\n " , count := [ 9, 2, 5, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "P", next := 432649, root := ~, start := 432645, stop := 432648 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "A", next := 432694, root := ~, start := 432680, stop := 432693 ), rec( content := " is not \ specified, then\n ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 432741, root := ~, start := 432722, stop := 432740 ), rec( content := " will de\ duce which decoder to use based on the filename\n extension of ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "A", next := 432829, root := ~, start := 432814, stop := 432828 ), rec( content := " (after \ removing the compression-related\n filename extensions ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 432904, root := ~, start := 432894, stop := 432903 ), rec( content := ", ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 432917, root := ~, start := 432906, stop := 432916 ), rec( content := ", and ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 432933, root := ~, start := 432923, stop := 432932 ), rec( content := "). For \ example,\n if the filename extension is ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ g6", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 432993, root := ~, start := 432983, stop := 432992 ), rec( content := ", then " , count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadDigraphs", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "C", next := 433019, root := ~, start := 433000, stop := 433018 ), rec( content := " will us\ e\n the graph6 decoder ", count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ FromGraph6String" ), content := 0, count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "Ref", next := 433089, root := ~, start := 433052, stop := 433088 ), rec( content := ".\n " , count := [ 9, 2, 5, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 5, 7 ], mathmode := "Text", name := "P", next := 433099, root := ~, start := 433095, stop := 433098 ), rec( content := "\n\n \ The currently supported file formats, and associated filename extensions,\n \ are:\n ", count := [ 9, 2, 5, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "graph6 (.g6)", count := [ 9, 2, 5, 8 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 8 ], name := "Mar\ k", next := 433230, root := ~, start := 433205, stop := 433229 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tA standard and widely-used format\ for undirected graphs, with no support\n\tfor loops or multiple edges. Only \ symmetric graphs are allowed -- each\n\tedge is combined with its converse edg\ e to produce a single undirected\n\tedge. This format is best used for \"dens\ e\" graphs -- those with many\n\tedges per vertex.\n ", count := [ 9, 2, \ 5, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 9 ], name := "Ite\ m", next := 433566, root := ~, start := 433237, stop := 433565 ), rec( attributes :\ = rec( ), content := [ rec( content := "sparse6 (.s6)", count := [ 9, 2, 5, 1\ 0 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 10 ], name := "Mar\ k", next := 433599, root := ~, start := 433573, stop := 433598 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tUnlike graph6, sparse6 has suppor\ t for loops and multiple edges.\n\tHowever, its use is still limited to symmet\ ric graphs. This format is\n\tbetter-suited to \"sparse\" graphs -- those wit\ h few edges per vertex.\n ", count := [ 9, 2, 5, 11 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 11 ], name := "Item", next := 4338\ 34, root := ~, start := 433606, stop := 433833 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph6 (.d6)", count := [ 9, 2, 5, 12 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 12 ], name := "Mark", next := 4338\ 68, root := ~, start := 433841, stop := 433867 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis format is based on graph6, but stores direction information -\ \n\ttherefore is not limited to symmetric graphs. Loops are allowed, but\n\tm\ ultiple edges are not. Best compression with \"dense\" graphs.\n ", coun\ t := [ 9, 2, 5, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 13 ], name := "Ite\ m", next := 434098, root := ~, start := 433875, stop := 434097 ), rec( attributes :\ = rec( ), content := [ rec( content := "disparse6 (.ds6)", count := [ 9, 2, 5\ , 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 14 ], name := "Mar\ k", next := 434134, root := ~, start := 434105, stop := 434133 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tAny type of digraph can be encode\ d in disparse6: directions, loops, and\n\tmultiple edges are all allowed. Sim\ ilar to sparse6, this has the best\n\tcompression rate with \"sparse\" graphs.\ \n ", count := [ 9, 2, 5, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 5, 15 ], name := "Item", next := 434346, root := ~, start := 434\ 141, stop := 434345 ), rec( attributes := rec( ), content := [ rec( content :\ = "plain text (.txt)", count := [ 9, 2, 5, 16 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 9, 2, 5, 16 ], name := "Mark", next := 4343\ 83, root := ~, start := 434353, stop := 434382 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis is a human-readable format which stores graphs in the form\n \ ", count := [ 9, 2, 5, 17 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "0 7 0 8 1\ 7 2 8 3 8 4 8 5 8 6 8", count := [ 9, 2, 5, 17 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 17 ], mathmode := "Text", name\ := "C", next := 434515, root := ~, start := 434470, stop := 434514 ), rec( cont\ ent := " i.e. pairs of vertices\n describing edges in a graph. More spe\ cifically, the vertices making up\n one edge must be separated by a sin\ gle space, and pairs of vertices must\n be separated by two spaces. ", coun\ t := [ 9, 2, 5, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 5, 18 ], mathmode := "Text", name\ := "P", next := 434739, root := ~, start := 434735, stop := 434738 ), rec( cont\ ent := "\n\n See ", count := [ 9, 2, 5, 18 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ReadPlainTextDigrap\ h" ), content := 0, count := [ 9, 2, 5, 18 ], mathmode := "Text", name := "Ref\ ", next := 434787, root := ~, start := 434753, stop := 434786 ), rec( content \ := " for a more flexible way to store\n digraphs in a plain text file. \ ", count := [ 9, 2, 5, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 5, 19 ], mathmode := "Text", name\ := "P", next := 434864, root := ~, start := 434860, stop := 434863 ), rec( cont\ ent := "\n", count := [ 9, 2, 5, 19 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( content := "\n Just so I remember:\n ReadPlainT\ extDigraph(\"data-local/soc-Epinions1.txt.gz\", \"\\t\", 1,\n '#');\ \n", count := [ 9, 2, 5, 19 ], mathmode := "Text", name := "XMLCOMMENT", root \ := ~ ), rec( content := "\n ", count := [ 9, 2, 5, 19 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 17 ], name := "Item", next := 4350\ 09, root := ~, start := 434390, stop := 435008 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "pickled (", count := [ 9, 2, 5, 20 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ p", count := [ 9, 2, 5, 20 ], mathmode := "Text", name := "PCDATA", root := ~ \ ) ], count := [ 9, 2, 5, 20 ], mathmode := "Text", name := "F", next := 435041 , root\ := ~, start := 435032, stop := 435040 ), rec( content := " or ", count := [ 9\ , 2, 5, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := ".pickle", count := [ 9, 2, 5, 20 \ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 20\ ], mathmode := "Text", name := "F", next := 435059, root := ~, start := 43504\ 5, stop := 435058 ), rec( content := ")", count := [ 9, 2, 5, 20 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 20 ], name := "Mar\ k", next := 435067, root := ~, start := 435017, stop := 435066 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n Digraphs are pickled using \ the ", count := [ 9, 2, 5, 21 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 9, 2, 5, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 5, 21 ], mathmode := "Text", name := "URL\ ", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 9, 2, 5, 21 ] , name\ := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes :=\ rec( Only := "Text" ), content := [ rec( attributes := rec( ), content := [ \ rec( content := "IO", count := [ 9, 2, 5, 21 ], name := "PCDATA", root := ~ ) \ ], count := [ 9, 2, 5, 21 ], name := "Package", next := 122, root := ~, start \ := 101, stop := 121 ) ], count := [ 9, 2, 5, 21 ], mathmode := "Text", name :=\ "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " \ package. This is particularly good\n when the ", count := [ 9, 2, 5, 21\ ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec(\ Attr := "DigraphGroup" ), content := 0, count := [ 9, 2, 5, 21 ], mathmode :=\ "Text", name := "Ref", next := 435203, root := ~, start := 435177, stop := 43\ 5202 ), rec( content := " is non-trivial.\n ", count := [ 9, 2, 5, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 21 ], name := "Ite\ m", next := 435233, root := ~, start := 435074, stop := 435232 ) ], count := [ 9, 2, 5, 8 ], mathmode := "Text", name := "List", next := 435245, root := ~, start := 433192, stop := 435244 ), rec( content := "\n\n \ ", count := [ 9, 2, 5, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"), \ 10);\n\ngap> ReadDigraphs(\n> Conc\ atenation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"), 17);\n\ngap> ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \ \"/data/tree9.4.txt\"));\n[ , \n \ , \n , \n , \n , \n , \n , \n , \n , \n , \n , \n , \ \n , \n ]", count := [ 9, 2, 5, 24 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 24 ], mathmode := "Text", name := "Example", next := 436291, root := ~, start := 435251, stop := 436290 ), rec( content := "\n ", count := [ 9, 2, 5, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 5, 3 ], mathmode := "Text", name := "Description", next := 436310, root := ~, start := 431712, stop := 436309 ) ], count := [ 9, 2, 5, 1 ], mathmode := "Text", name := "ManSection", next := 436324, root := ~, start := 431581, stop := 436323 ), rec( content := "\n\n ", count := [ 9, 2, 5, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename, digrap\ hs[, encoder] [, mode]", Name := "WriteDigraphs" ), content := 0, count := [ 9, 2, 6, 2 ], mathmode := "Text", name := "Func", next := 436418, root := ~, start := 436345, stop := 436417 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraphs", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "A", next := 436457, root := ~, start := 436442, stop := 436456 ), rec( content := " is a li\ st of digraphs or a digraph and ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "A", next := 436512, root := ~, start := 436497, stop := 436511 ), rec( content := "\n is\ a string or an ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "IO" ), content := [ rec( content := "\n http://gap-packages\ .github.io/io/\n", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17, stop := 77 ) ], count := [ 9, 2, 6, 3 ], name := "Alt", next := 84, root := ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "IO", count := [ 9, 2, 6, 3 ], name := \ "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], name := "Package", next := 122, root := ~, start := 101 , stop := 121 ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Alt", next := 128, root := ~, start := 84, stop := 127 ), rec( content := " file ob\ ject created using ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ File" ), content := 0, count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Ref", next := 436591, root := ~, start := 436566, stop := 436590 ), rec( content := ",\n t\ hen ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "W\ riteDigraphs", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 436622, root := ~, start := 436602, stop := 436621 ), rec( content := " writes \ the digraphs to the file represented by\n ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "A", next := 436689, root := ~, start := 436674, stop := 436688 ), rec( content := ". If th\ e supplied filename ends in one of the extensions\n ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 436761, root := ~, start := 436751, stop := 436760 ), rec( content := ", ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 436774, root := ~, start := 436763, stop := 436773 ), rec( content := ", or ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "C", next := 436789, root := ~, start := 436779, stop := 436788 ), rec( content := ", then t\ he file will be compressed\n appropriately. Excluding these extensions, if\ the file ends with an\n extension in the list below, the corresponding gra\ ph format will be used to\n encode it. If such an extension is not include\ d, an appropriate format\n will be chosen intelligently, and an extension a\ ppended, to minimise file\n size.\n ", count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "P", next := 437149, root := ~, start := 437145, stop := 437148 ), rec( content := "\n\n \ ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " TODO: e\ ncode and mode. ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "XMLCOMMENT" , root := ~ ), rec( content := "\n\n \ For more verbose information on the progress of the function, set the info\n \ level of ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nfoDigraphs", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "A", next := 437299, root := ~, start := 437280, stop := 437298 ), rec( content := " to 1 or\ higher, using ", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ etInfoLevel", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "C", next := 437341, root := ~, start := 437322, stop := 437340 ), rec( content := ".", count := [ 9, 2, 6, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 6, 5 ], mathmode := "Text", name := "P", next := 437346, root := ~, start := 437342, stop := 437345 ), rec( content := "\n\n \ The currently supported file formats are:\n ", count := [ 9, 2, 6, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "graph6 (.g6)", count := [ 9, 2, 6, 6 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 6 ], name := "Mar\ k", next := 437436, root := ~, start := 437411, stop := 437435 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tA standard and widely-used format\ for undirected graphs, with no support\n\tfor loops or multiple edges. Only \ symmetric graphs are allowed -- each\n\tedge is combined with its converse edg\ e to produce a single undirected\n\tedge. This format is best used for \"dens\ e\" graphs -- those with many\n\tedges per vertex.\n ", count := [ 9, 2, \ 6, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 7 ], name := "Ite\ m", next := 437772, root := ~, start := 437443, stop := 437771 ), rec( attributes :\ = rec( ), content := [ rec( content := "sparse6 (.s6)", count := [ 9, 2, 6, 8\ ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 8 ], name := "Mar\ k", next := 437805, root := ~, start := 437779, stop := 437804 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tUnlike graph6, sparse6 has suppor\ t for loops and multiple edges.\n\tHowever, its use is still limited to symmet\ ric graphs. This format is\n\tbetter-suited to \"sparse\" graphs -- those wit\ h few edges per vertex.\n ", count := [ 9, 2, 6, 9 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 9 ], name := "Item", next := 4380\ 40, root := ~, start := 437812, stop := 438039 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph6 (.d6)", count := [ 9, 2, 6, 10 ], mathmode := "Text", name :=\ "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 10 ], name := "Mark", next := 4380\ 74, root := ~, start := 438047, stop := 438073 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis format is based on graph6, but stores direction information -\ \n\ttherefore is not limited to symmetric graphs. Loops are allowed, but\n\tm\ ultiple edges are not. Best compression with \"dense\" graphs.\n ", coun\ t := [ 9, 2, 6, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 11 ], name := "Ite\ m", next := 438304, root := ~, start := 438081, stop := 438303 ), rec( attributes :\ = rec( ), content := [ rec( content := "disparse6 (.ds6)", count := [ 9, 2, 6\ , 12 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 12 ], name := "Mar\ k", next := 438340, root := ~, start := 438311, stop := 438339 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n\tAny type of digraph can be encode\ d in disparse6: directions, loops, and\n\tmultiple edges are all allowed. Sim\ ilar to sparse6, this has the best\n\tcompression rate with \"sparse\" graphs.\ \n ", count := [ 9, 2, 6, 13 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ 9, 2, 6, 13 ], name := "Item", next := 438552, root := ~, start := 438\ 347, stop := 438551 ), rec( attributes := rec( ), content := [ rec( content :\ = "plain text (.txt)", count := [ 9, 2, 6, 14 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ 9, 2, 6, 14 ], name := "Mark", next := 4385\ 90, root := ~, start := 438560, stop := 438589 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n\tThis is a human-readable format which stores graphs in the form\n \ ", count := [ 9, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "0 7 0 8 1\ 7 2 8 3 8 4 8 5 8 6 8", count := [ 9, 2, 6, 15 ], mathmode := "Text", na\ me := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 15 ], mathmode := "Text", name\ := "C", next := 438722, root := ~, start := 438677, stop := 438721 ), rec( cont\ ent := " i.e. pairs of vertices\n describing edges in a graph. More spe\ cifically, the vertices making up\n one edge must be separated by a sin\ gle space, and pairs of vertices must\n be separated by two spaces. ", coun\ t := [ 9, 2, 6, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 6, 16 ], mathmode := "Text", name\ := "P", next := 438946, root := ~, start := 438942, stop := 438945 ), rec( cont\ ent := "\n\n See ", count := [ 9, 2, 6, 16 ], mathmode := "Text", name \ := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "ReadPlainTextDigrap\ h" ), content := 0, count := [ 9, 2, 6, 16 ], mathmode := "Text", name := "Ref\ ", next := 438994, root := ~, start := 438960, stop := 438993 ), rec( content \ := " for a more flexible way to store\n digraphs in a plain text file. \ ", count := [ 9, 2, 6, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 6, 17 ], mathmode := "Text", name\ := "P", next := 439071, root := ~, start := 439067, stop := 439070 ), rec( cont\ ent := "\n", count := [ 9, 2, 6, 17 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( content := "\n Just so I remember:\n ReadPlainT\ extDigraph(\"data-local/soc-Epinions1.txt.gz\", \"\\t\", 1, '#');\n", coun\ t := [ 9, 2, 6, 17 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( cont\ ent := "\n ", count := [ 9, 2, 6, 17 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 6, 15 ], name := "Item", next := 439208, root := ~, start := 438\ 597, stop := 439207 ), rec( attributes := rec( ), content := [ rec( content :\ = "pickled (", count := [ 9, 2, 6, 18 ], mathmode := "Text", name := "PCDATA", root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := ".p", count \ := [ 9, 2, 6, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 9, 2, 6, 18 ], mathmode := "Text", name := "F", next := 439239, root := \ ~, start := 439230, stop := 439238 ), rec( content := " or ", count := [ 9, 2,\ 6, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := ".pickle", count := [ 9, 2, 6, 18 ], m\ athmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 18 ], math\ mode := "Text", name := "F", next := 439257, root := ~, start := 439243, stop \ := 439256 ), rec( content := ")", count := [ 9, 2, 6, 18 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 18 ], name := "Mark", next := 4392\ 65, root := ~, start := 439215, stop := 439264 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n Digraphs are pickled using the ", count := [ 9, 2, 6, 19 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := \ "Text" ), content := [ rec( attributes := rec( Text := "IO" ), content := [ re\ c( content := "\n http://gap-packages.github.io/io/\n", count := [ 9, 2, 6\ , 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, \ 6, 19 ], mathmode := "Text", name := "URL", next := 78, root := ~, start := 17\ , stop := 77 ) ], count := [ 9, 2, 6, 19 ], name := "Alt", next := 84, root :=\ ~, start := 1, stop := 83 ), rec( attributes := rec( Only := "Text" ), conten\ t := [ rec( attributes := rec( ), content := [ rec( content := "IO", count :=\ [ 9, 2, 6, 19 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 19 ], n\ ame := "Package", next := 122, root := ~, start := 101, stop := 121 ) ], count\ := [ 9, 2, 6, 19 ], mathmode := "Text", name := "Alt", next := 128, root := ~ , star\ t := 84, stop := 127 ), rec( content := " package. This is particularly good\n\ when the ", count := [ 9, 2, 6, 19 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphGroup" ), content :\ = 0, count := [ 9, 2, 6, 19 ], mathmode := "Text", name := "Ref", next := 4394\ 01, root := ~, start := 439375, stop := 439400 ), rec( content := " is non-tri\ vial.\n ", count := [ 9, 2, 6, 19 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ 9, 2, 6, 19 ], name := "Item", next := 439431, root := ~, start := 439\ 272, stop := 439430 ) ], count := [ 9, 2, 6, 6 ], mathmode := "Text", name := "List", next := 439443, root := ~, start := 437398, stop := 439442 ), rec( content := "\n\n \ ", count := [ 9, 2, 6, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> grs := [];;\ngap> grs[1] := Digraph([]);\n\ngap> grs[2] := Digraph([[1, 3], [2], [1, 2]]);\n\ngap> grs[3] := Digraph([\n> [6, 7], [6, 9], [1, 3\ , 4, 5, 8, 9],\n> [1, 2, 3, 4, 5, 6, 7, 10], [1, 5, 6, 7, 10], [2, 4, 5, 9, 10\ ],\n> [3, 4, 5, 6, 7, 8, 9, 10], [1, 3, 5, 7, 8, 9], [1, 2, 5],\n> [1, 2, 4, 6\ , 7, 8]]);\n\ngap> filename := C\ oncatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ngap> WriteDigraphs(fil\ ename, grs, \"w\");\nIO_OK\ngap> ReadDigraphs(filename);\n[ , \n , \n \ ]", count := [ 9, 2, 6, 22 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 6, 22 ], mathmode := "Text", name := "Example", next := 440210, root := ~, start := 439449, stop := 440209 ), rec( content := "\n ", count := [ 9, 2, 6, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 6, 3 ], mathmode := "Text", name := "Description", next := 440229, root := ~, start := 436421, stop := 440228 ) ], count := [ 9, 2, 6, 1 ], mathmode := "Text", name := "ManSection", next := 440243, root := ~, start := 436330, stop := 440242 ), rec( content := "\n\n ", count := [ 9, 2, 6, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename [, deco\ der]", Name := "IteratorFromDigraphFile" ), content := 0, count := [ 9, 2, 7, 2 ], mathmode := "Text", name := "Func", next := 440329, root := ~, start := 440264, stop := 440328 ), rec( attributes := rec( ), content := [ rec( content := "An iterat\ or.", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "Returns", next := 440363, root := ~, start := 440332, stop := 440362 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "A", next := 440402, root := ~, start := 440387, stop := 440401 ), rec( content := " is a st\ ring representing the name of a file containing\n encoded digraphs, then ", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ teratorFromDigraphFile", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "C", next := 440515, root := ~, start := 440485, stop := 440514 ), rec( content := " returns\ an iterator\n for which the value of ", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( BookName := "ref\ ", Oper := "NextIterator" ), content := 0, count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "Ref", next := 440604, root := ~, start := 440563, stop := 440603 ), rec( content := " is the\\ n next digraph encoded in the file.\n ", count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "P", next := 440658, root := ~, start := 440654, stop := 440657 ), rec( content := "\n\n \ If the optional argument ", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "A", next := 440703, root := ~, start := 440689, stop := 440702 ), rec( content := " is spec\ ified and is a function\n which decodes a string into a digraph, then ", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ teratorFromDigraphFile", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "C", next := 440813, root := ~, start := 440783, stop := 440812 ), rec( content := "\n wi\ ll use ", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ ecoder", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "A", next := 440841, root := ~, start := 440827, stop := 440840 ), rec( content := " to deco\ de the digraphs contained in ", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "A", next := 440893, root := ~, start := 440878, stop := 440892 ), rec( content := ".\n " , count := [ 9, 2, 7, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 5 ], mathmode := "Text", name := "P", next := 440903, root := ~, start := 440899, stop := 440902 ), rec( content := "\n\n \ The purpose of this function is to easily allow looping over digraphs\n enc\ oded in a file when loading all of the encoded digraphs would require\n too\ much memory. ", count := [ 9, 2, 7, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "P", next := 441081, root := ~, start := 441077, stop := 441080 ), rec( content := "\n\n \ To see what file types are available, see ", count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "WriteDi\ graphs" ), content := 0, count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "Ref", next := 441156, root := ~, start := 441129, stop := 441155 ), rec( content := ".\n\n \ ", count := [ 9, 2, 7, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "\\ ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out/man.d6.gz\");;\ngap\ > file := DigraphFile(filename, \"w\");;\ngap> for i in [1 .. 10] do\n> Writ\ eDigraphs(file, Digraph([[1, 3], [2], [1, 2]]));\n> od;\ngap> IO_Close(file);;\ \ngap> iter := IteratorFromDigraphFile(filename);\n\ngap> for x in i\ ter do od;\n", count := [ 9, 2, 7, 7 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 7 ], mathmode := "Text", name := "Example", next := 441505, root := ~, start := 441163, stop := 441504 ), rec( content := "\n ", count := [ 9, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 7, 3 ], mathmode := "Text", name := "Description", next := 441522, root := ~, start := 440366, stop := 441521 ) ], count := [ 9, 2, 7, 1 ], mathmode := "Text", name := "ManSection", next := 441536, root := ~, start := 440249, stop := 441535 ), rec( content := "\n\n ", count := [ 9, 2, 7, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "delimiter1, [del\ imiter2,] offset", Name := "DigraphPlainTextLineEncoder" ), content := 0, count := [ 9, 2, 8, 2 ], mathmode := "Text", name := "Func", next := 441638, root := ~, start := 441557, stop := 441637 ), rec( attributes := rec( Arg := "delimiter1, [del\ imiter2,] offset", Name := "DigraphPlainTextLineDecoder" ), content := 0, count := [ 9, 2, 8, 4 ], mathmode := "Text", name := "Oper", next := 441722, root := ~, start := 441641, stop := 441721 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "Returns", next := 441753, root := ~, start := 441725, stop := 441752 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se two functions return a function which encodes or decodes a digraph in\n \ a plain text format.", count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "P", next := 441878, root := ~, start := 441874, stop := 441877 ), rec( content := "\n\n \ ", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPlainTextLineEncoder", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "A", next := 441918, root := ~, start := 441884, stop := 441917 ), rec( content := " returns\ a function which takes a single\n digraph as an argument. The function re\ turns a string describing the edges\n of that digraph; each edge is written\ as a pair of integers separated by the\n string ", count := [ 9, 2, 8, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ elimiter2", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "A", next := 442148, root := ~, start := 442131, stop := 442147 ), rec( content := ", and th\ e edges themselves are separated by the\n string ", count := [ 9, 2, 8, 6 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ elimiter1", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "A", next := 442224, root := ~, start := 442207, stop := 442223 ), rec( content := ". ", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPlainTextLineDecoder", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "A", next := 442261, root := ~, start := 442227, stop := 442260 ), rec( content := " returns\ the\n corresponding decoder function, which takes a string argument in thi\ s format\n and returns a digraph.", count := [ 9, 2, 8, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "P", next := 442385, root := ~, start := 442381, stop := 442384 ), rec( content := "\n\n \ If only one delimiter is passed as an argument to\n ", count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphPlainTextLineDecoder", count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "A", next := 442479, root := ~, start := 442445, stop := 442478 ), rec( content := ", it wil\ l return a function which decodes\n a single edge, returning its contents a\ s a list of integers.", count := [ 9, 2, 8, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "P", next := 442589, root := ~, start := 442585, stop := 442588 ), rec( content := "\n\n \ The argument ", count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "A", next := 442621, root := ~, start := 442608, stop := 442620 ), rec( content := " should \ be an integer, which will describe a\n number to be added to each vertex be\ fore it is encoded, or after it is\n decoded. This may be used, for exampl\ e, to label vertices starting at 0\n instead of 1.", count := [ 9, 2, 8, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 8, 9 ], mathmode := "Text", name := "P", next := 442839, root := ~, start := 442835, stop := 442838 ), rec( content := "\n\n \ Note that the number of vertices of a digraph is not stored, and so vertices\n\ which are not connected to any edge may be lost.\n\n ", count := [ 9, 2, 8, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2, 3], [1], [1]]);\n\ngap> enc := DigraphPlainTextLineEncoder(\" \", \" \", -1);;\ngap> d\ ec := DigraphPlainTextLineDecoder(\" \", \" \", 1);;\ngap> enc(gr);\n\"0 1 0\ 2 1 0 2 0\"\ngap> dec(last);\n\ \n", count := [ 9, 2, 8, 10 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 10 ], mathmode := "Text", name := "Example", next := 443306, root := ~, start := 442980, stop := 443305 ), rec( content := "\n ", count := [ 9, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 8, 5 ], mathmode := "Text", name := "Description", next := 443323, root := ~, start := 441756, stop := 443322 ) ], count := [ 9, 2, 8, 1 ], mathmode := "Text", name := "ManSection", next := 443337, root := ~, start := 441542, stop := 443336 ), rec( content := "\n\n ", count := [ 9, 2, 8, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str", Name := "TournamentLineD\ ecoder" ), content := 0, count := [ 9, 2, 9, 2 ], mathmode := "Text", name := "Oper", next := 443404, root := ~, start := 443358, stop := 443403 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "Returns", next := 443436, root := ~, start := 443407, stop := 443435 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function takes a string ", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "A", next := 443496, root := ~, start := 443486, stop := 443495 ), rec( content := ", decode\ s it, and then returns the\n tournament [see ", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsTourn\ ament" ), content := 0, count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "Ref", next := 443577, root := ~, start := 443551, stop := 443576 ), rec( content := "] which \ it defines, according to\n the following rules. ", count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "P", next := 443639, root := ~, start := 443635, stop := 443638 ), rec( content := "\n\n \ The characters of the string ", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "A", next := 443684, root := ~, start := 443674, stop := 443683 ), rec( content := " represe\ nt the entries in the upper\n triangle of a tournament's adjacency matrix. \ The number of vertices\n ", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "C", next := 443804, root := ~, start := 443796, stop := 443803 ), rec( content := " will be\ detected from the length of the string and will be as\n large as possible.\ ", count := [ 9, 2, 9, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "P", next := 443895, root := ~, start := 443891, stop := 443894 ), rec( content := "\n\n \ The first character represents the possible edge ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 2", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 443963, root := ~, start := 443950, stop := 443962 ), rec( content := ", the se\ cond\n represents ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 3", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 444004, root := ~, start := 443991, stop := 444003 ), rec( content := " and so \ on until ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> n", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 444034, root := ~, start := 444021, stop := 444033 ), rec( content := "; then t\ he following\n character represents ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ -> 3", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 444093, root := ~, start := 444080, stop := 444092 ), rec( content := ", and so\ on up to the character which\n represents the edge ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ -1 -> n", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "C", next := 444170, root := ~, start := 444155, stop := 444169 ), rec( content := ". ", count := [ 9, 2, 9, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "P", next := 444176, root := ~, start := 444172, stop := 444175 ), rec( content := "\n\n \ If a character of the string with corresponding edge ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "C", next := 444248, root := ~, start := 444235, stop := 444247 ), rec( content := " is equa\ l\n to ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "C", next := 444273, root := ~, start := 444265, stop := 444272 ), rec( content := ", then t\ he edge ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "C", next := 444302, root := ~, start := 444289, stop := 444301 ), rec( content := " is pres\ ent in the tournament.\n Otherwise, the edge ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "C", next := 444370, root := ~, start := 444357, stop := 444369 ), rec( content := " is pres\ ent instead. In this way, all the\n possible edges are encoded one-by-one.\ \n\n ", count := [ 9, 2, 9, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := TournamentLineDecoder(\"100001\");\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ ], [ 1, 2, 4 ], [ 1, 2 ] \ ]\n", count := [ 9, 2, 9, 7 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 7 ], mathmode := "Text", name := "Example", next := 444645, root := ~, start := 444461, stop := 444644 ), rec( content := "\n ", count := [ 9, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 9, 3 ], mathmode := "Text", name := "Description", next := 444662, root := ~, start := 443439, stop := 444661 ) ], count := [ 9, 2, 9, 1 ], mathmode := "Text", name := "ManSection", next := 444676, root := ~, start := 443343, stop := 444675 ), rec( content := "\n\n ", count := [ 9, 2, 9, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str", Name := "AdjacencyMatrix\ UpperTriangleLineDecoder" ), content := 0, count := [ 9, 2, 10, 2 ], mathmode := "Text", name := "Oper", next := 444761, root := ~, start := 444697, stop := 444760 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "Returns", next := 444793, root := ~, start := 444764, stop := 444792 ), rec( attributes := rec( ), content := [ rec( content := "\n Thi\ s function takes a string ", count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "A", next := 444853, root := ~, start := 444843, stop := 444852 ), rec( content := ", decode\ s it, and then returns the\n topologically sorted digraph [see ", count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ TopologicalSort" ), content := 0, count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "Ref", next := 444962, root := ~, start := 444926, stop := 444961 ), rec( content := "]\n w\ hich it defines, according to the following rules. ", count := [ 9, 2, 10, 3 ] , mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "P", next := 445024, root := ~, start := 445020, stop := 445023 ), rec( content := "\n\n \ The characters of the string ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "A", next := 445069, root := ~, start := 445059, stop := 445068 ), rec( content := " represe\ nt the entries in the upper\n triangle of a digraph's adjacency matrix. The\ number of vertices ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "C", next := 445182, root := ~, start := 445174, stop := 445181 ), rec( content := "\n wi\ ll be detected from the length of the string and will be as large as\n poss\ ible. ", count := [ 9, 2, 10, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "P", next := 445277, root := ~, start := 445273, stop := 445276 ), rec( content := "\n\n \ The first character represents the possible edge ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 2", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 445345, root := ~, start := 445332, stop := 445344 ), rec( content := ", the se\ cond\n represents ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> 3", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 445386, root := ~, start := 445373, stop := 445385 ), rec( content := " and so \ on until ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ -> n", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 445416, root := ~, start := 445403, stop := 445415 ), rec( content := "; then t\ he following\n character represents ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ -> 3", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 445475, root := ~, start := 445462, stop := 445474 ), rec( content := ", and so\ on up to the character which\n represents the edge ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "n\ -1 -> n", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 445552, root := ~, start := 445537, stop := 445551 ), rec( content := ". If a \ character of the string with\n corresponding edge ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "i\ -> j", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 445625, root := ~, start := 445612, stop := 445624 ), rec( content := " is equa\ l to ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "C", next := 445646, root := ~, start := 445638, stop := 445645 ), rec( content := ", then t\ his edge is\n present in the digraph. Otherwise, it is not present. In thi\ s way, all the\n possible edges are encoded one-by-one. ", count := [ 9, 2, 10, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "P", next := 445793, root := ~, start := 445789, stop := 445792 ), rec( content := "\n\n \ In particular, note that there exists no edge ", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ i, j]", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "C", next := 445858, root := ~, start := 445845, stop := 445857 ), rec( content := " if ", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "j\ \\leq\n i", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "M", next := 445883, root := ~, start := 445862, stop := 445882 ), rec( content := ". In or\ der words, the digraph will be topologically sorted.\n\n ", count := [ 9, 2, 10, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := AdjacencyMatrixUpperTriangleLineDecoder(\"100001\");\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ ], [ 4 \ ], [ ] ]\ngap> gr := AdjacencyMatrixUpperTriangleLineDecoder(\"111111x111\");\ \n\ngap> OutNeighbours(gr);\n[ [ 2\ , 3, 4, 5 ], [ 3, 4 ], [ 4, 5 ], [ 5 ], [ ] ]\n", count := [ 9, 2, 10, 7 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 10, 7 ], mathmode := "Text", name := "Example", next := 446328, root := ~, start := 445949, stop := 446327 ), rec( content := "\n ", count := [ 9, 2, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 10, 3 ], mathmode := "Text", name := "Description", next := 446345, root := ~, start := 444796, stop := 446344 ) ], count := [ 9, 2, 10, 1 ], mathmode := "Text", name := "ManSection", next := 446359, root := ~, start := 444682, stop := 446358 ), rec( content := "\n\n ", count := [ 9, 2, 10, 8 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "str", Name := "TCodeDecoder" ) , content := 0, count := [ 9, 2, 11, 2 ], mathmode := "Text", name := "Oper", next := 446417, root := ~, start := 446380, stop := 446416 ), rec( attributes := rec( ), content := [ rec( content := "A digraph\ .", count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "Returns", next := 446449, root := ~, start := 446420, stop := 446448 ), rec( attributes := rec( ), content := [ rec( content := "\n If \ ", count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "A", next := 446483, root := ~, start := 446473, stop := 446482 ), rec( content := " is a st\ ring consisting of at least two non-negative integers\n separated by spaces\ , then this function will attempt to return the digraph\n which it defines \ as a TCode string. ", count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "P", next := 446668, root := ~, start := 446664, stop := 446667 ), rec( content := "\n\n \ The first integer of the string defines the number of vertices ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 446745, root := ~, start := 446737, stop := 446744 ), rec( content := " in\n \ the digraph, and the second defines the number of edges ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "e\ ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 446817, root := ~, start := 446809, stop := 446816 ), rec( content := ". The\n\ following ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ e", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 446847, root := ~, start := 446838, stop := 446846 ), rec( content := " integer\ s should be vertex numbers in the range ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 0 ..\n v-1]", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 446918, root := ~, start := 446895, stop := 446917 ), rec( content := ". These \ integers are read in pairs and define the digraph's\n edges. This function\ will return an error if ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "s\ tr", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "A", next := 447038, root := ~, start := 447028, stop := 447037 ), rec( content := " has few\ er than\n ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "2\ e+2", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "C", next := 447069, root := ~, start := 447058, stop := 447068 ), rec( content := " entries\ . ", count := [ 9, 2, 11, 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "P", next := 447083, root := ~, start := 447079, stop := 447082 ), rec( content := "\n\n \ Note that the vertex numbers will be incremented by 1 in the digraph\n retu\ rned. Hence the string fragment ", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "0\ 6", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "C", next := 447209, root := ~, start := 447199, stop := 447208 ), rec( content := " will de\ scribe the edge\n ", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "[\ 1,7]", count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "C", next := 447249, root := ~, start := 447237, stop := 447248 ), rec( content := ".\n " , count := [ 9, 2, 11, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := TCodeDecoder(\"3 2 0 2 2 1\");\n\ngap> OutNeighbours(gr);\n[ [ 3 ], [ ], [ 2 ] ]\ngap> gr := TCodeD\ ecoder(\"12 3 0 10 5 2 8 8\");\n\ \ngap> OutNeighbours(gr);\n[ [ 11 ], [ ], [ ], [ ], [ ], [ 3 ], [ ], [ ]\ , [ 9 ], [ ], \n [ ], [ ] ]\n", count := [ 9, 2, 11, 6 ], name := "PCDATA" , root := ~ ) ], count := [ 9, 2, 11, 6 ], mathmode := "Text", name := "Example", next := 447617, root := ~, start := 447255, stop := 447616 ), rec( content := "\n ", count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 11, 3 ], mathmode := "Text", name := "Description", next := 447634, root := ~, start := 446452, stop := 447633 ) ], count := [ 9, 2, 11, 1 ], mathmode := "Text", name := "ManSection", next := 447648, root := ~, start := 446365, stop := 447647 ), rec( content := "\n\n ", count := [ 9, 2, 11, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "digraph", Name := "PlainTextString\ " ), content := 0, count := [ 9, 2, 12, 2 ], mathmode := "Text", name := "Oper", next := 447713, root := ~, start := 447669, stop := 447712 ), rec( attributes := rec( Arg := "s", Name := "DigraphFromPlai\ nTextString" ), content := 0, count := [ 9, 2, 12, 4 ], mathmode := "Text", name := "Oper", next := 447765, root := ~, start := 447716, stop := 447764 ), rec( attributes := rec( ), content := [ rec( content := "A string.\ ", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "Returns", next := 447796, root := ~, start := 447768, stop := 447795 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ lainTextString", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "A", next := 447839, root := ~, start := 447817, stop := 447838 ), rec( content := " takes a\ single digraph, and returns a string\n describing the edges of that digrap\ h. ", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFromPlainTextString", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "A", next := 447961, root := ~, start := 447928, stop := 447960 ), rec( content := "\n ta\ kes such a string and returns the digraph which it describes. Each edge\n \ is written as a pair of integers separated by a single space. The edges\n \ themselves are separated by a double space. Vertex numbers are reduced by\n \ 1 when they are encoded, so that vertices in the string are labelled\n st\ arting at 0.", count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "P", next := 448292, root := ~, start := 448288, stop := 448291 ), rec( content := "\n\n \ Note that the number of vertices of a digraph is not stored, and so vertices\n\ which are not connected to any edge may be lost.", count := [ 9, 2, 12, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "P", next := 448431, root := ~, start := 448427, stop := 448430 ), rec( content := "\n\n \ \n The operation ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphFromPlainTextString", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "C", next := 448489, root := ~, start := 448456, stop := 448488 ), rec( content := "\n ta\ kes an optional first argument ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsMutab\ leDigraph" ), content := 0, count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "Ref", next := 448557, root := ~, start := 448527, stop := 448556 ), rec( content := "\n or\ ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsImmut\ ableDigraph" ), content := 0, count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "Ref", next := 448597, root := ~, start := 448565, stop := 448596 ), rec( content := ", which \ specifies whether the output digraph shall\n be mutable or immutable. If no\ first argument is provided, then an immutable\n digraph is returned by def\ ault.\n\n ", count := [ 9, 2, 12, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2, 3], [1], [1]]);\n\ngap> PlainTextString(gr);\n\"0 1 0 2 1 0 2 0\"\ngap> DigraphFromP\ lainTextString(last);\n\n", count := [ 9, 2, 12, 8 ], name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 12, 8 ], mathmode := "Text", name := "Example", next := 449018, root := ~, start := 448770, stop := 449017 ), rec( content := "\n ", count := [ 9, 2, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 12, 5 ], mathmode := "Text", name := "Description", next := 449035, root := ~, start := 447799, stop := 449034 ) ], count := [ 9, 2, 12, 1 ], mathmode := "Text", name := "ManSection", next := 449049, root := ~, start := 447654, stop := 449048 ), rec( content := "\n\n ", count := [ 9, 2, 12, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename, digrap\ h, delimiter, offset", Name := "WritePlainTextDigraph" ), content := 0, count := [ 9, 2, 13, 2 ], mathmode := "Text", name := "Func", next := 449149, root := ~, start := 449070, stop := 449148 ), rec( attributes := rec( Arg := "filename, delimi\ ter, offset, ignore", Name := "ReadPlainTextDigraph" ), content := 0, count := [ 9, 2, 13, 4 ], mathmode := "Text", name := "Oper", next := 449229, root := ~, start := 449152, stop := 449228 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se functions write and read a single digraph in a human-readable plain\n te\ xt format as follows: each line contains a single edge, and each edge is\n \ written as a pair of integers separated by the string ", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ elimiter", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "A", next := 449477, root := ~, start := 449461, stop := 449476 ), rec( content := ".", count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "P", next := 449482, root := ~, start := 449478, stop := 449481 ), rec( content := "\n\n \ ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "A", next := 449503, root := ~, start := 449488, stop := 449502 ), rec( content := " should \ be the name of a file which will be written to or\n read from, and ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "A", next := 449593, root := ~, start := 449580, stop := 449592 ), rec( content := " should \ be an integer which is added to each\n vertex number as it is written or re\ ad. For example, if\n ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "W\ ritePlainTextDigraph", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 449731, root := ~, start := 449703, stop := 449730 ), rec( content := " is call\ ed with ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "A", next := 449760, root := ~, start := 449747, stop := 449759 ), rec( content := " ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "-\ 1", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 449770, root := ~, start := 449761, stop := 449769 ), rec( content := ", then\n\ the vertices will be numbered in the file starting from 0 instead of 1 -\n\ ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ eadPlainTextDigraph", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 449885, root := ~, start := 449858, stop := 449884 ), rec( content := " would t\ hen need to be called with ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "o\ ffset", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "A", next := 449933, root := ~, start := 449920, stop := 449932 ), rec( content := "\n ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "1\ ", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "C", next := 449946, root := ~, start := 449938, stop := 449945 ), rec( content := " to conv\ ert back to the original graph.", count := [ 9, 2, 13, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "P", next := 449989, root := ~, start := 449985, stop := 449988 ), rec( content := "\n\n \ ", count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) , rec( attributes := rec( ), content := [ rec( content := "i\ gnore", count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "A", next := 450008, root := ~, start := 449995, stop := 450007 ), rec( content := " should \ be a list of characters which will be ignored when\n reading the graph.\n \ ", count := [ 9, 2, 13, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[1, 2, 3], [1, 1], [2]]);\n\ngap> filename := Concatenation(DIGRAPHS_Dir(), \"/tst/out\ /plain.txt\");;\ngap> WritePlainTextDigraph(filename, gr, \",\", -1);\ngap> Re\ adPlainTextDigraph(filename, \",\", 1, ['/', '%']);\n\n", count := [ 9, 2, 13, 8 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 8 ], mathmode := "Text", name := "Example", next := 450451, root := ~, start := 450094, stop := 450450 ), rec( content := "\n ", count := [ 9, 2, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 13, 5 ], mathmode := "Text", name := "Description", next := 450468, root := ~, start := 449232, stop := 450467 ) ], count := [ 9, 2, 13, 1 ], mathmode := "Text", name := "ManSection", next := 450482, root := ~, start := 449055, stop := 450481 ), rec( content := "\n\n ", count := [ 9, 2, 13, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes := rec( Arg := "filename, digrap\ h", Name := "WriteDIMACSDigraph" ), content := 0, count := [ 9, 2, 14, 2 ], mathmode := "Text", name := "Oper", next := 450560, root := ~, start := 450503, stop := 450559 ), rec( attributes := rec( Arg := "filename", Name := "ReadDIMACSDigra\ ph" ), content := 0, count := [ 9, 2, 14, 4 ], mathmode := "Text", name := "Oper", next := 450610, root := ~, start := 450563, stop := 450609 ), rec( attributes := rec( ), content := [ rec( content := "\n The\ se operations write or read the single symmetric digraph ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "A", next := 450705, root := ~, start := 450691, stop := 450704 ), rec( content := "\n to\ or from a file in DIMACS format, as appropriate. The operation\n ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "W\ riteDIMACSDigraph", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "C", next := 450805, root := ~, start := 450780, stop := 450804 ), rec( content := " records\ the vertices and edges of ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "A", next := 450854, root := ~, start := 450840, stop := 450853 ), rec( content := ".\n T\ he vertex labels of ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "d\ igraph", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "A", next := 450895, root := ~, start := 450881, stop := 450894 ), rec( content := " will be\ recorded only if they are\n integers. See ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymme\ tricDigraph" ), content := 0, count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "Ref", next := 450980, root := ~, start := 450948, stop := 450979 ), rec( content := " and ", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "Ref", next := 451024, root := ~, start := 450985, stop := 451023 ), rec( content := ".", count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "P", next := 451029, root := ~, start := 451025, stop := 451028 ), rec( content := "\n\n \ The first argument ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "A", next := 451069, root := ~, start := 451054, stop := 451068 ), rec( content := " should \ be the name of the file which will\n be written to or read from. A file ca\ n contain one symmetric digraph in\n DIMACS format. If ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "f\ ilename", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "A", next := 451226, root := ~, start := 451211, stop := 451225 ), rec( content := " ends in\ one of ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ gz", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "C", next := 451252, root := ~, start := 451242, stop := 451251 ), rec( content := ", ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ bz2", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "C", next := 451265, root := ~, start := 451254, stop := 451264 ), rec( content := ",\n o\ r ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := ".\ xz", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "C", next := 451284, root := ~, start := 451274, stop := 451283 ), rec( content := ", then t\ he file is compressed, or decompressed, appropriately.\n ", count := [ 9, 2, 14, 6 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 14, 7 ], mathmode := "Text", name := "P", next := 451355, root := ~, start := 451351, stop := 451354 ), rec( content := "\n\n \ The DIMACS format is described as follows. Each line in the DIMACS file has\n\ one of four types:\n ", count := [ 9, 2, 14, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "\n A line beginning with ", coun\ t := [ 9, 2, 14, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "c", count := [ 9, 2, 14, 9 ],\ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 9 ] , math\ mode := "Text", name := "C", next := 451523, root := ~, start := 451515, stop \ := 451522 ), rec( content := " and followed by any number of characters\n \ is a comment line, and is ignored.\n ", count := [ 9, 2, 14, 9 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 9 ], name := "Ite\ m", next := 451621, root := ~, start := 451478, stop := 451620 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n A line beginning with ", coun\ t := [ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "p", count := [ 9, 2, 14, 11 ]\ , mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 11\ ], mathmode := "Text", name := "C", next := 451673, root := ~, start := 45166\ 5, stop := 451672 ), rec( content := " defines the numbers of vertices and edg\ es\n the digraph. This line has the format ", count := [ 9, 2, 14, 11 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "p edge \n ", count :\ = [ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count\ := [ 9, 2, 14, 11 ], mathmode := "Text", name := "C", next := 451823, root :=\ ~, start := 451763, stop := 451822 ), rec( content := ", where ", count := [ \ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "", count := [ 9, 2\ , 14, 11 ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], coun\ t := [ 9, 2, 14, 11 ], mathmode := "Text", name := "C", next := 451857, root :\ = ~, start := 451831, stop := 451856 ), rec( content := " and\n ", coun\ t := [ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "", count := [ 9, 2,\ 14, 11 ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], coun\ t := [ 9, 2, 14, 11 ], mathmode := "Text", name := "C", next := 451893, root :\ = ~, start := 451870, stop := 451892 ), rec( content := " are replaced by the \ relevant integers. There\n must be exactly one such line in the file, a\ nd it must occur before any\n of the following kinds of line.", count :\ = [ 9, 2, 14, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := 0, count := [ 9, 2, 14, 12 ], mathmode := "Text" , name\ := "P", next := 452062, root := ~, start := 452058, stop := 452061 ), rec( cont\ ent := "\n\n Although it is required to be present, the value of\n \ ", count := [ 9, 2, 14, 12 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "", c\ ount := [ 9, 2, 14, 12 ], mathmode := "Text", name := "PCDATA", next := 6, roo\ t := ~ ) ], count := [ 9, 2, 14, 12 ], mathmode := "Text", name := "C", next :\ = 452155, root := ~, start := 452132, stop := 452154 ), rec( content := " will\ be ignored. The correct number of edges\n will be deduced from the res\ t of the information in the file.\n ", count := [ 9, 2, 14, 12 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 11 ], name := "Ite\ m", next := 452284, root := ~, start := 451628, stop := 452283 ), rec( attributes :\ = rec( ), content := [ rec( content := "\n A line of the form ", count\ := [ 9, 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "e ", count := [ 9, 2, \ 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2\ , 14, 14 ], mathmode := "Text", name := "C", next := 452353, root := ~, start \ := 452325, stop := 452352 ), rec( content := ", where ", count := [ 9, 2, 14, \ 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := re\ c( ), content := [ rec( content := "", count := [ 9, 2, 14, 14 ], mathmode\ := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 14, 1\ 4 ], mathmode := "Text", name := "C", next := 452377, root := ~, start := 4523\ 61, stop := 452376 ), rec( content := "\n and ", count := [ 9, 2, 14, 1\ 4 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec\ ( ), content := [ rec( content := "", count := [ 9, 2, 14, 14 ], mathmode \ := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 14, 14\ ], mathmode := "Text", name := "C", next := 452406, root := ~, start := 45239\ 0, stop := 452405 ), rec( content := " are integers in the range ", count := [ \ 9, 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attrib\ utes := rec( ), content := [ rec( content := "[1 ..\n ]\ ", count := [ 9, 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ \ ) ], count := [ 9, 2, 14, 14 ], mathmode := "Text", name := "C", next := 45247\ 6, root := ~, start := 452433, stop := 452475 ), rec( content := ", specifies \ that there is a (symmetric) edge\n in the digraph between the vertices \ ", count := [ 9, 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "", count := [ 9, 2, 14, 14\ ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ \ 9, 2, 14, 14 ], mathmode := "Text", name := "C", next := 452581, root := ~, star\ t := 452565, stop := 452580 ), rec( content := " and\n ", count := [ 9,\ 2, 14, 14 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attribut\ es := rec( ), content := [ rec( content := "", count := [ 9, 2, 14, 14 ], \ mathmode := "Text", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, \ 2, 14, 14 ], mathmode := "Text", name := "C", next := 452610, root := ~, start\ := 452594, stop := 452609 ), rec( content := ". A symmetric edge only needs t\ o be defined once; an\n additional line ", count := [ 9, 2, 14, 14 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "e ", count := [ 9, 2, 14, 14 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 14 ], mathmode :=\ "Text", name := "C", next := 452715, root := ~, start := 452687, stop := 4527\ 14 ), rec( content := ", or ", count := [ 9, 2, 14, 14 ], mathmode := "Text", name\ := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( conte\ nt := "e \n ", count := [ 9, 2, 14, 14 ], mathmode := "Text", n\ ame := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 14 ], mathmode := "Text", name\ := "C", next := 452758, root := ~, start := 452720, stop := 452757 ), rec( cont\ ent := ", will be interpreted as an additional, multiple, edge.\n Loops\ are permitted.\n ", count := [ 9, 2, 14, 14 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 14 ], name := "Item", next := 4528\ 56, root := ~, start := 452291, stop := 452855 ), rec( attributes := rec( ), content := [ rec( cont\ ent := "\n A line of the form ", count := [ 9, 2, 14, 16 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :=\ [ rec( content := "n ", count := [ 9, 2, 14, 16 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 16 ], mathmode := "\ Text", name := "C", next := 452929, root := ~, start := 452897, stop := 452928\ ), rec( content := ", where\n ", count := [ 9, 2, 14, 16 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content :\ = [ rec( content := "", count := [ 9, 2, 14, 16 ], mathmode := "Text", name\ := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 14, 16 ], mathmode :=\ "Text", name := "C", next := 452961, root := ~, start := 452945, stop := 4529\ 60 ), rec( content := " is an integer in the range ", count := [ 9, 2, 14, 16 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "[1 ..\n ]", count := [ 9, 2, 1\ 4, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2,\ 14, 16 ], mathmode := "Text", name := "C", next := 453032, root := ~, start :\ = 452989, stop := 453031 ), rec( content := " and ", count := [ 9, 2, 14, 16 ] , math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( content := "", count := [ 9, 2, 14, 16 ], mathmode := "Te\ xt", name := "PCDATA", next := 6, root := ~ ) ], count := [ 9, 2, 14, 16 ], math\ mode := "Text", name := "C", next := 453057, root := ~, start := 453037, stop \ := 453056 ), rec( content := " is an integer,\n signifies that the vert\ ex ", count := [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", root :=\ ~ ), rec( attributes := rec( ), content := [ rec( content := "", count :=\ [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", next := 6, root := ~ \ ) ], count := [ 9, 2, 14, 16 ], mathmode := "Text", name := "C", next := 45312\ 3, root := ~, start := 453107, stop := 453122 ), rec( content := " has the lab\ el\n ", count := [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ), rec( attributes := rec( ), content := [ rec( content := "", c\ ount := [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", next := 6, roo\ t := ~ ) ], count := [ 9, 2, 14, 16 ], mathmode := "Text", name := "C", next :\ = 453166, root := ~, start := 453146, stop := 453165 ), rec( content := " in t\ he digraph. If a label is not specified for a\n vertex, then ", count :\ = [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attr\ ibutes := rec( ), content := [ rec( content := "ReadDIMACSDigraph", count := \ [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count :\ = [ 9, 2, 14, 16 ], mathmode := "Text", name := "C", next := 453262, root := ~ , star\ t := 453238, stop := 453261 ), rec( content := " will assign the label\n \ ", count := [ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", root := \ ~ ), rec( attributes := rec( ), content := [ rec( content := "1", count := [ \ 9, 2, 14, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ \ 9, 2, 14, 16 ], mathmode := "Text", name := "C", next := 453301, root := ~, star\ t := 453293, stop := 453300 ), rec( content := ", according to the DIMACS spec\ ification.\n ", count := [ 9, 2, 14, 16 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ 9, 2, 14, 16 ], name := "Item", next := 4533\ 55, root := ~, start := 452863, stop := 453354 ) ], count := [ 9, 2, 14, 8 ], mathmode := "Text", name := "List", next := 453367, root := ~, start := 451465, stop := 453366 ), rec( content := "\n\n \ A detailed definition of the DIMACS format can be found at\n ", count := [ 9, 2, 14, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "h\ ttp://mat.gsia.cmu.edu/COLOR/general/ccformat.ps", count := [ 9, 2, 14, 18 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 18 ], mathmode := "Text", name := "URL", next := 453496, root := ~, start := 453436, stop := 453495 ), rec( content := ", in Sec\ tion\n 2.1. Note that optional descriptor lines, as described in Section 2.\ 1, will\n be ignored.\n ", count := [ 9, 2, 14, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0, count := [ 9, 2, 14, 19 ], mathmode := "Text", name := "P", next := 453613, root := ~, start := 453609, stop := 453612 ), rec( content := "\n\n\n \ ", count := [ 9, 2, 14, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\\ ngap> gr := Digraph([[2], [1, 3, 4], [2, 4], [2, 3]]);\n\ngap> filename := Concatenation(DIGRAPHS_Dir(),\n> \ \"/tst/out/dimacs.dimacs\");;\ngap> WriteDIMACSDigrap\ h(filename, gr);;\ngap> ReadDIMACSDigraph(filename);\n", count := [ 9, 2, 14, 20 ], name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 20 ], mathmode := "Text", name := "Example", next := 453975, root := ~, start := 453620, stop := 453974 ), rec( content := "\n ", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 14, 5 ], mathmode := "Text", name := "Description", next := 453992, root := ~, start := 450613, stop := 453991 ) ], count := [ 9, 2, 14, 1 ], mathmode := "Text", name := "ManSection", next := 454006, root := ~, start := 450488, stop := 454005 ), rec( content := "\n\n ", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 2, 0, 1 ], mathmode := "Text", name := "Section", next := 454020, root := ~, start := 421151, stop := 454019 ) , rec( content := "\n", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 9, 0, 0, 1 ], mathmode := "Text", name := "Chapter", next := 454031, root := ~, start := 399111, stop := 454030 ), rec( content := "\n ", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( content := " visualising, io ", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( content := "\n", count := [ 9, 2, 14, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ 0, 0, 4, 1 ], mathmode := "Text", name := "Body", next := 454065, root := ~, start := 3969, stop := 454064 ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 0, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Grape to Digraphs Command M\ ap\n ", count := [ "A", 0, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 2 ], mathmode := "Text", name := "Heading", next := 454136, root := ~, start := 454080, stop := 454135 ), rec( content := "\n Below is a table of ", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.ac.\ uk/~leonard/grape/", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 0, 0, 3 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ "A", 0, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " commands with the ", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Digraphs", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " counterparts.\n The sections in this c\ hapter correspond to the chapters in the ", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.ac.\ uk/~leonard/grape/", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 0, 0, 3 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ), content := [ rec( attributes := rec( ), content := [ rec( content := "Grape", count := [ "A", 0, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 3 ], name := "Package", next := 137, root := ~, start := 113, stop := 136 ) ], count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := "\n manual.\n\n ", count := [ "A", 0, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 1, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Functions to cons\ truct and modify graphs\n ", count := [ "A", 1, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 2 ], mathmode := "Text", name := "Heading", next := 454381, root := ~, start := 454310, stop := 454380 ), rec( content := "\n ", count := [ "A", 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 1, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 1, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 3 ], name := "E", next := 454519, root := ~, start := 454409, stop := 454518 ), rec( content := "\n ", count := [ "A", 1, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 3 ], mathmode := "Text", name := "Alt", next := 454530, root := ~, start := 454386, stop := 454529 ), rec( content := "\n\n ", count := [ "A", 1, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 4 ], name := "HorLine", next := 454576, root := ~, start := 454566, stop := 454575 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 1, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 1, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 1, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 1, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 5 ], name := "Item", next := 454645, root := ~, start := 454597, stop := 454644 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 1, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 7 ], name := "Item", next := 454705, root := ~, start := 454654, stop := 454704 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 1, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 9 ], name := "Alt", next := 454786, root := ~, start := 454731, stop := 454785 ), rec( content := "\n \ ", count := [ "A", 1, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 9 ], name := "Item", next := 454802, root := ~, start := 454714, stop := 454801 ) ], count := [ "A", 1, 0, 4 ], name := "Row", next := 454815, root := ~, start := 454583, stop := 454814 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 10 ], name := "HorLine", next := 454832, root := ~, start := 454822, stop := 454831 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raph", count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "C", next := 454883, root := ~, start := 454871, stop := 454882 ), rec( content := "\n \ ", count := [ "A", 1, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 11 ], name := "Item", next := 454899, root := ~, start := 454854, stop := 454898 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ " ), content := 0, count := [ "A", 1, 0, 13 ], mathmode := "Text", name := "Ref", next := 454946, root := ~, start := 454925, stop := 454945 ), rec( content := "\n \ ", count := [ "A", 1, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 13 ], name := "Item", next := 454962, root := ~, start := 454908, stop := 454961 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Used with the arguments of ", count := [ "A", 1, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Graph", count := [ "A", 1, 0, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "C", next := 4550\ 55, root := ~, start := 455043, stop := 455054 ), rec( HTML := true, content := "\ in ", count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , ", count := [ "A", 1, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Oper := "Digraph" ), content := 0, count := [ "A", 1, 0, 15 ], mathmode := \ "Text", name := "Ref", next := 455101, root := ~, start := 455068, stop := 4551\ 00 ), rec( HTML := true, content := " will return an equivalent digraph. ", count := [ "A", 1, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := 0 , count := [ "A", 1, 0, 16 ], mathmode := \ "Text", name := "P", next := 455141, root := ~, start := 455137, stop := 45514\ 0 ), rec( HTML := true, content := "\n ", count := [ "A", 1, 0, 16 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 15 ], name := "Alt", next := 455158, root := ~, start := 454988, stop := 455157 ), rec( content := "\n \ ", count := [ "A", 1, 0, 16 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 15 ], name := "Item", next := 455174, root := ~, start := 454971, stop := 455173 ) ], count := [ "A", 1, 0, 10 ], name := "Row", next := 455187, root := ~, start := 454840, stop := 455186 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 17 ], name := "HorLine", next := 455204, root := ~, start := 455194, stop := 455203 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ dgeOrbitsGraph", count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "C", next := 455265, root := ~, start := 455243, stop := 455264 ), rec( content := "\n \ ", count := [ "A", 1, 0, 18 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 18 ], name := "Item", next := 455281, root := ~, start := 455226, stop := 455280 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EdgeOrb\ itsDigraph" ), content := 0, count := [ "A", 1, 0, 20 ], mathmode := "Text", name := "Ref", next := 455338, root := ~, start := 455307, stop := 455337 ), rec( content := "\n \ ", count := [ "A", 1, 0, 20 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 20 ], name := "Item", next := 455354, root := ~, start := 455290, stop := 455353 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "EdgeOrbitsDigraph" ), content := 0 , count := [ "A", 1, 0, 22 ], mathmode := \ "Text", name := "Ref", next := 455453, root := ~, start := 455422, stop := 4554\ 52 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 1, 0, 22 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "EdgeOrbitsDigraph", count := [ "A", 1, 0, 22 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 22 ], mathmode := \ "Text", name := "C", next := 455508, root := ~, start := 455484, stop := 45550\ 7 ), rec( HTML := true, content := " in ", count := [ "A", 1, 0, 22 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 22 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 22 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 22 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 22 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 22 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 22 ], name := "Alt", next := 455537, root := ~, start := 455380, stop := 455536 ), rec( content := "\n \ ", count := [ "A", 1, 0, 22 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 22 ], name := "Item", next := 455553, root := ~, start := 455363, stop := 455552 ) ], count := [ "A", 1, 0, 17 ], name := "Row", next := 455566, root := ~, start := 455212, stop := 455565 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 23 ], name := "HorLine", next := 455583, root := ~, start := 455573, stop := 455582 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ ullGraph", count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "C", next := 455638, root := ~, start := 455622, stop := 455637 ), rec( content := "\n \ ", count := [ "A", 1, 0, 24 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 24 ], name := "Item", next := 455654, root := ~, start := 455605, stop := 455653 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "NullDig\ raph" ), content := 0, count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "Ref", next := 455705, root := ~, start := 455680, stop := 455704 ), rec( content := "\n \ ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n or\n ", count := [ "A", 1, 0, 26 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EmptyDigrap\ h" ), content := 0, count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "Ref\ ", next := 455787, root := ~, start := 455761, stop := 455786 ), rec( HTML := true , content := "\ \n ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 1, 0, 26 ], name := "Alt", next := 455804, root := ~, start := 455716, stop := 455803 ), rec( content := "\n \ ", count := [ "A", 1, 0, 26 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 26 ], name := "Item", next := 455820, root := ~, start := 455663, stop := 455819 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Note that, unlike in ", count := [ "A", 1, 0, 28 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 28 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 28 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 28 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 28 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 28 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , the arguments of ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "NullDigraph" ), content := 0 , count := [ "A", 1, 0, 28 ], mathmode := \ "Text", name := "Ref", next := 455963, root := ~, start := 455924, stop := 4559\ 62 ), rec( HTML := true, content := " and ", count := [ "A", 1, 0, 28 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EmptyD\ igraph" ), content := 0, count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "Ref\ ", next := 455994, root := ~, start := 455968, stop := 455993 ), rec( HTML := true , content := "\ are just\n non-negative integers. Since digraphs in ", count := [ "A", 1, 0, 28 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 1, 0, 28 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 28 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ are immutable,\n there is no reason (although it is possible) to c\ reate a null digraph\n and then add edges, as you might be used to \ doing in ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 28 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 28 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 28 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 28 ], name := "Alt", next := 456255, root := ~, start := 455846, stop := 456254 ), rec( content := "\n \ ", count := [ "A", 1, 0, 28 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 28 ], name := "Item", next := 456271, root := ~, start := 455829, stop := 456270 ) ], count := [ "A", 1, 0, 23 ], name := "Row", next := 456284, root := ~, start := 455591, stop := 456283 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 29 ], name := "HorLine", next := 456301, root := ~, start := 456291, stop := 456300 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ompleteGraph", count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "C", next := 456360, root := ~, start := 456340, stop := 456359 ), rec( content := "\n \ ", count := [ "A", 1, 0, 30 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 30 ], name := "Item", next := 456376, root := ~, start := 456323, stop := 456375 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Complet\ eDigraph" ), content := 0, count := [ "A", 1, 0, 32 ], mathmode := "Text", name := "Ref", next := 456431, root := ~, start := 456402, stop := 456430 ), rec( content := "\n \ ", count := [ "A", 1, 0, 32 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 32 ], name := "Item", next := 456447, root := ~, start := 456385, stop := 456446 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Note that, unlike in ", count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 34 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 34 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 34 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 34 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , the argument to ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "CompleteDigraph" ), content := 0 , count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "Ref", next := 456592, root := ~, start := 456549, stop := 4565\ 91 ), rec( HTML := true, content := " is simply a non-negative integer\n \ indicating the number of vertices. By default a complete digraph in\n \ ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ does not have any loops. If you want a complete digraph\n with loo\ ps at every vertex, then use ", count := [ "A", 1, 0, 34 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "DigraphAddAllLoops" ), content := 0 , count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "Ref", next := 456880, root := ~, start := 456834, stop := 4568\ 79 ), rec( HTML := true, content := ".\n ", count := [ "A", 1, 0, 34 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], name := "Alt", next := 456898, root := ~, start := 456473, stop := 456897 ), rec( content := "\n \ ", count := [ "A", 1, 0, 34 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 34 ], name := "Item", next := 456914, root := ~, start := 456456, stop := 456913 ) ], count := [ "A", 1, 0, 29 ], name := "Row", next := 456927, root := ~, start := 456309, stop := 456926 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 35 ], name := "HorLine", next := 456944, root := ~, start := 456934, stop := 456943 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "J\ ohnsonGraph", count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "C", next := 457002, root := ~, start := 456983, stop := 457001 ), rec( content := "\n \ ", count := [ "A", 1, 0, 36 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 36 ], name := "Item", next := 457018, root := ~, start := 456966, stop := 457017 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Johnson\ Digraph" ), content := 0, count := [ "A", 1, 0, 38 ], mathmode := "Text", name := "Ref", next := 457072, root := ~, start := 457044, stop := 457071 ), rec( content := "\n \ ", count := [ "A", 1, 0, 38 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 38 ], name := "Item", next := 457088, root := ~, start := 457027, stop := 457087 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 40 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 40 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 40 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 40 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 40 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "JohnsonGraph", count :=\ [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 40 ], mathmode := \ "Text", name := "C", next := 457184, root := ~, start := 457165, stop := 45718\ 3 ), rec( HTML := true, content := " is equivalent to\n ", count := [ "A", 1, 0, 40 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Johnso\ nDigraph" ), content := 0, count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "Ref\ ", next := 457242, root := ~, start := 457214, stop := 457241 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 40 ], name := "Alt", next := 457260, root := ~, start := 457114, stop := 457259 ), rec( content := "\n \ ", count := [ "A", 1, 0, 40 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 40 ], name := "Item", next := 457276, root := ~, start := 457097, stop := 457275 ) ], count := [ "A", 1, 0, 35 ], name := "Row", next := 457289, root := ~, start := 456952, stop := 457288 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 41 ], name := "HorLine", next := 457306, root := ~, start := 457296, stop := 457305 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ayleyGraph", count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "C", next := 457363, root := ~, start := 457345, stop := 457362 ), rec( content := "\n \ ", count := [ "A", 1, 0, 42 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 42 ], name := "Item", next := 457379, root := ~, start := 457328, stop := 457378 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 44 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "CayleyD\ igraph" ), content := 0, count := [ "A", 1, 0, 44 ], mathmode := "Text", name := "Ref", next := 457432, root := ~, start := 457405, stop := 457431 ), rec( content := "\n \ ", count := [ "A", 1, 0, 44 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 44 ], name := "Item", next := 457448, root := ~, start := 457388, stop := 457447 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "CayleyDigraph" ), content := 0 , count := [ "A", 1, 0, 46 ], mathmode := \ "Text", name := "Ref", next := 457545, root := ~, start := 457518, stop := 4575\ 44 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 1, 0, 46 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "CayleyGraph", count := [ "A", 1, 0, 46 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 46 ], mathmode := \ "Text", name := "C", next := 457596, root := ~, start := 457578, stop := 45759\ 5 ), rec( HTML := true, content := " in ", count := [ "A", 1, 0, 46 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 46 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 46 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 46 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 46 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 46 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 1, 0, 46 ], name := "Alt", next := 457625, root := ~, start := 457474, stop := 457624 ), rec( content := "\n \ ", count := [ "A", 1, 0, 46 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 46 ], name := "Item", next := 457641, root := ~, start := 457457, stop := 457640 ) ], count := [ "A", 1, 0, 41 ], name := "Row", next := 457654, root := ~, start := 457314, stop := 457653 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 47 ], name := "HorLine", next := 457671, root := ~, start := 457661, stop := 457670 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ ddEdgeOrbit", count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "C", next := 457729, root := ~, start := 457710, stop := 457728 ), rec( content := "\n \ ", count := [ "A", 1, 0, 48 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 48 ], name := "Item", next := 457745, root := ~, start := 457693, stop := 457744 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 50 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ AddEdgeOrbit" ), content := 0, count := [ "A", 1, 0, 50 ], mathmode := "Text", name := "Ref", next := 457804, root := ~, start := 457771, stop := 457803 ), rec( content := "\n \ ", count := [ "A", 1, 0, 50 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 50 ], name := "Item", next := 457820, root := ~, start := 457754, stop := 457819 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Note that, unlike in ", count := [ "A", 1, 0, 52 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 52 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 52 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 52 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 52 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , digraphs in ", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Digraphs", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "Pac\ kage", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true, content := "\ are immutable,\n and this operation returns a new digraph, i.e. it\ does not modify the\n digraph given as an argument.\n ", count := [ "A", 1, 0, 52 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], name := "Alt", next := 458084, root := ~, start := 457846, stop := 458083 ), rec( content := "\n \ ", count := [ "A", 1, 0, 52 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 52 ], name := "Item", next := 458100, root := ~, start := 457829, stop := 458099 ) ], count := [ "A", 1, 0, 47 ], name := "Row", next := 458113, root := ~, start := 457679, stop := 458112 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 53 ], name := "HorLine", next := 458130, root := ~, start := 458120, stop := 458129 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "R\ emoveEdgeOrbit", count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "C", next := 458191, root := ~, start := 458169, stop := 458190 ), rec( content := "\n \ ", count := [ "A", 1, 0, 54 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 54 ], name := "Item", next := 458207, root := ~, start := 458152, stop := 458206 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 56 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ RemoveEdgeOrbit" ), content := 0, count := [ "A", 1, 0, 56 ], mathmode := "Text", name := "Ref", next := 458269, root := ~, start := 458233, stop := 458268 ), rec( content := "\n \ ", count := [ "A", 1, 0, 56 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 56 ], name := "Item", next := 458285, root := ~, start := 458216, stop := 458284 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Note that, unlike in ", count := [ "A", 1, 0, 58 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 1, 0, 58 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 1, 0, 58 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 1, 0, 58 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 1, 0, 58 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , digraphs in ", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Digraphs", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "Pac\ kage", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true, content := "\ are immutable,\n and this operation returns a new digraph, i.e. it\ does not modify the\n digraph given as an argument.\n ", count := [ "A", 1, 0, 58 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], name := "Alt", next := 458549, root := ~, start := 458311, stop := 458548 ), rec( content := "\n \ ", count := [ "A", 1, 0, 58 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 58 ], name := "Item", next := 458565, root := ~, start := 458294, stop := 458564 ) ], count := [ "A", 1, 0, 53 ], name := "Row", next := 458578, root := ~, start := 458138, stop := 458577 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 59 ], name := "HorLine", next := 458595, root := ~, start := 458585, stop := 458594 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ ssignVertexNames", count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "C", next := 458658, root := ~, start := 458634, stop := 458657 ), rec( content := "\n \ ", count := [ "A", 1, 0, 60 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 60 ], name := "Item", next := 458674, root := ~, start := 458617, stop := 458673 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "SetDigr\ aphVertexLabels" ), content := 0, count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "Ref", next := 458736, root := ~, start := 458700, stop := 458735 ), rec( content := "\n \ ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n and\n ", count := [ "A", 1, 0, 62 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "SetDigraph\ VertexLabel" ), content := 0, count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "Ref\ ", next := 458828, root := ~, start := 458793, stop := 458827 ), rec( HTML := true , content := "\ \n ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 1, 0, 62 ], name := "Alt", next := 458845, root := ~, start := 458747, stop := 458844 ), rec( content := "\n \ ", count := [ "A", 1, 0, 62 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 62 ], name := "Item", next := 458861, root := ~, start := 458683, stop := 458860 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Use ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "SetDigraphVertexLabel" ), content := 0 , count := [ "A", 1, 0, 64 ], mathmode := \ "Text", name := "Ref", next := 458957, root := ~, start := 458922, stop := 4589\ 56 ), rec( HTML := true, content := " to set an individual\n vertex\ label and ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( Oper := "SetDigraphVertexLabels" ), content := 0, count := [ "A", 1, 0, 64 ], mathmode := \ "Text", name := "Ref", next := 459044, root := ~, start := 459008, stop := 4590\ 43 ), rec( HTML := true, content := " to set all of\n the vertex la\ bels.\n ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 1, 0, 64 ], name := "Alt", next := 459106, root := ~, start := 458887, stop := 459105 ), rec( content := "\n \ ", count := [ "A", 1, 0, 64 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 64 ], name := "Item", next := 459122, root := ~, start := 458870, stop := 459121 ) ], count := [ "A", 1, 0, 59 ], name := "Row", next := 459135, root := ~, start := 458603, stop := 459134 ), rec( attributes := rec( ), content := 0, count := [ "A", 1, 0, 65 ], name := "HorLine", next := 459152, root := ~, start := 459142, stop := 459151 ) ], count := [ "A", 1, 0, 4 ], mathmode := "Text", name := "Table", next := 459165, root := ~, start := 454536, stop := 459164 ), rec( content := "\n ", count := [ "A", 1, 0, 66 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 1, 0, 1 ], mathmode := "Text", name := "Section", next := 459178, root := ~, start := 454296, stop := 459177 ), rec( content := "\n\n ", count := [ "A", 1, 0, 66 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 2, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Functions to insp\ ect graphs, vertices and edges\n ", count := [ "A", 2, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 2 ], mathmode := "Text", name := "Heading", next := 459274, root := ~, start := 459196, stop := 459273 ), rec( content := "\n ", count := [ "A", 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 2, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 3 ], name := "E", next := 459412, root := ~, start := 459302, stop := 459411 ), rec( content := "\n ", count := [ "A", 2, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 3 ], mathmode := "Text", name := "Alt", next := 459423, root := ~, start := 459279, stop := 459422 ), rec( content := "\n\n ", count := [ "A", 2, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 4 ], name := "HorLine", next := 459469, root := ~, start := 459459, stop := 459468 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 2, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 2, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 2, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 2, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 5 ], name := "Item", next := 459539, root := ~, start := 459491, stop := 459538 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 2, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 7 ], name := "Item", next := 459599, root := ~, start := 459548, stop := 459598 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 2, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 9 ], name := "Alt", next := 459680, root := ~, start := 459625, stop := 459679 ), rec( content := "\n \ ", count := [ "A", 2, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 9 ], name := "Item", next := 459696, root := ~, start := 459608, stop := 459695 ) ], count := [ "A", 2, 0, 4 ], name := "Row", next := 459709, root := ~, start := 459477, stop := 459708 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 10 ], name := "HorLine", next := 459726, root := ~, start := 459716, stop := 459725 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sGraph", count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "C", next := 459779, root := ~, start := 459765, stop := 459778 ), rec( content := "\n \ ", count := [ "A", 2, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 11 ], name := "Item", next := 459795, root := ~, start := 459748, stop := 459794 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Filt := "IsDigra\ ph" ), content := 0, count := [ "A", 2, 0, 13 ], mathmode := "Text", name := "Ref", next := 459844, root := ~, start := 459821, stop := 459843 ), rec( content := "\n \ ", count := [ "A", 2, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 13 ], name := "Item", next := 459860, root := ~, start := 459804, stop := 459859 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Filt := "IsDigraph" ), content := 0, count := [ "A", 2, 0, 15 ], mathmode := \ "Text", name := "Ref", next := 459939, root := ~, start := 459916, stop := 4599\ 38 ), rec( HTML := true, content := " is a category and not a function like\n \ ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsGraph", count := [ "A\ ", 2, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 15 ], mathmode := \ "Text", name := "C", next := 460004, root := ~, start := 459990, stop := 46000\ 3 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 15 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 15 ], name := "Alt", next := 460022, root := ~, start := 459886, stop := 460021 ), rec( content := "\n \ ", count := [ "A", 2, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 15 ], name := "Item", next := 460038, root := ~, start := 459869, stop := 460037 ) ], count := [ "A", 2, 0, 10 ], name := "Row", next := 460051, root := ~, start := 459734, stop := 460050 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 16 ], name := "HorLine", next := 460068, root := ~, start := 460058, stop := 460067 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ rderGraph", count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "C", next := 460124, root := ~, start := 460107, stop := 460123 ), rec( content := "\n \ ", count := [ "A", 2, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 17 ], name := "Item", next := 460140, root := ~, start := 460090, stop := 460139 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ NrVertices" ), content := 0, count := [ "A", 2, 0, 19 ], mathmode := "Text", name := "Ref", next := 460197, root := ~, start := 460166, stop := 460196 ), rec( content := "\n \ ", count := [ "A", 2, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 19 ], name := "Item", next := 460213, root := ~, start := 460149, stop := 460212 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphNrVertices" ), content := 0 , count := [ "A", 2, 0, 21 ], mathmode := \ "Text", name := "Ref", next := 460314, root := ~, start := 460283, stop := 4603\ 13 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "OrderGraph", count := [ "A", 2, 0, 21 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 21 ], mathmode := \ "Text", name := "C", next := 460364, root := ~, start := 460347, stop := 46036\ 3 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 21 ], name := "Alt", next := 460393, root := ~, start := 460239, stop := 460392 ), rec( content := "\n \ ", count := [ "A", 2, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 21 ], name := "Item", next := 460409, root := ~, start := 460222, stop := 460408 ) ], count := [ "A", 2, 0, 16 ], name := "Row", next := 460422, root := ~, start := 460076, stop := 460421 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 22 ], name := "HorLine", next := 460439, root := ~, start := 460429, stop := 460438 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sVertex(", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( content := "graph", count := [ "A", 2, 0, 23 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 23 ], mathmode := \ "Text", name := "A", next := 460502, root := ~, start := 460490, stop := 46050\ 1 ), rec( content := ", ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "v\ ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~\ ) ], count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "A", next := 4605\ 12, root := ~, start := 460504, stop := 460511 ), rec( content := ")", count := [ "A", 2, 0, 23 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "C", next := 460517, root := ~, start := 460478, stop := 460516 ), rec( content := "\n \ ", count := [ "A", 2, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 23 ], name := "Item", next := 460533, root := ~, start := 460461, stop := 460532 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( attributes :=\ rec( ), content := [ rec( content := "v", count := [ "A", 2, 0, 25 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], mathmode := \ "Text", name := "A", next := 460570, root := ~, start := 460562, stop := 46056\ 9 ), rec( content := " in DigraphVertices(", count := [ "A", 2, 0, 25 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( cont\ ent := "digraph", count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "A", next := 4606\ 04, root := ~, start := 460590, stop := 460603 ), rec( content := ")", count := [ "A", 2, 0, 25 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "C", next := 460609, root := ~, start := 460559, stop := 460608 ), rec( content := "\n \ ", count := [ "A", 2, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 25 ], name := "Item", next := 460625, root := ~, start := 460542, stop := 460624 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphVertices" ), content := 0 , count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "Ref", next := 460724, root := ~, start := 460695, stop := 4607\ 23 ), rec( HTML := true, content := " can be used to return\n the l\ ist of vertices of a digraph, or if ", count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "v", count := [ "A", 2, 0, 27 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "A", next := 4608\ 08, root := ~, start := 460800, stop := 460807 ), rec( HTML := true, content := "\ is known to be\n a positive integer, then you can simply do\n \ ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( attributes := rec( ), content := [ rec( HTML := \ true, content := "v", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], mathmode := "Text", name \ := "A", next := 460902, root := ~, start := 460894, stop := 460901 ), rec( HTML\ := true, content := " <= DigraphNrVertices(", count := [ "A", 2, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), cont\ ent := [ rec( HTML := true, content := "digraph", count := [ "A", 2, 0, 27 ], \ mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ] , math\ mode := "Text", name := "A", next := 460941, root := ~, start := 460927, stop \ := 460940 ), rec( HTML := true, content := ")", count := [ "A", 2, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "C", next := 460946, root := ~, start := 460891, stop := 46094\ 5 ), rec( HTML := true, content := "\n ", count := [ "A", 2, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], name := "Alt", next := 460963, root := ~, start := 460651, stop := 460962 ), rec( content := "\n \ ", count := [ "A", 2, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 27 ], name := "Item", next := 460979, root := ~, start := 460634, stop := 460978 ) ], count := [ "A", 2, 0, 22 ], name := "Row", next := 460992, root := ~, start := 460447, stop := 460991 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 28 ], name := "HorLine", next := 461009, root := ~, start := 460999, stop := 461008 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexName", count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "C", next := 461065, root := ~, start := 461048, stop := 461064 ), rec( content := "\n \ ", count := [ "A", 2, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 29 ], name := "Item", next := 461081, root := ~, start := 461031, stop := 461080 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabel" ), content := 0, count := [ "A", 2, 0, 31 ], mathmode := "Text", name := "Ref", next := 461139, root := ~, start := 461107, stop := 461138 ), rec( content := "\n \ ", count := [ "A", 2, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 31 ], name := "Item", next := 461155, root := ~, start := 461090, stop := 461154 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Unlike ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "VertexName", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "\ PCDATA", root := ~ ) ], count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "C", next := 4612\ 35, root := ~, start := 461218, stop := 461234 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 33 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 33 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 33 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , the return value of\n ", count := [ "A", 2, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digrap\ hVertexLabel" ), content := 0, count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "Ref\ ", next := 461312, root := ~, start := 461280, stop := 461311 ), rec( HTML := true , content := "\ is mutable.\n ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 33 ], name := "Alt", next := 461341, root := ~, start := 461181, stop := 461340 ), rec( content := "\n \ ", count := [ "A", 2, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 33 ], name := "Item", next := 461357, root := ~, start := 461164, stop := 461356 ) ], count := [ "A", 2, 0, 28 ], name := "Row", next := 461370, root := ~, start := 461017, stop := 461369 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 34 ], name := "HorLine", next := 461387, root := ~, start := 461377, stop := 461386 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexNames", count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "C", next := 461444, root := ~, start := 461426, stop := 461443 ), rec( content := "\n \ ", count := [ "A", 2, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 35 ], name := "Item", next := 461460, root := ~, start := 461409, stop := 461459 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ VertexLabels" ), content := 0, count := [ "A", 2, 0, 37 ], mathmode := "Text", name := "Ref", next := 461519, root := ~, start := 461486, stop := 461518 ), rec( content := "\n \ ", count := [ "A", 2, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 37 ], name := "Item", next := 461535, root := ~, start := 461469, stop := 461534 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Unlike ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "VertexNames", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "C", next := 4616\ 16, root := ~, start := 461598, stop := 461615 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 39 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 39 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 39 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ , the return value of\n ", count := [ "A", 2, 0, 39 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digrap\ hVertexLabels" ), content := 0, count := [ "A", 2, 0, 39 ], mathmode := "Text" , name := "Ref\ ", next := 461694, root := ~, start := 461661, stop := 461693 ), rec( HTML := true , content := "\ is mutable.\n ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 39 ], name := "Alt", next := 461723, root := ~, start := 461561, stop := 461722 ), rec( content := "\n \ ", count := [ "A", 2, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 39 ], name := "Item", next := 461739, root := ~, start := 461544, stop := 461738 ) ], count := [ "A", 2, 0, 34 ], name := "Row", next := 461752, root := ~, start := 461395, stop := 461751 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 40 ], name := "HorLine", next := 461769, root := ~, start := 461759, stop := 461768 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertices", count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "C", next := 461823, root := ~, start := 461808, stop := 461822 ), rec( content := "\n \ ", count := [ "A", 2, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 41 ], name := "Item", next := 461839, root := ~, start := 461791, stop := 461838 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Vertices" ), content := 0, count := [ "A", 2, 0, 43 ], mathmode := "Text", name := "Ref", next := 461894, root := ~, start := 461865, stop := 461893 ), rec( content := "\n \ ", count := [ "A", 2, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 43 ], name := "Item", next := 461910, root := ~, start := 461848, stop := 461909 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The vertices of a digraph in ", count := [ "A", 2, 0, 45 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 2, 0, 45 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ is always the range ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "[1\n .. n]", count := [ "A", 2, 0, 45 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "\ Text", name := "C", next := 462055, root := ~, start := 462026, stop := 462054\ ), rec( HTML := true, content := " where ", count := [ "A", 2, 0, 45 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "n", count := [ "A", 2, 0, 45 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "C", next := 4620\ 70, root := ~, start := 462062, stop := 462069 ), rec( HTML := true, content := "\ is the number of vertices of the\n digraph. The operation ", count := [ "A", 2, 0, 45 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digrap\ hVertices" ), content := 0, count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "Ref\ ", next := 462169, root := ~, start := 462140, stop := 462168 ), rec( HTML := true , content := "\ behaves the\n same as ", count := [ "A", 2, 0, 45 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Vertices", count := [ "A", 2, 0, 45 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "T\ ext", name := "C", next := 462217, root := ~, start := 462202, stop := 462216 ), rec( HTML := true , content := "\ in ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 45 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 45 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 45 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 45 ], name := "Alt", next := 462246, root := ~, start := 461936, stop := 462245 ), rec( content := "\n \ ", count := [ "A", 2, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 45 ], name := "Item", next := 462262, root := ~, start := 461919, stop := 462261 ) ], count := [ "A", 2, 0, 40 ], name := "Row", next := 462275, root := ~, start := 461777, stop := 462274 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 46 ], name := "HorLine", next := 462292, root := ~, start := 462282, stop := 462291 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexDegree", count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "C", next := 462350, root := ~, start := 462331, stop := 462349 ), rec( content := "\n \ ", count := [ "A", 2, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 47 ], name := "Item", next := 462366, root := ~, start := 462314, stop := 462365 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutDegr\ eeOfVertex" ), content := 0, count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "Ref", next := 462423, root := ~, start := 462392, stop := 462422 ), rec( content := "\n \ ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n (see also: ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "InDegreeOfVertex" ), content := 0 , count := [ "A", 2, 0, 49 ], mathmode := \ "Text", name := "Ref", next := 462505, root := ~, start := 462475, stop := 4625\ 04 ), rec( HTML := true, content := ")\n ", count := [ "A", 2, 0, 49 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 49 ], name := "Alt", next := 462523, root := ~, start := 462434, stop := 462522 ), rec( content := "\n \ ", count := [ "A", 2, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 49 ], name := "Item", next := 462539, root := ~, start := 462375, stop := 462538 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "OutDegreeOfVertex" ), content := 0 , count := [ "A", 2, 0, 51 ], mathmode := \ "Text", name := "Ref", next := 462640, root := ~, start := 462609, stop := 4626\ 39 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "VertexDegree", count := [ "A", 2, 0, 51 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 51 ], mathmode := \ "Text", name := "C", next := 462692, root := ~, start := 462673, stop := 46269\ 1 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 51 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 51 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 51 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 51 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 51 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 51 ], name := "Alt", next := 462721, root := ~, start := 462565, stop := 462720 ), rec( content := "\n \ ", count := [ "A", 2, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 51 ], name := "Item", next := 462737, root := ~, start := 462548, stop := 462736 ) ], count := [ "A", 2, 0, 46 ], name := "Row", next := 462750, root := ~, start := 462300, stop := 462749 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 52 ], name := "HorLine", next := 462767, root := ~, start := 462757, stop := 462766 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexDegrees", count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "C", next := 462826, root := ~, start := 462806, stop := 462825 ), rec( content := "\n \ ", count := [ "A", 2, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 53 ], name := "Item", next := 462842, root := ~, start := 462789, stop := 462841 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutDegr\ eeSet" ), content := 0, count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "Ref", next := 462894, root := ~, start := 462868, stop := 462893 ), rec( content := "\n \ ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n (see also:\n ", count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "OutDeg\ reeSequence" ), content := 0, count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "Ref\ ", next := 462989, root := ~, start := 462958, stop := 462988 ), rec( HTML := true , content := "\ ,\n ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "OutDegrees" ), content := 0 , count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "Ref", next := 463027, root := ~, start := 463003, stop := 4630\ 26 ), rec( HTML := true, content := ",\n ", count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "InDegr\ ees" ), content := 0, count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "Ref\ ", next := 463064, root := ~, start := 463041, stop := 463063 ), rec( HTML := true , content := "\ ,\n ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "InDegreeSequence" ), content := 0 , count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "Ref", next := 463108, root := ~, start := 463078, stop := 4631\ 07 ), rec( HTML := true, content := ",\n ", count := [ "A", 2, 0, 55 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "InDegr\ eeSet" ), content := 0, count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "Ref\ ", next := 463147, root := ~, start := 463122, stop := 463146 ), rec( HTML := true , content := "\ )\n ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 55 ], name := "Alt", next := 463165, root := ~, start := 462905, stop := 463164 ), rec( content := "\n \ ", count := [ "A", 2, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 55 ], name := "Item", next := 463181, root := ~, start := 462851, stop := 463180 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "OutDegreeSet" ), content := 0 , count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "Ref", next := 463277, root := ~, start := 463251, stop := 4632\ 76 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "VertexDegrees", count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "C", next := 463330, root := ~, start := 463310, stop := 46332\ 9 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 57 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 57 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 57 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 57 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 57 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 57 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 57 ], name := "Alt", next := 463359, root := ~, start := 463207, stop := 463358 ), rec( content := "\n \ ", count := [ "A", 2, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 57 ], name := "Item", next := 463375, root := ~, start := 463190, stop := 463374 ) ], count := [ "A", 2, 0, 52 ], name := "Row", next := 463388, root := ~, start := 462775, stop := 463387 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 58 ], name := "HorLine", next := 463405, root := ~, start := 463395, stop := 463404 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sLoopy", count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "C", next := 463458, root := ~, start := 463444, stop := 463457 ), rec( content := "\n \ ", count := [ "A", 2, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 59 ], name := "Item", next := 463474, root := ~, start := 463427, stop := 463473 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digraph\ HasLoops" ), content := 0, count := [ "A", 2, 0, 61 ], mathmode := "Text", name := "Ref", next := 463529, root := ~, start := 463500, stop := 463528 ), rec( content := "\n \ ", count := [ "A", 2, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 61 ], name := "Item", next := 463545, root := ~, start := 463483, stop := 463544 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "DigraphHasLoops" ), content := 0 , count := [ "A", 2, 0, 63 ], mathmode := \ "Text", name := "Ref", next := 463644, root := ~, start := 463615, stop := 4636\ 43 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 63 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "IsLoopy", count := [ "A", 2, 0, 63 ], mathmode := "Text" , name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "C", next := 4636\ 91, root := ~, start := 463677, stop := 463690 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 63 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 63 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 63 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 63 ], name := "Alt", next := 463720, root := ~, start := 463571, stop := 463719 ), rec( content := "\n \ ", count := [ "A", 2, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 63 ], name := "Item", next := 463736, root := ~, start := 463554, stop := 463735 ) ], count := [ "A", 2, 0, 58 ], name := "Row", next := 463749, root := ~, start := 463413, stop := 463748 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 64 ], name := "HorLine", next := 463766, root := ~, start := 463756, stop := 463765 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sSimpleGraph", count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "C", next := 463825, root := ~, start := 463805, stop := 463824 ), rec( content := "\n \ ", count := [ "A", 2, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 65 ], name := "Item", next := 463841, root := ~, start := 463788, stop := 463840 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Prop := "IsSymmetricDigraph" ), content := 0 , count := [ "A", 2, 0, 67 ], mathmode := \ "Text", name := "Ref", next := 463929, root := ~, start := 463897, stop := 4639\ 28 ), rec( HTML := true, content := "\n and not ", count := [ "A", 2, 0, 67 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "Digrap\ hHasLoops" ), content := 0, count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "Ref\ ", next := 463980, root := ~, start := 463951, stop := 463979 ), rec( HTML := true , content := "\ \n ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 67 ], name := "Alt", next := 463997, root := ~, start := 463867, stop := 463996 ), rec( content := "\n \ ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\\ n ", count := [ "A", 2, 0, 67 ], name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Prop := "IsSymmetricDigraph" ), content := 0, count := [ "A", 2, 0, 67 ], name := "Ref\ ", next := 464069, root := ~, start := 464037, stop := 464068 ), rec( content := "\ \n ", count := [ "A", 2, 0, 67 ], name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "Alt", next := 464086, root := ~, start := 464008, stop := 464085 ), rec( content := "\n \ ", count := [ "A", 2, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 67 ], name := "Item", next := 464102, root := ~, start := 463850, stop := 464101 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 69 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 69 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 69 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 69 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsSimpleGraph", count :\ = [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "C", next := 464199, root := ~, start := 464179, stop := 46419\ 8 ), rec( HTML := true, content := " is equivalent to\n ", count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsSymm\ etricDigraph" ), content := 0, count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "Ref\ ", next := 464261, root := ~, start := 464229, stop := 464260 ), rec( HTML := true , content := "\ \n and not ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "DigraphHasLoops" ), content := 0 , count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "Ref", next := 464312, root := ~, start := 464283, stop := 4643\ 11 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 69 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], name := "Alt", next := 464330, root := ~, start := 464128, stop := 464329 ), rec( content := "\n \ ", count := [ "A", 2, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 69 ], name := "Item", next := 464346, root := ~, start := 464111, stop := 464345 ) ], count := [ "A", 2, 0, 64 ], name := "Row", next := 464359, root := ~, start := 463774, stop := 464358 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 70 ], name := "HorLine", next := 464376, root := ~, start := 464366, stop := 464375 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ djacency", count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "C", next := 464431, root := ~, start := 464415, stop := 464430 ), rec( content := "\n \ ", count := [ "A", 2, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 71 ], name := "Item", next := 464447, root := ~, start := 464398, stop := 464446 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "OutNeig\ hboursOfVertex" ), content := 0, count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "Ref", next := 464508, root := ~, start := 464473, stop := 464507 ), rec( content := "\n \ ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n (see also ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "InNeighboursOfVertex" ), content := 0 , count := [ "A", 2, 0, 73 ], mathmode := \ "Text", name := "Ref", next := 464593, root := ~, start := 464559, stop := 4645\ 92 ), rec( HTML := true, content := ")\n ", count := [ "A", 2, 0, 73 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 73 ], name := "Alt", next := 464611, root := ~, start := 464519, stop := 464610 ), rec( content := "\n \ ", count := [ "A", 2, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 73 ], name := "Item", next := 464627, root := ~, start := 464456, stop := 464626 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "OutNeighboursOfVertex" ), content := 0 , count := [ "A", 2, 0, 75 ], mathmode := \ "Text", name := "Ref", next := 464732, root := ~, start := 464697, stop := 4647\ 31 ), rec( HTML := true, content := " behaves the same\n as ", count := [ "A", 2, 0, 75 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Adjacency", count := [ "A", 2, 0, 75 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 75 ], mathmode := "\ Text", name := "C", next := 464781, root := ~, start := 464765, stop := 464780\ ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 75 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 75 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 75 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 75 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 75 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 75 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 75 ], name := "Alt", next := 464810, root := ~, start := 464653, stop := 464809 ), rec( content := "\n \ ", count := [ "A", 2, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 75 ], name := "Item", next := 464826, root := ~, start := 464636, stop := 464825 ) ], count := [ "A", 2, 0, 70 ], name := "Row", next := 464839, root := ~, start := 464384, stop := 464838 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 76 ], name := "HorLine", next := 464856, root := ~, start := 464846, stop := 464855 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sEdge", count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "C", next := 464908, root := ~, start := 464895, stop := 464907 ), rec( content := "\n \ ", count := [ "A", 2, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 77 ], name := "Item", next := 464924, root := ~, start := 464878, stop := 464923 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 79 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graph and list", Oper := "IsDigraphEdge" ), content := 0, count := [ "A", 2, 0, 79 ], mathmode := "Text", name := "Ref", next := 465006, root := ~, start := 464950, stop := 465005 ), rec( content := "\n \ ", count := [ "A", 2, 0, 79 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 79 ], name := "Item", next := 465022, root := ~, start := 464933, stop := 465021 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The arguments of\n ", count := [ "A", 2, 0, 81 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for d\ igraph and list", Oper := "IsDigraphEdge" ), content := 0, count := [ "A", 2, 0, 81 ], mathmode := \ "Text", name := "Ref", next := 465163, root := ~, start := 465107, stop := 4651\ 62 ), rec( HTML := true, content := "\n can be a digraph and an edg\ e, or a digraph and two vertices. In the\n first form the operation\ \n ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Label := "for digraph and list", Oper := \ "IsDigraphEdge" ), content := 0, count := [ "A", 2, 0, 81 ], mathmode := "Text\ ", name := "Ref", next := 465349, root := ~, start := 465293, stop := 465348 ) , rec( HTML := true , content := "\ \n behaves the same as ", count := [ "A", 2, 0, 81 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "IsEdge", count := [ "A", 2, 0, 81 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "C", next := 4653\ 95, root := ~, start := 465382, stop := 465394 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 81 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 81 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 81 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 81 ], name := "Alt", next := 465424, root := ~, start := 465048, stop := 465423 ), rec( content := "\n \ ", count := [ "A", 2, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 81 ], name := "Item", next := 465440, root := ~, start := 465031, stop := 465439 ) ], count := [ "A", 2, 0, 76 ], name := "Row", next := 465453, root := ~, start := 464864, stop := 465452 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 82 ], name := "HorLine", next := 465470, root := ~, start := 465460, stop := 465469 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ irectedEdges", count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "C", next := 465529, root := ~, start := 465509, stop := 465528 ), rec( content := "\n \ ", count := [ "A", 2, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 83 ], name := "Item", next := 465545, root := ~, start := 465492, stop := 465544 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 85 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Edges" ), content := 0, count := [ "A", 2, 0, 85 ], mathmode := "Text", name := "Ref", next := 465597, root := ~, start := 465571, stop := 465596 ), rec( content := "\n \ ", count := [ "A", 2, 0, 85 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 85 ], name := "Item", next := 465613, root := ~, start := 465554, stop := 465612 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphEdges" ), content := 0 , count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "Ref", next := 465709, root := ~, start := 465683, stop := 4657\ 08 ), rec( HTML := true, content := " behaves the same\n as ", count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "DirectedEdges", count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "C", next := 465762, root := ~, start := 465742, stop := 46576\ 1 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 87 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 87 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 87 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 87 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 87 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 87 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 87 ], name := "Alt", next := 465791, root := ~, start := 465639, stop := 465790 ), rec( content := "\n \ ", count := [ "A", 2, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 87 ], name := "Item", next := 465807, root := ~, start := 465622, stop := 465806 ) ], count := [ "A", 2, 0, 82 ], name := "Row", next := 465820, root := ~, start := 465478, stop := 465819 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 88 ], name := "HorLine", next := 465837, root := ~, start := 465827, stop := 465836 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ ndirectedEdges", count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "C", next := 465898, root := ~, start := 465876, stop := 465897 ), rec( content := "\n \ ", count := [ "A", 2, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 89 ], name := "Item", next := 465914, root := ~, start := 465859, stop := 465913 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 2, 0, 91 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 91 ], name := "Item", next := 465960, root := ~, start := 465923, stop := 465959 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 2, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 2, 0, 93 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 93 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 2, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 93 ], name := "Alt", next := 466119, root := ~, start := 465986, stop := 466118 ), rec( content := "\n \ ", count := [ "A", 2, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 93 ], name := "Item", next := 466135, root := ~, start := 465969, stop := 466134 ) ], count := [ "A", 2, 0, 88 ], name := "Row", next := 466148, root := ~, start := 465845, stop := 466147 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 94 ], name := "HorLine", next := 466165, root := ~, start := 466155, stop := 466164 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ istance", count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "C", next := 466219, root := ~, start := 466204, stop := 466218 ), rec( content := "\n \ ", count := [ "A", 2, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 95 ], name := "Item", next := 466235, root := ~, start := 466187, stop := 466234 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 97 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph and two vertices", Oper := "DigraphShortestDistance" ), content := 0, count := [ "A", 2, 0, 97 ], mathmode := "Text", name := "Ref", next := 466352, root := ~, start := 466261, stop := 466351 ), rec( content := "\n \ ", count := [ "A", 2, 0, 97 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 97 ], name := "Item", next := 466368, root := ~, start := 466244, stop := 466367 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The operation ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Label := "for a digraph and two ver\ tices", Oper := "DigraphShortestDistance" ), content := 0, count := [ "A", 2, 0, 99 ], mathmode := \ "Text", name := "Ref", next := 466526, root := ~, start := 466438, stop := 4665\ 25 ), rec( HTML := true, content := " behaves the same as\n ", count := [ "A", 2, 0, 99 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Distance", count := [ "A", 2, 0, 99 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 99 ], mathmode := "T\ ext", name := "C", next := 466574, root := ~, start := 466559, stop := 466573 ), rec( HTML := true , content := "\ in ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 99 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 99 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 2, 0, 99 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 2, 0, 99 ], name := "Alt", next := 466603, root := ~, start := 466394, stop := 466602 ), rec( content := "\n \ ", count := [ "A", 2, 0, 99 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 99 ], name := "Item", next := 466619, root := ~, start := 466377, stop := 466618 ) ], count := [ "A", 2, 0, 94 ], name := "Row", next := 466632, root := ~, start := 466173, stop := 466631 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 100 ], name := "HorLine", next := 466649, root := ~, start := 466639, stop := 466648 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 101 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ iameter", count := [ "A", 2, 0, 101 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 101 ] , mathmode := "Text", name := "C", next := 466703, root := ~, start := 466688, stop := 466702 ), rec( content := "\n \ ", count := [ "A", 2, 0, 101 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 101 ], name := "Item", next := 466719, root := ~, start := 466671, stop := 466718 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 103 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Diameter" ), content := 0, count := [ "A", 2, 0, 103 ], mathmode := "Text", name := "Ref", next := 466774, root := ~, start := 466745, stop := 466773 ), rec( content := "\n \ ", count := [ "A", 2, 0, 103 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 103 ], name := "Item", next := 466790, root := ~, start := 466728, stop := 466789 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 105 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 105 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphDiameter" ), content := 0 , count := [ "A", 2, 0, 105 ] , mathmode := \ "Text", name := "Ref", next := 466889, root := ~, start := 466860, stop := 4668\ 88 ), rec( HTML := true, content := " behaves the same\n as ", count := [ "A", 2, 0, 105 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Diameter", count := [ "A", 2, 0, 105 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 105 ], mathmode := \ "Text", name := "C", next := 466937, root := ~, start := 466922, stop := 46693\ 6 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 105 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 105 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 105 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 105 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 105 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 105 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 105 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 2, 0, 105 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 2, 0, 105 ], name := "Alt", next := 466966, root := ~, start := 466816, stop := 466965 ), rec( content := "\n \ ", count := [ "A", 2, 0, 105 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 105 ], name := "Item", next := 466982, root := ~, start := 466799, stop := 466981 ) ], count := [ "A", 2, 0, 100 ], name := "Row", next := 466995, root := ~, start := 466657, stop := 466994 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 106 ], name := "HorLine", next := 467012, root := ~, start := 467002, stop := 467011 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 107 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ irth", count := [ "A", 2, 0, 107 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 107 ] , mathmode := "Text", name := "C", next := 467063, root := ~, start := 467051, stop := 467062 ), rec( content := "\n \ ", count := [ "A", 2, 0, 107 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 107 ], name := "Item", next := 467079, root := ~, start := 467034, stop := 467078 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 109 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ UndirectedGirth" ), content := 0, count := [ "A", 2, 0, 109 ], mathmode := "Text", name := "Ref", next := 467141, root := ~, start := 467105, stop := 467140 ), rec( content := "\n \ ", count := [ "A", 2, 0, 109 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 109 ], name := "Item", next := 467157, root := ~, start := 467088, stop := 467156 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 111 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphUndirectedGirth" ), content := 0 , count := [ "A", 2, 0, 111 ] , mathmode := \ "Text", name := "Ref", next := 467263, root := ~, start := 467227, stop := 4672\ 62 ), rec( HTML := true, content := " behaves the same\n as ", count := [ "A", 2, 0, 111 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Girth", count := [ "A", 2, 0, 111 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "C", next := 4673\ 08, root := ~, start := 467296, stop := 467307 ), rec( HTML := true, content := "\ in ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", roo\ t := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 111 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 111 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 111 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 111 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ except that in the case that the\n argument is a forest, ", count := [ "A", 2, 0, 111 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "infinity", count := [ "A", 2, 0, 111 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := \ "Text", name := "K", next := 467402, root := ~, start := 467387, stop := 46740\ 1 ), rec( HTML := true, content := " is returned by ", count := [ "A", 2, 0, 111 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 2, 0, 111 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ \n whereas ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "-1", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA" , root\ := ~ ) ], count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "C", next := 4674\ 58, root := ~, start := 467449, stop := 467457 ), rec( HTML := true, content := "\ is returned by ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 111 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 111 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 111 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 111 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 111 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 2, 0, 111 ], name := "Alt", next := 467499, root := ~, start := 467183, stop := 467498 ), rec( content := "\n \ ", count := [ "A", 2, 0, 111 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 111 ], name := "Item", next := 467515, root := ~, start := 467166, stop := 467514 ) ], count := [ "A", 2, 0, 106 ], name := "Row", next := 467528, root := ~, start := 467020, stop := 467527 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 112 ], name := "HorLine", next := 467545, root := ~, start := 467535, stop := 467544 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sConnectedGraph", count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "C", next := 467607, root := ~, start := 467584, stop := 467606 ), rec( content := "\n \ ", count := [ "A", 2, 0, 113 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 113 ], name := "Item", next := 467623, root := ~, start := 467567, stop := 467622 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 115 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsStron\ glyConnectedDigraph" ), content := 0, count := [ "A", 2, 0, 115 ], mathmode := "Text", name := "Ref", next := 467689, root := ~, start := 467649, stop := 467688 ), rec( content := "\n \ ", count := [ "A", 2, 0, 115 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 115 ], name := "Item", next := 467705, root := ~, start := 467632, stop := 467704 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The attribute ", count := [ "A", 2, 0, 117 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "IsStronglyConnectedDigraph\ " ), content := 0, count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "Ref\ ", next := 467815, root := ~, start := 467775, stop := 467814 ), rec( HTML := true , content := "\ behaves the\n same as ", count := [ "A", 2, 0, 117 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "IsConnectedGraph", count := [ "A", 2, 0, 117 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 117 ], mathmode := \ "Text", name := "C", next := 467871, root := ~, start := 467848, stop := 46787\ 0 ), rec( HTML := true, content := " in ", count := [ "A", 2, 0, 117 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 117 ], mat\ hmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 117 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 117 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 117 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 117 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 117 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 2, 0, 117 ], name := "Alt", next := 467900, root := ~, start := 467731, stop := 467899 ), rec( content := "\n \ ", count := [ "A", 2, 0, 117 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 117 ], name := "Item", next := 467916, root := ~, start := 467714, stop := 467915 ) ], count := [ "A", 2, 0, 112 ], name := "Row", next := 467929, root := ~, start := 467553, stop := 467928 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 118 ], name := "HorLine", next := 467946, root := ~, start := 467936, stop := 467945 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 119 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sBipartite", count := [ "A", 2, 0, 119 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 2, 0, 119 ] , mathmode := "Text", name := "C", next := 468003, root := ~, start := 467985, stop := 468002 ), rec( content := "\n \ ", count := [ "A", 2, 0, 119 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 119 ], name := "Item", next := 468019, root := ~, start := 467968, stop := 468018 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipar\ titeDigraph" ), content := 0, count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "Ref", next := 468077, root := ~, start := 468045, stop := 468076 ), rec( content := "\n \ ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n and ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "IsSymmetricDigraph" ), content := 0 , count := [ "A", 2, 0, 121 ] , mathmode := \ "Text", name := "Ref", next := 468154, root := ~, start := 468122, stop := 4681\ 53 ), rec( HTML := true, content := "\n ", count := [ "A", 2, 0, 121 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 121 ], name := "Alt", next := 468171, root := ~, start := 468088, stop := 468170 ), rec( content := "\n \ ", count := [ "A", 2, 0, 121 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 121 ], name := "Item", next := 468187, root := ~, start := 468028, stop := 468186 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 123 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 123 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 123 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 123 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 123 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ function ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsBipartite", count := [ \ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ] , mathmode := \ "Text", name := "C", next := 468282, root := ~, start := 468264, stop := 46828\ 1 ), rec( HTML := true, content := " is equivalent to\n ", count := [ "A", 2, 0, 123 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipa\ rtiteDigraph" ), content := 0, count := [ "A", 2, 0, 123 ], mathmode := "Text" , name := "Ref\ ", next := 468344, root := ~, start := 468312, stop := 468343 ), rec( HTML := true , content := "\ and ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Prop := "IsSymmetricDigraph" ), content := 0, count := [ "A", 2, 0, 123 ], mathmode := \ "Text", name := "Ref", next := 468395, root := ~, start := 468349, stop := 4683\ 94 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 123 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ], name := "Alt", next := 468413, root := ~, start := 468213, stop := 468412 ), rec( content := "\n \ ", count := [ "A", 2, 0, 123 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 123 ], name := "Item", next := 468429, root := ~, start := 468196, stop := 468428 ) ], count := [ "A", 2, 0, 118 ], name := "Row", next := 468442, root := ~, start := 467954, stop := 468441 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 124 ], name := "HorLine", next := 468459, root := ~, start := 468449, stop := 468458 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 125 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sNullGraph", count := [ "A", 2, 0, 125 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 2, 0, 125 ] , mathmode := "Text", name := "C", next := 468516, root := ~, start := 468498, stop := 468515 ), rec( content := "\n \ ", count := [ "A", 2, 0, 125 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 125 ], name := "Item", next := 468532, root := ~, start := 468481, stop := 468531 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsNullD\ igraph" ), content := 0, count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "Ref", next := 468585, root := ~, start := 468558, stop := 468584 ), rec( content := "\n \ ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n or ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "IsEmptyDigraph" ), content := 0 , count := [ "A", 2, 0, 127 ] , mathmode := \ "Text", name := "Ref", next := 468657, root := ~, start := 468629, stop := 4686\ 56 ), rec( HTML := true, content := "\n ", count := [ "A", 2, 0, 127 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 127 ], name := "Alt", next := 468674, root := ~, start := 468596, stop := 468673 ), rec( content := "\n \ ", count := [ "A", 2, 0, 127 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 127 ], name := "Item", next := 468690, root := ~, start := 468541, stop := 468689 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 129 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 129 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 129 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 129 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 129 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ function ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsNullGraph", count := [ \ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ] , mathmode := \ "Text", name := "C", next := 468785, root := ~, start := 468767, stop := 46878\ 4 ), rec( HTML := true, content := " is equivalent to\n ", count := [ "A", 2, 0, 129 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsEmpt\ yDigraph" ), content := 0, count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "Ref\ ", next := 468843, root := ~, start := 468815, stop := 468842 ), rec( HTML := true , content := "\ and its synonym ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Prop := "IsNullDigraph" ), content := 0 , count := [ "A", 2, 0, 129 ] , mathmode := \ "Text", name := "Ref", next := 468901, root := ~, start := 468860, stop := 4689\ 00 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 129 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ], name := "Alt", next := 468919, root := ~, start := 468716, stop := 468918 ), rec( content := "\n \ ", count := [ "A", 2, 0, 129 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 129 ], name := "Item", next := 468935, root := ~, start := 468699, stop := 468934 ) ], count := [ "A", 2, 0, 124 ], name := "Row", next := 468948, root := ~, start := 468467, stop := 468947 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 130 ], name := "HorLine", next := 468965, root := ~, start := 468955, stop := 468964 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sCompleteGraph", count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "C", next := 469026, root := ~, start := 469004, stop := 469025 ), rec( content := "\n \ ", count := [ "A", 2, 0, 131 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 131 ], name := "Item", next := 469042, root := ~, start := 468987, stop := 469041 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 133 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsCompl\ eteDigraph" ), content := 0, count := [ "A", 2, 0, 133 ], mathmode := "Text", name := "Ref", next := 469099, root := ~, start := 469068, stop := 469098 ), rec( content := "\n \ ", count := [ "A", 2, 0, 133 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 133 ], name := "Item", next := 469115, root := ~, start := 469051, stop := 469114 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 2, 0, 135 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 2, 0, 135 ] , name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 2, 0, 135 ], name := "PCDATA", root :\ = ~ ) ], count := [ "A", 2, 0, 135 ], name := "Package", next := 137, root := \ ~, start := 113, stop := 136 ) ], count := [ "A", 2, 0, 135 ], mathmode := "Te\ xt", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true , content := "\ function ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsCompleteGraph", count\ := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ] , mathmode := \ "Text", name := "C", next := 469214, root := ~, start := 469192, stop := 46921\ 3 ), rec( HTML := true, content := " is roughly\n equivalent to\n \ ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA" , root := ~ ), rec( attributes :\ = rec( Prop := "IsCompleteDigraph" ), content := 0, count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "Ref", next := 469295, root := ~, start := 469264, stop := 4692\ 94 ), rec( HTML := true, content := " except that in ", count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 2, 0, 135 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ \n a complete digraph has ", count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "no loops", count := [ "A", 2, 0, 135 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "E", next := 469372, root := ~, start := 469357, stop := 46937\ 1 ), rec( HTML := true, content := ".\n ", count := [ "A", 2, 0, 135 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], name := "Alt", next := 469390, root := ~, start := 469141, stop := 469389 ), rec( content := "\n \ ", count := [ "A", 2, 0, 135 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 135 ], name := "Item", next := 469406, root := ~, start := 469124, stop := 469405 ) ], count := [ "A", 2, 0, 130 ], name := "Row", next := 469419, root := ~, start := 468973, stop := 469418 ), rec( attributes := rec( ), content := 0, count := [ "A", 2, 0, 136 ], name := "HorLine", next := 469436, root := ~, start := 469426, stop := 469435 ) ], count := [ "A", 2, 0, 4 ], mathmode := "Text", name := "Table", next := 469449, root := ~, start := 459429, stop := 469448 ), rec( content := "\n ", count := [ "A", 2, 0, 137 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 2, 0, 1 ], mathmode := "Text", name := "Section", next := 469462, root := ~, start := 459182, stop := 469461 ), rec( content := "\n\n ", count := [ "A", 2, 0, 137 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 3, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Functions to dete\ rmine regularity properties of graphs\n ", count := [ "A", 3, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 2 ], mathmode := "Text", name := "Heading", next := 469565, root := ~, start := 469480, stop := 469564 ), rec( content := "\n ", count := [ "A", 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 3, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 3, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 3 ], name := "E", next := 469703, root := ~, start := 469593, stop := 469702 ), rec( content := "\n ", count := [ "A", 3, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 3 ], mathmode := "Text", name := "Alt", next := 469714, root := ~, start := 469570, stop := 469713 ), rec( content := "\n\n ", count := [ "A", 3, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 4 ], name := "HorLine", next := 469760, root := ~, start := 469750, stop := 469759 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 3, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 3, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 3, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 3, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 5 ], name := "Item", next := 469830, root := ~, start := 469782, stop := 469829 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 3, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 3, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 7 ], name := "Item", next := 469890, root := ~, start := 469839, stop := 469889 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 3, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 9 ], name := "Alt", next := 469971, root := ~, start := 469916, stop := 469970 ), rec( content := "\n \ ", count := [ "A", 3, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 9 ], name := "Item", next := 469987, root := ~, start := 469899, stop := 469986 ) ], count := [ "A", 3, 0, 4 ], name := "Row", next := 470000, root := ~, start := 469768, stop := 469999 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 10 ], name := "HorLine", next := 470017, root := ~, start := 470007, stop := 470016 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sRegularGraph", count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "C", next := 470077, root := ~, start := 470056, stop := 470076 ), rec( content := "\n \ ", count := [ "A", 3, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 11 ], name := "Item", next := 470093, root := ~, start := 470039, stop := 470092 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOutRe\ gularDigraph" ), content := 0, count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "Ref", next := 470152, root := ~, start := 470119, stop := 470151 ), rec( content := "\n \ ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n (see also\n ", count := [ "A", 3, 0, 13 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsInRe\ gularDigraph" ), content := 0, count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "Ref\ ", next := 470247, root := ~, start := 470215, stop := 470246 ), rec( HTML := true , content := "\ \n and\n ", count := [ "A", 3, 0, 13 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsRegularD\ igraph" ), content := 0, count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "Ref\ ", next := 470306, root := ~, start := 470276, stop := 470305 ), rec( HTML := true , content := "\ )\n ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 3, 0, 13 ], name := "Alt", next := 470324, root := ~, start := 470163, stop := 470323 ), rec( content := "\n \ ", count := [ "A", 3, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 13 ], name := "Item", next := 470340, root := ~, start := 470102, stop := 470339 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 15 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 15 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsRegularGraph", count \ := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 15 ], mathmode := \ "Text", name := "C", next := 470438, root := ~, start := 470417, stop := 47043\ 7 ), rec( HTML := true, content := " is\n equivalent to ", count := [ "A", 3, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsOutR\ egularDigraph" ), content := 0, count := [ "A", 3, 0, 15 ], mathmode := "Text" , name := "Ref\ ", next := 470501, root := ~, start := 470468, stop := 470500 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 3, 0, 15 ], name := "Alt", next := 470519, root := ~, start := 470366, stop := 470518 ), rec( content := "\n \ ", count := [ "A", 3, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 15 ], name := "Item", next := 470535, root := ~, start := 470349, stop := 470534 ) ], count := [ "A", 3, 0, 10 ], name := "Row", next := 470548, root := ~, start := 470025, stop := 470547 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 16 ], name := "HorLine", next := 470565, root := ~, start := 470555, stop := 470564 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ocalParameters", count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "C", next := 470626, root := ~, start := 470604, stop := 470625 ), rec( content := "\n \ ", count := [ "A", 3, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 17 ], name := "Item", next := 470642, root := ~, start := 470587, stop := 470641 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 19 ], name := "Item", next := 470688, root := ~, start := 470651, stop := 470687 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 21 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "LocalParameters", count\ := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], mathmode := \ "Text", name := "C", next := 470786, root := ~, start := 470764, stop := 47078\ 5 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], name := "Alt", next := 470899, root := ~, start := 470713, stop := 470898 ), rec( content := "\n \ ", count := [ "A", 3, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 21 ], name := "Item", next := 470915, root := ~, start := 470697, stop := 470914 ) ], count := [ "A", 3, 0, 16 ], name := "Row", next := 470928, root := ~, start := 470573, stop := 470927 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 22 ], name := "HorLine", next := 470945, root := ~, start := 470935, stop := 470944 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ lobalParameters", count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "C", next := 471007, root := ~, start := 470984, stop := 471006 ), rec( content := "\n \ ", count := [ "A", 3, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 23 ], name := "Item", next := 471023, root := ~, start := 470967, stop := 471022 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 25 ], name := "Item", next := 471069, root := ~, start := 471032, stop := 471068 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 27 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "GlobalParameters", coun\ t := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], mathmode := \ "Text", name := "C", next := 471168, root := ~, start := 471145, stop := 47116\ 7 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], name := "Alt", next := 471281, root := ~, start := 471094, stop := 471280 ), rec( content := "\n \ ", count := [ "A", 3, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 27 ], name := "Item", next := 471297, root := ~, start := 471078, stop := 471296 ) ], count := [ "A", 3, 0, 22 ], name := "Row", next := 471310, root := ~, start := 470953, stop := 471309 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 28 ], name := "HorLine", next := 471327, root := ~, start := 471317, stop := 471326 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sDistanceRegular", count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "C", next := 471390, root := ~, start := 471366, stop := 471389 ), rec( content := "\n \ ", count := [ "A", 3, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 29 ], name := "Item", next := 471406, root := ~, start := 471349, stop := 471405 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsDista\ nceRegularDigraph" ), content := 0, count := [ "A", 3, 0, 31 ], mathmode := "Text", name := "Ref", next := 471470, root := ~, start := 471432, stop := 471469 ), rec( content := "\n \ ", count := [ "A", 3, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 31 ], name := "Item", next := 471486, root := ~, start := 471415, stop := 471485 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 33 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 33 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 33 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 33 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "IsDistanceRegular", coun\ t := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], mathmode := \ "Text", name := "C", next := 471587, root := ~, start := 471563, stop := 47158\ 6 ), rec( HTML := true, content := " is\n equivalent to ", count := [ "A", 3, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsDist\ anceRegularDigraph" ), content := 0, count := [ "A", 3, 0, 33 ], mathmode := "\ Text", name := "Ref", next := 471655, root := ~, start := 471617, stop := 4716\ 54 ), rec( HTML := true, content := ".\n ", count := [ "A", 3, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], name := "Alt", next := 471673, root := ~, start := 471512, stop := 471672 ), rec( content := "\n \ ", count := [ "A", 3, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 33 ], name := "Item", next := 471689, root := ~, start := 471495, stop := 471688 ) ], count := [ "A", 3, 0, 28 ], name := "Row", next := 471702, root := ~, start := 471335, stop := 471701 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 34 ], name := "HorLine", next := 471719, root := ~, start := 471709, stop := 471718 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ollapsedAdjacencyMat", count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "C", next := 471786, root := ~, start := 471758, stop := 471785 ), rec( content := "\n \ ", count := [ "A", 3, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 35 ], name := "Item", next := 471802, root := ~, start := 471741, stop := 471801 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 37 ], name := "Item", next := 471848, root := ~, start := 471811, stop := 471847 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 39 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 39 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 39 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 39 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "CollapsedAdjacencyMat", coun\ t := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], mathmode := \ "Text", name := "C", next := 471952, root := ~, start := 471924, stop := 47195\ 1 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 39 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], name := "Alt", next := 472065, root := ~, start := 471873, stop := 472064 ), rec( content := "\n \ ", count := [ "A", 3, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 39 ], name := "Item", next := 472081, root := ~, start := 471857, stop := 472080 ) ], count := [ "A", 3, 0, 34 ], name := "Row", next := 472094, root := ~, start := 471727, stop := 472093 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 40 ], name := "HorLine", next := 472111, root := ~, start := 472101, stop := 472110 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "O\ rbitalGraphColadjMats", count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "C", next := 472179, root := ~, start := 472150, stop := 472178 ), rec( content := "\n \ ", count := [ "A", 3, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 41 ], name := "Item", next := 472195, root := ~, start := 472133, stop := 472194 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 43 ], name := "Item", next := 472241, root := ~, start := 472204, stop := 472240 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 45 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 45 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 45 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 45 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "OrbitalGraphColadjMats" , coun\ t := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], mathmode := \ "Text", name := "C", next := 472346, root := ~, start := 472317, stop := 47234\ 5 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 45 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], name := "Alt", next := 472459, root := ~, start := 472266, stop := 472458 ), rec( content := "\n \ ", count := [ "A", 3, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 45 ], name := "Item", next := 472475, root := ~, start := 472250, stop := 472474 ) ], count := [ "A", 3, 0, 40 ], name := "Row", next := 472488, root := ~, start := 472119, stop := 472487 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 46 ], name := "HorLine", next := 472505, root := ~, start := 472495, stop := 472504 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexTransitiveDRGs", count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "C", next := 472571, root := ~, start := 472544, stop := 472570 ), rec( content := "\n \ ", count := [ "A", 3, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 47 ], name := "Item", next := 472587, root := ~, start := 472527, stop := 472586 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 3, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 49 ], name := "Item", next := 472633, root := ~, start := 472596, stop := 472632 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attri\ butes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "h\ ttp://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 3, 0, 51 ], mathm\ ode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 3, 0, 51 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 3, 0, 51 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 3, 0, 51 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ function ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "VertexTransitiveDRGs", coun\ t := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], mathmode := \ "Text", name := "C", next := 472736, root := ~, start := 472709, stop := 47273\ 5 ), rec( HTML := true, content := " has no equivalent\n function i\ n ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this might change in the future.\n ", count := [ "A", 3, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], name := "Alt", next := 472849, root := ~, start := 472658, stop := 472848 ), rec( content := "\n \ ", count := [ "A", 3, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 51 ], name := "Item", next := 472865, root := ~, start := 472642, stop := 472864 ) ], count := [ "A", 3, 0, 46 ], name := "Row", next := 472878, root := ~, start := 472513, stop := 472877 ), rec( attributes := rec( ), content := 0, count := [ "A", 3, 0, 52 ], name := "HorLine", next := 472895, root := ~, start := 472885, stop := 472894 ) ], count := [ "A", 3, 0, 4 ], mathmode := "Text", name := "Table", next := 472909, root := ~, start := 469720, stop := 472908 ), rec( content := "\n\n ", count := [ "A", 3, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 3, 0, 1 ], mathmode := "Text", name := "Section", next := 472923, root := ~, start := 469466, stop := 472922 ), rec( content := "\n\n ", count := [ "A", 3, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 4, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Some special vert\ ex subsets of a graph\n ", count := [ "A", 4, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 2 ], mathmode := "Text", name := "Heading", next := 473010, root := ~, start := 472941, stop := 473009 ), rec( content := "\n ", count := [ "A", 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 4, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 4, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 3 ], name := "E", next := 473148, root := ~, start := 473038, stop := 473147 ), rec( content := "\n ", count := [ "A", 4, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 3 ], mathmode := "Text", name := "Alt", next := 473159, root := ~, start := 473015, stop := 473158 ), rec( content := "\n\n ", count := [ "A", 4, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 4 ], name := "HorLine", next := 473205, root := ~, start := 473195, stop := 473204 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 4, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 4, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 4, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 4, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 5 ], name := "Item", next := 473275, root := ~, start := 473227, stop := 473274 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 4, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 4, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 7 ], name := "Item", next := 473335, root := ~, start := 473284, stop := 473334 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 4, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 9 ], name := "Alt", next := 473416, root := ~, start := 473361, stop := 473415 ), rec( content := "\n \ ", count := [ "A", 4, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 9 ], name := "Item", next := 473432, root := ~, start := 473344, stop := 473431 ) ], count := [ "A", 4, 0, 4 ], name := "Row", next := 473445, root := ~, start := 473213, stop := 473444 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 10 ], name := "HorLine", next := 473462, root := ~, start := 473452, stop := 473461 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ onnectedComponent", count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "C", next := 473526, root := ~, start := 473501, stop := 473525 ), rec( content := "\n \ ", count := [ "A", 4, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 11 ], name := "Item", next := 473542, root := ~, start := 473484, stop := 473541 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ ConnectedComponent" ), content := 0, count := [ "A", 4, 0, 13 ], mathmode := "Text", name := "Ref", next := 473607, root := ~, start := 473568, stop := 473606 ), rec( content := "\n \ ", count := [ "A", 4, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 13 ], name := "Item", next := 473623, root := ~, start := 473551, stop := 473622 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n When applied to a simple di/graph and a vertex,\n ", count := [ "A", 4, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "ConnectedComponent", count := [ "A", 4, 0, 15 ], mathmod\ e := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], mathmode := \ "Text", name := "C", next := 473764, root := ~, start := 473739, stop := 47376\ 3 ), rec( HTML := true, content := " in ", count := [ "A", 4, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 15 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphCon\ nectedComponent" ), content := 0, count := [ "A", 4, 0, 15 ], mathmode := "Tex\ t", name := "Ref", next := 473844, root := ~, start := 473805, stop := 473843 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Oper := "DigraphConnectedComponent"\ ), content := 0, count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "Ref\ ", next := 473897, root := ~, start := 473858, stop := 473896 ), rec( HTML := true , content := "\ can also be applied to\n non-simple graphs (in the language of ", count := [ "A", 4, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 15 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ ).\n ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 4, 0, 15 ], name := "Alt", next := 473997, root := ~, start := 473649, stop := 473996 ), rec( content := "\n \ ", count := [ "A", 4, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 15 ], name := "Item", next := 474013, root := ~, start := 473632, stop := 474012 ) ], count := [ "A", 4, 0, 10 ], name := "Row", next := 474026, root := ~, start := 473470, stop := 474025 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 16 ], name := "HorLine", next := 474043, root := ~, start := 474033, stop := 474042 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ onnectedComponents", count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "C", next := 474108, root := ~, start := 474082, stop := 474107 ), rec( content := "\n \ ", count := [ "A", 4, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 17 ], name := "Item", next := 474124, root := ~, start := 474065, stop := 474123 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ ConnectedComponents" ), content := 0, count := [ "A", 4, 0, 19 ], mathmode := "Text", name := "Ref", next := 474190, root := ~, start := 474150, stop := 474189 ), rec( content := "\n \ ", count := [ "A", 4, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 19 ], name := "Item", next := 474206, root := ~, start := 474133, stop := 474205 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n When applied to a simple di/graph,\n ", count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "ConnectedComponents", count := [ "A", 4, 0, 21 ], mathmo\ de := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "C", next := 474335, root := ~, start := 474309, stop := 47433\ 4 ), rec( HTML := true, content := " in ", count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 21 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DigraphCon\ nectedComponents" ), content := 0, count := [ "A", 4, 0, 21 ], mathmode := "Te\ xt", name := "Ref", next := 474416, root := ~, start := 474376, stop := 474415\ ), rec( HTML := true, content := ".\n ", count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digrap\ hConnectedComponents" ), content := 0, count := [ "A", 4, 0, 21 ], mathmode := \ "Text", name := "Ref", next := 474470, root := ~, start := 474430, stop := 4744\ 69 ), rec( HTML := true, content := " can also be applied to\n non-\ simple graphs (in the language of ", count := [ "A", 4, 0, 21 ], mathmode := "\ Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ ).\n ", count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 4, 0, 21 ], name := "Alt", next := 474570, root := ~, start := 474232, stop := 474569 ), rec( content := "\n \ ", count := [ "A", 4, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 21 ], name := "Item", next := 474586, root := ~, start := 474215, stop := 474585 ) ], count := [ "A", 4, 0, 16 ], name := "Row", next := 474599, root := ~, start := 474051, stop := 474598 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 22 ], name := "HorLine", next := 474616, root := ~, start := 474606, stop := 474615 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ icomponents", count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "PCDATA" , root := ~ ) ], count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "C", next := 474674, root := ~, start := 474655, stop := 474673 ), rec( content := "\n \ ", count := [ "A", 4, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 23 ], name := "Item", next := 474690, root := ~, start := 474638, stop := 474689 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ Bicomponents" ), content := 0, count := [ "A", 4, 0, 25 ], mathmode := "Text", name := "Ref", next := 474749, root := ~, start := 474716, stop := 474748 ), rec( content := "\n \ ", count := [ "A", 4, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 25 ], name := "Item", next := 474765, root := ~, start := 474699, stop := 474764 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n When applied to a simple bipartite di/graph,\n ", count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Bicomponents", count := [ "A", 4, 0, 27 ], mathmode := "\ Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "C", next := 474897, root := ~, start := 474878, stop := 47489\ 6 ), rec( HTML := true, content := " in ", count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 27 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "DigraphBic\ omponents" ), content := 0, count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Ref\ ", next := 474971, root := ~, start := 474938, stop := 474970 ), rec( HTML := true , content := "\ .\n The definition of ", count := [ "A", 4, 0, 27 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( Prop := "IsBipart\ iteDigraph" ), content := 0, count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Ref\ ", next := 475035, root := ~, start := 475003, stop := 475034 ), rec( HTML := true , content := "\ does not require\n its argument to be simple (in the language of " , count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ ), i.e. the\n bipartite graphs in ", count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ are a proper subset of the bipartite\n digraphs in ", count := [ "A", 4, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 4, 0, 27 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 27 ], name := "Alt", next := 475259, root := ~, start := 474791, stop := 475258 ), rec( content := "\n \ ", count := [ "A", 4, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 27 ], name := "Item", next := 475275, root := ~, start := 474774, stop := 475274 ) ], count := [ "A", 4, 0, 22 ], name := "Row", next := 475288, root := ~, start := 474624, stop := 475287 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 28 ], name := "HorLine", next := 475305, root := ~, start := 475295, stop := 475304 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ istanceSet", count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "C", next := 475362, root := ~, start := 475344, stop := 475361 ), rec( content := "\n \ ", count := [ "A", 4, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 29 ], name := "Item", next := 475378, root := ~, start := 475327, stop := 475377 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a \ digraph, a pos int, and an int", Oper := "DigraphDistanceSet" ), content := 0, count := [ "A", 4, 0, 31 ], mathmode := "Text", name := "Ref", next := 475497, root := ~, start := 475406, stop := 475496 ), rec( content := "\n \ ", count := [ "A", 4, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 31 ], name := "Item", next := 475513, root := ~, start := 475387, stop := 475512 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "DistanceSet", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "C", next := 4755\ 87, root := ~, start := 475569, stop := 475586 ), rec( HTML := true, content := "\ in ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 33 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 33 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 33 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 33 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for a dig\ raph, a pos int, and an int", Oper := "DigraphDistanceSet" ), content := 0, count := [ "A", 4, 0, 33 ], mathmode := \ "Text", name := "Ref", next := 475719, root := ~, start := 475628, stop := 4757\ 18 ), rec( HTML := true, content := " in ", count := [ "A", 4, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 4, 0, 33 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 33 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 33 ], name := "Alt", next := 475751, root := ~, start := 475539, stop := 475750 ), rec( content := "\n \ ", count := [ "A", 4, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 33 ], name := "Item", next := 475767, root := ~, start := 475522, stop := 475766 ) ], count := [ "A", 4, 0, 28 ], name := "Row", next := 475780, root := ~, start := 475313, stop := 475779 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 34 ], name := "HorLine", next := 475797, root := ~, start := 475787, stop := 475796 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "L\ ayers", count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "C", next := 475849, root := ~, start := 475836, stop := 475848 ), rec( content := "\n \ ", count := [ "A", 4, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 35 ], name := "Item", next := 475865, root := ~, start := 475819, stop := 475864 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Layers" ), content := 0, count := [ "A", 4, 0, 37 ], mathmode := "Text", name := "Ref", next := 475918, root := ~, start := 475891, stop := 475917 ), rec( content := "\n \ ", count := [ "A", 4, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 37 ], name := "Item", next := 475934, root := ~, start := 475874, stop := 475933 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Layers", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "C", next := 4760\ 03, root := ~, start := 475990, stop := 476002 ), rec( HTML := true, content := "\ in ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 39 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 39 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 39 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 39 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphLay\ ers" ), content := 0, count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "Ref\ ", next := 476071, root := ~, start := 476044, stop := 476070 ), rec( HTML := true , content := "\ in ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 39 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 39 ], name := "Alt", next := 476103, root := ~, start := 475960, stop := 476102 ), rec( content := "\n \ ", count := [ "A", 4, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 39 ], name := "Item", next := 476119, root := ~, start := 475943, stop := 476118 ) ], count := [ "A", 4, 0, 34 ], name := "Row", next := 476132, root := ~, start := 475805, stop := 476131 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 40 ], name := "HorLine", next := 476149, root := ~, start := 476139, stop := 476148 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ ndependentSet", count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "C", next := 476209, root := ~, start := 476188, stop := 476208 ), rec( content := "\n \ ", count := [ "A", 4, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 41 ], name := "Item", next := 476225, root := ~, start := 476171, stop := 476224 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ IndependentSet" ), content := 0, count := [ "A", 4, 0, 43 ], mathmode := "Text", name := "Ref", next := 476286, root := ~, start := 476251, stop := 476285 ), rec( content := "\n \ ", count := [ "A", 4, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 43 ], name := "Item", next := 476302, root := ~, start := 476234, stop := 476301 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "IndependentSet", count := [ "A", 4, 0, 45 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "C", next := 4763\ 79, root := ~, start := 476358, stop := 476378 ), rec( HTML := true, content := "\ in ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 4, 0, 45 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 4, 0, 45 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 4, 0, 45 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 4, 0, 45 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "DigraphInd\ ependentSet" ), content := 0, count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "Ref\ ", next := 476455, root := ~, start := 476420, stop := 476454 ), rec( HTML := true , content := "\ in ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 45 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 4, 0, 45 ], name := "Alt", next := 476487, root := ~, start := 476328, stop := 476486 ), rec( content := "\n \ ", count := [ "A", 4, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 45 ], name := "Item", next := 476503, root := ~, start := 476311, stop := 476502 ) ], count := [ "A", 4, 0, 40 ], name := "Row", next := 476516, root := ~, start := 476157, stop := 476515 ), rec( attributes := rec( ), content := 0, count := [ "A", 4, 0, 46 ], name := "HorLine", next := 476533, root := ~, start := 476523, stop := 476532 ) ], count := [ "A", 4, 0, 4 ], mathmode := "Text", name := "Table", next := 476547, root := ~, start := 473165, stop := 476546 ), rec( content := "\n\n ", count := [ "A", 4, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 4, 0, 1 ], mathmode := "Text", name := "Section", next := 476561, root := ~, start := 472927, stop := 476560 ), rec( content := "\n\n ", count := [ "A", 4, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 5, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Functions to cons\ truct new graphs from old\n ", count := [ "A", 5, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 2 ], mathmode := "Text", name := "Heading", next := 476652, root := ~, start := 476579, stop := 476651 ), rec( content := "\n ", count := [ "A", 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 5, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 5, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 3 ], name := "E", next := 476790, root := ~, start := 476680, stop := 476789 ), rec( content := "\n ", count := [ "A", 5, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 3 ], mathmode := "Text", name := "Alt", next := 476801, root := ~, start := 476657, stop := 476800 ), rec( content := "\n\n ", count := [ "A", 5, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 4 ], name := "HorLine", next := 476847, root := ~, start := 476837, stop := 476846 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 5, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 5, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 5, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 5, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 5 ], name := "Item", next := 476917, root := ~, start := 476869, stop := 476916 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 5, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 5, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 7 ], name := "Item", next := 476977, root := ~, start := 476926, stop := 476976 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 5, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 9 ], name := "Alt", next := 477058, root := ~, start := 477003, stop := 477057 ), rec( content := "\n \ ", count := [ "A", 5, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 9 ], name := "Item", next := 477074, root := ~, start := 476986, stop := 477073 ) ], count := [ "A", 5, 0, 4 ], name := "Row", next := 477087, root := ~, start := 476855, stop := 477086 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 10 ], name := "HorLine", next := 477104, root := ~, start := 477094, stop := 477103 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ nducedSubgraph", count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "C", next := 477165, root := ~, start := 477143, stop := 477164 ), rec( content := "\n \ ", count := [ "A", 5, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 11 ], name := "Item", next := 477181, root := ~, start := 477126, stop := 477180 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Induced\ Subdigraph" ), content := 0, count := [ "A", 5, 0, 13 ], mathmode := "Text", name := "Ref", next := 477238, root := ~, start := 477207, stop := 477237 ), rec( content := "\n \ ", count := [ "A", 5, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 13 ], name := "Item", next := 477254, root := ~, start := 477190, stop := 477253 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Oper := "InducedSubdigraph" ), content := 0 , count := [ "A", 5, 0, 15 ], mathmode := \ "Text", name := "Ref", next := 477341, root := ~, start := 477310, stop := 4773\ 40 ), rec( HTML := true, content := " in ", count := [ "A", 5, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 15 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ does not allow a third\n argument ", count := [ "A", 5, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "G", count := [ "A", 5, 0, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "C", next := 4774\ 08, root := ~, start := 477400, stop := 477407 ), rec( HTML := true, content := "\ specifying a group of automorphisms on the\n resultant graph. This\ may be included in a future release.\n ", count := [ "A", 5, 0, 15 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], name := "Alt", next := 477539, root := ~, start := 477280, stop := 477538 ), rec( content := "\n \ ", count := [ "A", 5, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 15 ], name := "Item", next := 477555, root := ~, start := 477263, stop := 477554 ) ], count := [ "A", 5, 0, 10 ], name := "Row", next := 477568, root := ~, start := 477112, stop := 477567 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 16 ], name := "HorLine", next := 477585, root := ~, start := 477575, stop := 477584 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ istanceSetInduced", count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "C", next := 477649, root := ~, start := 477624, stop := 477648 ), rec( content := "\n \ ", count := [ "A", 5, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 17 ], name := "Item", next := 477665, root := ~, start := 477607, stop := 477664 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 19 ], name := "Item", next := 477711, root := ~, start := 477674, stop := 477710 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 21 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 21 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 5, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 21 ], name := "Alt", next := 477870, root := ~, start := 477737, stop := 477869 ), rec( content := "\n \ ", count := [ "A", 5, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 21 ], name := "Item", next := 477886, root := ~, start := 477720, stop := 477885 ) ], count := [ "A", 5, 0, 16 ], name := "Row", next := 477899, root := ~, start := 477593, stop := 477898 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 22 ], name := "HorLine", next := 477916, root := ~, start := 477906, stop := 477915 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ istanceGraph", count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "C", next := 477975, root := ~, start := 477955, stop := 477974 ), rec( content := "\n \ ", count := [ "A", 5, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 23 ], name := "Item", next := 477991, root := ~, start := 477938, stop := 477990 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graph and list", Oper := "DistanceDigraph" ), content := 0, count := [ "A", 5, 0, 25 ], mathmode := "Text", name := "Ref", next := 478075, root := ~, start := 478017, stop := 478074 ), rec( content := "\n \ ", count := [ "A", 5, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 25 ], name := "Item", next := 478091, root := ~, start := 478000, stop := 478090 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "DistanceGraph", count := [ "A", 5, 0, 27 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "C", next := 4781\ 67, root := ~, start := 478147, stop := 478166 ), rec( HTML := true, content := "\ in ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 5, 0, 27 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for digra\ ph and list", Oper := "DistanceDigraph" ), content := 0, count := [ "A", 5, 0, 27 ], mathmode := \ "Text", name := "Ref", next := 478266, root := ~, start := 478208, stop := 4782\ 65 ), rec( HTML := true, content := " in\n ", count := [ "A", 5, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 27 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 27 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 27 ], name := "Alt", next := 478310, root := ~, start := 478117, stop := 478309 ), rec( content := "\n \ ", count := [ "A", 5, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 27 ], name := "Item", next := 478326, root := ~, start := 478100, stop := 478325 ) ], count := [ "A", 5, 0, 22 ], name := "Row", next := 478339, root := ~, start := 477924, stop := 478338 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 28 ], name := "HorLine", next := 478356, root := ~, start := 478346, stop := 478355 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ omplementGraph", count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "C", next := 478417, root := ~, start := 478395, stop := 478416 ), rec( content := "\n \ ", count := [ "A", 5, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 29 ], name := "Item", next := 478433, root := ~, start := 478378, stop := 478432 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ Dual" ), content := 0, count := [ "A", 5, 0, 31 ], mathmode := "Text", name := "Ref", next := 478484, root := ~, start := 478459, stop := 478483 ), rec( content := "\n \ ", count := [ "A", 5, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 31 ], name := "Item", next := 478500, root := ~, start := 478442, stop := 478499 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Oper := "DigraphDual" ), content := 0 , count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "Ref", next := 478581, root := ~, start := 478556, stop := 4785\ 80 ), rec( HTML := true, content := " in ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 33 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ is equivalent to\n ", count := [ "A", 5, 0, 33 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "ComplementGraph", count := [ "A", 5, 0, 33 ], mathmode :=\ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "C", next := 478647, root := ~, start := 478625, stop := 47864\ 6 ), rec( HTML := true, content := " in ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 5, 0, 33 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 33 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 33 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 33 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ with the optional argument\n ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "comploops", count := [ "A", 5, 0, 33 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], mathmode := "\ Text", name := "A", next := 478714, root := ~, start := 478698, stop := 478713\ ), rec( HTML := true, content := " set to ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "true", count := [ "A", 5, 0, 33 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "K", next := 4787\ 33, root := ~, start := 478722, stop := 478732 ), rec( HTML := true, content := "\ (loops/nonloops are\n complemented).\n ", count := [ "A", 5, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], name := "Alt", next := 478797, root := ~, start := 478526, stop := 478796 ), rec( content := "\n \ ", count := [ "A", 5, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 33 ], name := "Item", next := 478813, root := ~, start := 478509, stop := 478812 ) ], count := [ "A", 5, 0, 28 ], name := "Row", next := 478826, root := ~, start := 478364, stop := 478825 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 34 ], name := "HorLine", next := 478843, root := ~, start := 478833, stop := 478842 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "P\ ointGraph", count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "C", next := 478899, root := ~, start := 478882, stop := 478898 ), rec( content := "\n \ ", count := [ "A", 5, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 35 ], name := "Item", next := 478915, root := ~, start := 478865, stop := 478914 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 37 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 37 ], name := "Item", next := 478961, root := ~, start := 478924, stop := 478960 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 39 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 39 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 39 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 5, 0, 39 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 39 ], name := "Alt", next := 479120, root := ~, start := 478987, stop := 479119 ), rec( content := "\n \ ", count := [ "A", 5, 0, 39 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 39 ], name := "Item", next := 479136, root := ~, start := 478970, stop := 479135 ) ], count := [ "A", 5, 0, 34 ], name := "Row", next := 479149, root := ~, start := 478851, stop := 479148 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 40 ], name := "HorLine", next := 479166, root := ~, start := 479156, stop := 479165 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "E\ dgeGraph", count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "C", next := 479221, root := ~, start := 479205, stop := 479220 ), rec( content := "\n \ ", count := [ "A", 5, 0, 41 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 41 ], name := "Item", next := 479237, root := ~, start := 479188, stop := 479236 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EdgeUnd\ irectedDigraph" ), content := 0, count := [ "A", 5, 0, 43 ], mathmode := "Text", name := "Ref", next := 479298, root := ~, start := 479263, stop := 479297 ), rec( content := "\n \ ", count := [ "A", 5, 0, 43 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 43 ], name := "Item", next := 479314, root := ~, start := 479246, stop := 479313 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "EdgeGraph", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "P\ CDATA", root := ~ ) ], count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "C", next := 4793\ 86, root := ~, start := 479370, stop := 479385 ), rec( HTML := true, content := "\ in ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 45 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 45 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 45 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 5, 0, 45 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "EdgeUndire\ ctedDigraph" ), content := 0, count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "Ref\ ", next := 479462, root := ~, start := 479427, stop := 479461 ), rec( HTML := true , content := "\ in ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 45 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 45 ], name := "Alt", next := 479494, root := ~, start := 479340, stop := 479493 ), rec( content := "\n \ ", count := [ "A", 5, 0, 45 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 45 ], name := "Item", next := 479510, root := ~, start := 479323, stop := 479509 ) ], count := [ "A", 5, 0, 40 ], name := "Row", next := 479523, root := ~, start := 479174, stop := 479522 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 46 ], name := "HorLine", next := 479540, root := ~, start := 479530, stop := 479539 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "S\ witchedGraph", count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "C", next := 479599, root := ~, start := 479579, stop := 479598 ), rec( content := "\n \ ", count := [ "A", 5, 0, 47 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 47 ], name := "Item", next := 479615, root := ~, start := 479562, stop := 479614 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 49 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 49 ], name := "Item", next := 479661, root := ~, start := 479624, stop := 479660 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 51 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 51 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 5, 0, 51 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 51 ], name := "Alt", next := 479820, root := ~, start := 479687, stop := 479819 ), rec( content := "\n \ ", count := [ "A", 5, 0, 51 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 51 ], name := "Item", next := 479836, root := ~, start := 479670, stop := 479835 ) ], count := [ "A", 5, 0, 46 ], name := "Row", next := 479849, root := ~, start := 479548, stop := 479848 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 52 ], name := "HorLine", next := 479866, root := ~, start := 479856, stop := 479865 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "U\ nderlyingGraph", count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "C", next := 479927, root := ~, start := 479905, stop := 479926 ), rec( content := "\n \ ", count := [ "A", 5, 0, 53 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 53 ], name := "Item", next := 479943, root := ~, start := 479888, stop := 479942 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digraph\ SymmetricClosure" ), content := 0, count := [ "A", 5, 0, 55 ], mathmode := "Text", name := "Ref", next := 480006, root := ~, start := 479969, stop := 480005 ), rec( content := "\n \ ", count := [ "A", 5, 0, 55 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 55 ], name := "Item", next := 480022, root := ~, start := 479952, stop := 480021 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "UnderlyingGraph", count := [ "A", 5, 0, 57 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "C", next := 4801\ 00, root := ~, start := 480078, stop := 480099 ), rec( HTML := true, content := "\ in ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 57 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 57 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 57 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 5, 0, 57 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "DigraphSym\ metricClosure" ), content := 0, count := [ "A", 5, 0, 57 ], mathmode := "Text" , name := "Ref\ ", next := 480178, root := ~, start := 480141, stop := 480177 ), rec( HTML := true , content := "\ in ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 57 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 57 ], name := "Alt", next := 480210, root := ~, start := 480048, stop := 480209 ), rec( content := "\n \ ", count := [ "A", 5, 0, 57 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 57 ], name := "Item", next := 480226, root := ~, start := 480031, stop := 480225 ) ], count := [ "A", 5, 0, 52 ], name := "Row", next := 480239, root := ~, start := 479874, stop := 480238 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 58 ], name := "HorLine", next := 480256, root := ~, start := 480246, stop := 480255 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "Q\ uotientGraph", count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "C", next := 480315, root := ~, start := 480295, stop := 480314 ), rec( content := "\n \ ", count := [ "A", 5, 0, 59 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 59 ], name := "Item", next := 480331, root := ~, start := 480278, stop := 480330 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Quotien\ tDigraph" ), content := 0, count := [ "A", 5, 0, 61 ], mathmode := "Text", name := "Ref", next := 480386, root := ~, start := 480357, stop := 480385 ), rec( content := "\n \ ", count := [ "A", 5, 0, 61 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 61 ], name := "Item", next := 480402, root := ~, start := 480340, stop := 480401 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( Oper := "QuotientDigraph" ), content := 0 , count := [ "A", 5, 0, 63 ], mathmode := \ "Text", name := "Ref", next := 480487, root := ~, start := 480458, stop := 4804\ 86 ), rec( HTML := true, content := " in ", count := [ "A", 5, 0, 63 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 63 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ takes a partition\n ", count := [ "A", 5, 0, 63 ], mathmode := "Te\ xt", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "p", count := [ "A", 5, 0, 63 ], mathmode := "Text", name \ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "C", next := 4805\ 40, root := ~, start := 480532, stop := 480539 ), rec( HTML := true, content := "\ of the vertices of the digraph, rather than a set of\n generating \ pairs. Hence, the equivalence might not be\n group-invariant. Als\ o note that in ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Digraphs", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "Pac\ kage", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true, content := "\ multiple edges may be\n created, whereas in ", count := [ "A", 5, 0, 63 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 5, 0, 63 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 63 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 63 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 63 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ they are effectively combined into a\n single edge.\n ", count := [ "A", 5, 0, 63 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], name := "Alt", next := 480860, root := ~, start := 480428, stop := 480859 ), rec( content := "\n \ ", count := [ "A", 5, 0, 63 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 63 ], name := "Item", next := 480876, root := ~, start := 480411, stop := 480875 ) ], count := [ "A", 5, 0, 58 ], name := "Row", next := 480889, root := ~, start := 480264, stop := 480888 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 64 ], name := "HorLine", next := 480906, root := ~, start := 480896, stop := 480905 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "B\ ipartiteDouble", count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "C", next := 480967, root := ~, start := 480945, stop := 480966 ), rec( content := "\n \ ", count := [ "A", 5, 0, 65 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 65 ], name := "Item", next := 480983, root := ~, start := 480928, stop := 480982 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Biparti\ teDoubleDigraph" ), content := 0, count := [ "A", 5, 0, 67 ], mathmode := "Text", name := "Ref", next := 481045, root := ~, start := 481009, stop := 481044 ), rec( content := "\n \ ", count := [ "A", 5, 0, 67 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 67 ], name := "Item", next := 481061, root := ~, start := 480992, stop := 481060 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "BipartiteDouble", count := [ "A", 5, 0, 69 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "C", next := 4811\ 39, root := ~, start := 481117, stop := 481138 ), rec( HTML := true, content := "\ in ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 5, 0, 69 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 5, 0, 69 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 5, 0, 69 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 5, 0, 69 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "BipartiteD\ oubleDigraph" ), content := 0, count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "Ref\ ", next := 481216, root := ~, start := 481180, stop := 481215 ), rec( HTML := true , content := "\ in ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 69 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 69 ], name := "Alt", next := 481248, root := ~, start := 481087, stop := 481247 ), rec( content := "\n \ ", count := [ "A", 5, 0, 69 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 69 ], name := "Item", next := 481264, root := ~, start := 481070, stop := 481263 ) ], count := [ "A", 5, 0, 64 ], name := "Row", next := 481277, root := ~, start := 480914, stop := 481276 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 70 ], name := "HorLine", next := 481294, root := ~, start := 481284, stop := 481293 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ eodesicsGraph", count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "C", next := 481354, root := ~, start := 481333, stop := 481353 ), rec( content := "\n \ ", count := [ "A", 5, 0, 71 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 71 ], name := "Item", next := 481370, root := ~, start := 481316, stop := 481369 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 73 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 73 ], name := "Item", next := 481416, root := ~, start := 481379, stop := 481415 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 75 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 75 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 75 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , and no plans to\n include one.\n ", count := [ "A", 5, 0, 75 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 75 ], name := "Alt", next := 481575, root := ~, start := 481442, stop := 481574 ), rec( content := "\n \ ", count := [ "A", 5, 0, 75 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 75 ], name := "Item", next := 481591, root := ~, start := 481425, stop := 481590 ) ], count := [ "A", 5, 0, 70 ], name := "Row", next := 481604, root := ~, start := 481302, stop := 481603 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 76 ], name := "HorLine", next := 481621, root := ~, start := 481611, stop := 481620 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ollapsedIndependentOrbitsGraph", count := [ "A", 5, 0, 77 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 77 ], mathmode := "Text", name := "C", next := 481698, root := ~, start := 481660, stop := 481697 ), rec( content := "\n \ ", count := [ "A", 5, 0, 77 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 77 ], name := "Item", next := 481714, root := ~, start := 481643, stop := 481713 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 79 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 79 ], name := "Item", next := 481760, root := ~, start := 481723, stop := 481759 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 81 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 81 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 81 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this feature\n might be implemented in the future.\n \ ", count := [ "A", 5, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 81 ], name := "Alt", next := 481948, root := ~, start := 481786, stop := 481947 ), rec( content := "\n \ ", count := [ "A", 5, 0, 81 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 81 ], name := "Item", next := 481964, root := ~, start := 481769, stop := 481963 ) ], count := [ "A", 5, 0, 76 ], name := "Row", next := 481977, root := ~, start := 481629, stop := 481976 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 82 ], name := "HorLine", next := 481994, root := ~, start := 481984, stop := 481993 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ollapsedCompleteOrbitsGraph", count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "C", next := 482068, root := ~, start := 482033, stop := 482067 ), rec( content := "\n \ ", count := [ "A", 5, 0, 83 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 83 ], name := "Item", next := 482084, root := ~, start := 482016, stop := 482083 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 85 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 85 ], name := "Item", next := 482130, root := ~, start := 482093, stop := 482129 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 87 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 87 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 87 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this feature\n might be implemented in the future.\n \ ", count := [ "A", 5, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 87 ], name := "Alt", next := 482318, root := ~, start := 482156, stop := 482317 ), rec( content := "\n \ ", count := [ "A", 5, 0, 87 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 87 ], name := "Item", next := 482334, root := ~, start := 482139, stop := 482333 ) ], count := [ "A", 5, 0, 82 ], name := "Row", next := 482347, root := ~, start := 482002, stop := 482346 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 88 ], name := "HorLine", next := 482364, root := ~, start := 482354, stop := 482363 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "N\ ewGroupGraph", count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "C", next := 482423, root := ~, start := 482403, stop := 482422 ), rec( content := "\n \ ", count := [ "A", 5, 0, 89 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 89 ], name := "Item", next := 482439, root := ~, start := 482386, stop := 482438 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 5, 0, 91 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 91 ], name := "Item", next := 482485, root := ~, start := 482448, stop := 482484 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 5, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 5, 0, 93 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ . However, the same\n results would be possible by calling ", count := [ "A", 5, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Oper := "Digrap\ hCopy" ), content := 0, count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "Ref\ ", next := 482680, root := ~, start := 482655, stop := 482679 ), rec( HTML := true , content := "\ on\n the digraph to produce a copy ", count := [ "A", 5, 0, 93 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "gr", count := [ "A", 5, 0, 93 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "C", next := 4827\ 35, root := ~, start := 482726, stop := 482734 ), rec( HTML := true, content := "\ , and then applying\n ", count := [ "A", 5, 0, 93 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "SetDigraphGroup(gr, G)", count := [ "A", 5, 0, 93 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], mathmode := \ "Text", name := "C", next := 482796, root := ~, start := 482767, stop := 48279\ 5 ), rec( HTML := true, content := ".\n ", count := [ "A", 5, 0, 93 ] , mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], name := "Alt", next := 482814, root := ~, start := 482511, stop := 482813 ), rec( content := "\n \ ", count := [ "A", 5, 0, 93 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 93 ], name := "Item", next := 482830, root := ~, start := 482494, stop := 482829 ) ], count := [ "A", 5, 0, 88 ], name := "Row", next := 482843, root := ~, start := 482372, stop := 482842 ), rec( attributes := rec( ), content := 0, count := [ "A", 5, 0, 94 ], name := "HorLine", next := 482860, root := ~, start := 482850, stop := 482859 ) ], count := [ "A", 5, 0, 4 ], mathmode := "Text", name := "Table", next := 482874, root := ~, start := 476807, stop := 482873 ), rec( content := "\n\n ", count := [ "A", 5, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 5, 0, 1 ], mathmode := "Text", name := "Section", next := 482888, root := ~, start := 476565, stop := 482887 ), rec( content := "\n\n ", count := [ "A", 5, 0, 95 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 6, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Vertex-Colouring \ and Complete Subgraphs\n ", count := [ "A", 6, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 2 ], mathmode := "Text", name := "Heading", next := 482976, root := ~, start := 482906, stop := 482975 ), rec( content := "\n ", count := [ "A", 6, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 6, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 6, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 3 ], name := "E", next := 483114, root := ~, start := 483004, stop := 483113 ), rec( content := "\n ", count := [ "A", 6, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 3 ], mathmode := "Text", name := "Alt", next := 483125, root := ~, start := 482981, stop := 483124 ), rec( content := "\n\n ", count := [ "A", 6, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 4 ], name := "HorLine", next := 483171, root := ~, start := 483161, stop := 483170 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 6, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 6, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 6, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 6, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 6, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 5 ], name := "Item", next := 483241, root := ~, start := 483193, stop := 483240 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 6, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 6, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 7 ], name := "Item", next := 483301, root := ~, start := 483250, stop := 483300 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 6, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 9 ], name := "Alt", next := 483382, root := ~, start := 483327, stop := 483381 ), rec( content := "\n \ ", count := [ "A", 6, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 9 ], name := "Item", next := 483398, root := ~, start := 483310, stop := 483397 ) ], count := [ "A", 6, 0, 4 ], name := "Row", next := 483411, root := ~, start := 483179, stop := 483410 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 10 ], name := "HorLine", next := 483428, root := ~, start := 483418, stop := 483427 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "V\ ertexColouring", count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ) ], count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "C", next := 483489, root := ~, start := 483467, stop := 483488 ), rec( content := "\n \ ", count := [ "A", 6, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 11 ], name := "Item", next := 483505, root := ~, start := 483450, stop := 483504 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Digraph\ GreedyColouring", Label := "for a digraph" ), content := 0, count := [ "A", 6, 0, 13 ], mathmode := "Text", name := "Ref", next := 483589, root := ~, start := 483531, stop := 483588 ), rec( content := "\n \ ", count := [ "A", 6, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 13 ], name := "Item", next := 483605, root := ~, start := 483514, stop := 483604 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "VertexColouring", count := [ "A", 6, 0, 15 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "C", next := 4836\ 83, root := ~, start := 483661, stop := 483682 ), rec( HTML := true, content := "\ in ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 6, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 6, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 6, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Attr := "DigraphGreedyColouring", Label :=\ "for a digraph" ), content := 0, count := [ "A", 6, 0, 15 ], mathmode := "Tex\ t", name := "Ref", next := 483782, root := ~, start := 483712, stop := 483781 ), rec( HTML := true , content := "\ in ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ ,\n except it returns a transformation rather than a list of vertex\ \n colors.\n ", count := [ "A", 6, 0, 15 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], name := "Alt", next := 483910, root := ~, start := 483631, stop := 483909 ), rec( content := "\n \ ", count := [ "A", 6, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 15 ], name := "Item", next := 483926, root := ~, start := 483614, stop := 483925 ) ], count := [ "A", 6, 0, 10 ], name := "Row", next := 483939, root := ~, start := 483436, stop := 483938 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 16 ], name := "HorLine", next := 483956, root := ~, start := 483946, stop := 483955 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ompleteSubgraphs", count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "C", next := 484019, root := ~, start := 483995, stop := 484018 ), rec( content := "\n \ ", count := [ "A", 6, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 17 ], name := "Item", next := 484035, root := ~, start := 483978, stop := 484034 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ Cliques" ), content := 0, count := [ "A", 6, 0, 19 ], mathmode := "Text", name := "Ref", next := 484089, root := ~, start := 484061, stop := 484088 ), rec( content := "\n \ ", count := [ "A", 6, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 19 ], name := "Item", next := 484105, root := ~, start := 484044, stop := 484104 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n The functionality of ", count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "CompleteSubgraphs", count := [ "A", 6, 0, 21 ], mathmode\ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "C", next := 484206, root := ~, start := 484182, stop := 48420\ 5 ), rec( HTML := true, content := " in ", count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 6, 0, 21 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 6, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 6, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 6, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is provided\n by ", count := [ "A", 6, 0, 21 ], mathmode := "Text" , name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Func := "DigraphCliques" ), content := 0 , count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "Ref", next := 484273, root := ~, start := 484245, stop := 4842\ 72 ), rec( HTML := true, content := " and ", count := [ "A", 6, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digrap\ hMaximalCliques" ), content := 0, count := [ "A", 6, 0, 21 ], mathmode := "Tex\ t", name := "Ref", next := 484313, root := ~, start := 484278, stop := 484312 ), rec( HTML := true , content := "\ \n in ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "Digraphs", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PC\ DATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "Pac\ kage", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true, content := "\ .\n ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 6, 0, 21 ], name := "Alt", next := 484357, root := ~, start := 484131, stop := 484356 ), rec( content := "\n \ ", count := [ "A", 6, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 21 ], name := "Item", next := 484373, root := ~, start := 484114, stop := 484372 ) ], count := [ "A", 6, 0, 16 ], name := "Row", next := 484386, root := ~, start := 483964, stop := 484385 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 22 ], name := "HorLine", next := 484403, root := ~, start := 484393, stop := 484402 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "C\ ompleteSubgraphsOfGivenSize", count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "C", next := 484477, root := ~, start := 484442, stop := 484476 ), rec( content := "\n \ ", count := [ "A", 6, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 23 ], name := "Item", next := 484493, root := ~, start := 484425, stop := 484492 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digraph\ Cliques" ), content := 0, count := [ "A", 6, 0, 25 ], mathmode := "Text", name := "Ref", next := 484547, root := ~, start := 484519, stop := 484546 ), rec( content := "\n \ ", count := [ "A", 6, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 25 ], name := "Item", next := 484563, root := ~, start := 484502, stop := 484562 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Much of the functionality of ", count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "CompleteSubgraphsOfGivenSize", count := [ "A", 6, 0, 27 ] , math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "C", next := 484683, root := ~, start := 484648, stop := 48468\ 2 ), rec( HTML := true, content := " in\n ", count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ) , content := [ rec( attr\ ibutes := rec( Text := "Grape" ), content := [ rec( HTML := true, content := "\ http://www.maths.qmul.ac.uk/~leonard/grape/", count := [ "A", 6, 0, 27 ], math\ mode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], math\ mode := "Text", name := "URL", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 6, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 6, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 6, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is provided by ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Func := "DigraphCliques" ), content := 0 , count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "Ref", next := 484750, root := ~, start := 484722, stop := 4847\ 49 ), rec( HTML := true, content := " and ", count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Func := "Digrap\ hMaximalCliques" ), content := 0, count := [ "A", 6, 0, 27 ], mathmode := "Tex\ t", name := "Ref", next := 484804, root := ~, start := 484755, stop := 484803 ), rec( HTML := true , content := "\ in ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( ), content := [ rec( HTML := true, content := "Digraphs", count := [ "\ A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , except that the\n functionality relating to vertex weights and pa\ rtial colourings is\n not implemented.\n ", count := [ "A", 6, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], name := "Alt", next := 484960, root := ~, start := 484589, stop := 484959 ), rec( content := "\n \ ", count := [ "A", 6, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 27 ], name := "Item", next := 484976, root := ~, start := 484572, stop := 484975 ) ], count := [ "A", 6, 0, 22 ], name := "Row", next := 484989, root := ~, start := 484411, stop := 484988 ), rec( attributes := rec( ), content := 0, count := [ "A", 6, 0, 28 ], name := "HorLine", next := 485006, root := ~, start := 484996, stop := 485005 ) ], count := [ "A", 6, 0, 4 ], mathmode := "Text", name := "Table", next := 485020, root := ~, start := 483131, stop := 485019 ), rec( content := "\n ", count := [ "A", 6, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 6, 0, 1 ], mathmode := "Text", name := "Section", next := 485033, root := ~, start := 482892, stop := 485032 ), rec( content := "\n\n ", count := [ "A", 6, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n ", count := [ "A", 7, 0, 1 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n Automorphism grou\ ps and isomorphism testing for graphs\n ", count := [ "A", 7, 0, 2 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 2 ], mathmode := "Text", name := "Heading", next := 485136, root := ~, start := 485051, stop := 485135 ), rec( content := "\n ", count := [ "A", 7, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "HTML" ), content := [ rec( content := "\n ", count := [ "A", 7, 0, 3 ], name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "\n The tab\ le in this section contains more information when viewed in\n html form\ at.\n ", count := [ "A", 7, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 3 ], name := "E", next := 485274, root := ~, start := 485164, stop := 485273 ), rec( content := "\n ", count := [ "A", 7, 0, 3 ], name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 3 ], mathmode := "Text", name := "Alt", next := 485285, root := ~, start := 485141, stop := 485284 ), rec( content := "\n\n ", count := [ "A", 7, 0, 3 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Align := "|l|l|l|" ), content := [ rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 4 ], name := "HorLine", next := 485331, root := ~, start := 485321, stop := 485330 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Not := "Text" ), content := [ rec( attributes :=\ rec( Text := "Grape" ), content := [ rec( content := "http://www.maths.qmul.a\ c.uk/~leonard/grape/", count := [ "A", 7, 0, 5 ], mathmode := "Text", name := \ "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 5 ], mathmode := "Text", name := "URL\ ", next := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 7, 0, 5 ], name := "Alt", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := "Text" ) , content := [ rec( attributes :=\ rec( ), content := [ rec( content := "Grape", count := [ "A", 7, 0, 5 ], name\ := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 5 ], name := "Package", next := 137, root := ~, start := 113 , stop := 136 ) ], count := [ "A", 7, 0, 5 ], mathmode := "Text", name := "Alt", next := 143, root := ~, start := 93, stop := 142 ), rec( content := " command\ \n ", count := [ "A", 7, 0, 5 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 5 ], name := "Item", next := 485401, root := ~, start := 485353, stop := 485400 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "D\ igraphs", count := [ "A", 7, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 7 ], mathmode := "Text", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( content := " command\ \n ", count := [ "A", 7, 0, 7 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 7 ], name := "Item", next := 485461, root := ~, start := 485410, stop := 485460 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n Comments\n ", count := [ "A", 7, 0, 9 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 9 ], name := "Alt", next := 485542, root := ~, start := 485487, stop := 485541 ), rec( content := "\n \ ", count := [ "A", 7, 0, 9 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 9 ], name := "Item", next := 485558, root := ~, start := 485470, stop := 485557 ) ], count := [ "A", 7, 0, 4 ], name := "Row", next := 485571, root := ~, start := 485339, stop := 485570 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 10 ], name := "HorLine", next := 485588, root := ~, start := 485578, stop := 485587 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "A\ utGroupGraph", count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "PCDATA\ ", root := ~ ) ], count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "C", next := 485647, root := ~, start := 485627, stop := 485646 ), rec( content := "\n \ ", count := [ "A", 7, 0, 11 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 11 ], name := "Item", next := 485663, root := ~, start := 485610, stop := 485662 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automor\ phismGroup", Label := "for a digraph" ), content := 0, count := [ "A", 7, 0, 13 ], mathmode := "Text", name := "Ref", next := 485742, root := ~, start := 485689, stop := 485741 ), rec( content := "\n \ ", count := [ "A", 7, 0, 13 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 13 ], name := "Item", next := 485758, root := ~, start := 485672, stop := 485757 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "AutGroupGraph", count := [ "A", 7, 0, 15 ], mathmode := "Text", name :\ = "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "C", next := 4858\ 34, root := ~, start := 485814, stop := 485833 ), rec( HTML := true, content := "\ in ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 7, 0, 15 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 7, 0, 15 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 7, 0, 15 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to\n ", count := [ "A", 7, 0, 15 ], mathmode := "Tex\ t", name := "PCDATA", root := ~ ), rec( attributes := rec( Attr := "Automorphi\ smGroup", Label := "for a digraph" ), content := 0, count := [ "A", 7, 0, 15 ] , mathmode := \ "Text", name := "Ref", next := 485928, root := ~, start := 485875, stop := 4859\ 27 ), rec( HTML := true, content := " in ", count := [ "A", 7, 0, 15 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 7, 0, 15 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 15 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDAT\ A", root := ~ ) ], count := [ "A", 7, 0, 15 ], name := "Alt", next := 485960, root := ~, start := 485784, stop := 485959 ), rec( content := "\n \ ", count := [ "A", 7, 0, 15 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 15 ], name := "Item", next := 485976, root := ~, start := 485767, stop := 485975 ) ], count := [ "A", 7, 0, 10 ], name := "Row", next := 485989, root := ~, start := 485596, stop := 485988 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 16 ], name := "HorLine", next := 486006, root := ~, start := 485996, stop := 486005 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphIsomorphism", count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "C", next := 486068, root := ~, start := 486045, stop := 486067 ), rec( content := "\n \ ", count := [ "A", 7, 0, 17 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 17 ], name := "Item", next := 486084, root := ~, start := 486028, stop := 486083 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsomorphismDigraphs" ), content := 0, count := [ "A", 7, 0, 19 ], mathmode := "Text", name := "Ref", next := 486164, root := ~, start := 486110, stop := 486163 ), rec( content := "\n \ ", count := [ "A", 7, 0, 19 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 19 ], name := "Item", next := 486180, root := ~, start := 486093, stop := 486179 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "GraphIsomorphism", count := [ "A", 7, 0, 21 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "C", next := 4862\ 59, root := ~, start := 486236, stop := 486258 ), rec( HTML := true, content := "\ in ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 7, 0, 21 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 7, 0, 21 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 7, 0, 21 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Label := "for digraphs", Oper := "I\ somorphismDigraphs" ), content := 0, count := [ "A", 7, 0, 21 ], mathmode := "\ Text", name := "Ref", next := 486354, root := ~, start := 486288, stop := 4863\ 53 ), rec( HTML := true, content := " in ", count := [ "A", 7, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 7, 0, 21 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n Note however that ", count := [ "A", 7, 0, 21 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "GraphIsomorphism", count := [ "A", 7, 0, 21 ], mathmode :\ = "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := \ "Text", name := "C", next := 486423, root := ~, start := 486400, stop := 48642\ 2 ), rec( HTML := true, content := "'s optional argument\n ", count := [ "A", 7, 0, 21 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "firstunbindcanon", count := [ "A", 7, 0, 21 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], mathmode := \ "Text", name := "A", next := 486479, root := ~, start := 486456, stop := 48647\ 8 ), rec( HTML := true, content := " is not applicable, since the canonical\n \ labellings of a digraph are not portable between sessions.\n \ ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], name := "Alt", next := 486606, root := ~, start := 486206, stop := 486605 ), rec( content := "\n \ ", count := [ "A", 7, 0, 21 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 21 ], name := "Item", next := 486622, root := ~, start := 486189, stop := 486621 ) ], count := [ "A", 7, 0, 16 ], name := "Row", next := 486635, root := ~, start := 486014, stop := 486634 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 22 ], name := "HorLine", next := 486652, root := ~, start := 486642, stop := 486651 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "I\ sIsomorphicGraph", count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ) ], count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "C", next := 486715, root := ~, start := 486691, stop := 486714 ), rec( content := "\n \ ", count := [ "A", 7, 0, 23 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 23 ], name := "Item", next := 486731, root := ~, start := 486674, stop := 486730 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Label := "for di\ graphs", Oper := "IsIsomorphicDigraph" ), content := 0, count := [ "A", 7, 0, 25 ], mathmode := "Text", name := "Ref", next := 486811, root := ~, start := 486757, stop := 486810 ), rec( content := "\n \ ", count := [ "A", 7, 0, 25 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 25 ], name := "Item", next := 486827, root := ~, start := 486740, stop := 486826 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDA\ TA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML := true, cont\ ent := "IsIsomorphicGraph", count := [ "A", 7, 0, 27 ], mathmode := "Text", name\ := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "C", next := 4869\ 07, root := ~, start := 486883, stop := 486906 ), rec( HTML := true, content := "\ in ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes :\ = rec( Not := "Text" ), content := [ rec( attributes := rec( Text := "Grape" ) , cont\ ent := [ rec( HTML := true, content := "http://www.maths.qmul.ac.uk/~leonard/g\ rape/", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root\ := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "URL", next\ := 87, root := ~, start := 17, stop := 86 ) ], count := [ "A", 7, 0, 27 ], name := "Alt\ ", next := 93, root := ~, start := 1, stop := 92 ), rec( attributes := rec( Only := \ "Text" ), content := [ rec( attributes := rec( ), content := [ rec( HTML := t\ rue, content := "Grape", count := [ "A", 7, 0, 27 ], name := "PCDATA", root :=\ ~ ) ], count := [ "A", 7, 0, 27 ], name := "Package", next := 137, root := ~, star\ t := 113, stop := 136 ) ], count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "Alt\ ", next := 143, root := ~, start := 93, stop := 142 ), rec( HTML := true, content := "\ is equivalent to ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCD\ ATA", root := ~ ), rec( attributes := rec( Label := "for digraphs", Oper := "I\ sIsomorphicDigraph" ), content := 0, count := [ "A", 7, 0, 27 ], mathmode := "\ Text", name := "Ref", next := 487002, root := ~, start := 486936, stop := 4870\ 01 ), rec( HTML := true, content := " in ", count := [ "A", 7, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 7, 0, 27 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ .\n Note however that ", count := [ "A", 7, 0, 27 ], mathmode := "T\ ext", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML \ := true, content := "IsIsomorphicGraph", count := [ "A", 7, 0, 27 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := \ "Text", name := "C", next := 487072, root := ~, start := 487048, stop := 48707\ 1 ), rec( HTML := true, content := "'s optional argument\n ", count := [ "A", 7, 0, 27 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "firstunbindcanon", count := [ "A", 7, 0, 27 ], mathmode \ := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], mathmode := \ "Text", name := "A", next := 487128, root := ~, start := 487105, stop := 48712\ 7 ), rec( HTML := true, content := " is not applicable, since the canonical\n \ labellings of a digraph are not portable between sessions.\n \ ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], name := "Alt", next := 487255, root := ~, start := 486853, stop := 487254 ), rec( content := "\n \ ", count := [ "A", 7, 0, 27 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 27 ], name := "Item", next := 487271, root := ~, start := 486836, stop := 487270 ) ], count := [ "A", 7, 0, 22 ], name := "Row", next := 487284, root := ~, start := 486660, stop := 487283 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 28 ], name := "HorLine", next := 487301, root := ~, start := 487291, stop := 487300 ), rec( attributes := rec( ), content := [ rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( content := "G\ raphIsomorphismClassRepresentatives", count := [ "A", 7, 0, 29 ], mathmode := \ "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 29 ], mathmode := "Text", name := "C", next := 487383, root := ~, start := 487340, stop := 487382 ), rec( content := "\n \ ", count := [ "A", 7, 0, 29 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 29 ], name := "Item", next := 487399, root := ~, start := 487323, stop := 487398 ), rec( attributes := rec( ), content := [ rec( content := "\n \ None\n ", count := [ "A", 7, 0, 31 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 31 ], name := "Item", next := 487445, root := ~, start := 487408, stop := 487444 ), rec( attributes := rec( ), content := [ rec( content := "\n \ ", count := [ "A", 7, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( Only := "HTML" ) , content := [ rec( HTML := true, content := "\ \n There is no analogous function in ", count := [ "A", 7, 0, 33 ], mathmode := \ "Text", name := "PCDATA", root := ~ ), rec( attributes := rec( ), content := [ rec( HTML\ := true, content := "Digraphs", count := [ "A", 7, 0, 33 ], mathmode := "Text\ ", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 33 ], mathmode := "T\ ext", name := "Package", next := 28, root := ~, start := 1, stop := 27 ), rec( HTML := true , content := "\ , although this feature\n might be implemented in the future.\n \ ", count := [ "A", 7, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 33 ], name := "Alt", next := 487633, root := ~, start := 487471, stop := 487632 ), rec( content := "\n \ ", count := [ "A", 7, 0, 33 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 33 ], name := "Item", next := 487649, root := ~, start := 487454, stop := 487648 ) ], count := [ "A", 7, 0, 28 ], name := "Row", next := 487662, root := ~, start := 487309, stop := 487661 ), rec( attributes := rec( ), content := 0, count := [ "A", 7, 0, 34 ], name := "HorLine", next := 487679, root := ~, start := 487669, stop := 487678 ) ], count := [ "A", 7, 0, 4 ], mathmode := "Text", name := "Table", next := 487693, root := ~, start := 485291, stop := 487692 ), rec( content := "\n ", count := [ "A", 7, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 7, 0, 1 ], mathmode := "Text", name := "Section", next := 487706, root := ~, start := 485037, stop := 487705 ), rec( content := "\n", count := [ "A", 7, 0, 35 ], mathmode := "Text", name := "PCDATA", root := ~ ) ], count := [ "A", 0, 0, 1 ], mathmode := "Text", name := "Appendix", next := 487718, root := ~, start := 454067, stop := 487717 ), rec( content := " Grape to Digraphs command map ", count := [ "A", 7, 0, 35 ], mathmode := "Text", name := "XMLCOMMENT", root := ~ ), rec( attributes := rec( Databases := "digraphs" ), content := 0, count := [ "Bib", 0, 0, 1 ], mathmode := "Text", name := "Bibliography", next := 487798, root := ~, start := 487761, stop := 487797 ), rec( attributes := rec( ), content := 0, count := [ "Ind", 0, 0, 1 ], mathmode := "Text", name := "TheIndex", next := 487810, root := ~, start := 487799, stop := 487809 ) ], count := [ 0, 0, 0, 1 ], name := "Book", next := 487819, root := ~, start := 1962, stop := 487818 ) ], count := [ 0, 0, 0, 1 ], indent := "", indextext := "< (for digraphs) 4.1
\n= \ (for digraphs) 4.1
\n<\ code class=\"func\">AdjacencyMatrix 5.2-1
\nAdjacencyMatrixMutableCopy 5.2-1 \nAdjacencyMatrixUpperTriangleLineDecoder 9.2-10 \nArticulati\ onPoints 5.3-14 \nAsBinaryRelation 3.2-1 \nAsDigraph 3.2-2 \nAsGraph 3.2-4 \ \nAsMonoid 5.5-1 \nAsSemigroup, for a filter\ and a digraph 5.5-1 \ \n    for a filter, semilattice digraph, and two lists 5.5-2 \nAsTransformation 3.2-5\ \nAutomorphismGroup, for a digraph 7.2-2 \n   \ ; for a digraph and a homogeneous list 7.2-5 \n    for a digraph, homogeneo\ us list, and list 7.2-6 \nBipartiteDoubleDigraph 3.3-37 \nBlissAutomorphi\ smGroup, for a digraph 7.2\ -3 \n    for a digraph and homogenous list 7.2-3 \n   \ for a digraph, homogenous list, and list 7.2-3 \nBlissCanonicalDigraph 7.2-9 \nBlissCanonicalLabelling, for a digraph 7.2-7 \n    for a digr\ aph and a list 7.2-8 \ \nBooleanAdjacencyMatrix 5.2-3 \nBooleanAdjacencyMa\ trixMutableCopy 5.2-3 \ \nBridges 5.3-15 \nCayleyDigraph 3.1-12 \nChainDigraph 3.5-1\ \nCharacteristicPolynomial 5.2-2 \nChromati\ cNumber 7.3-17 \ \nCliqueNumber 8.1-5 \nCliquesFinder 8.1-2 \n\ CompleteBipartiteDigraph 3\ .5-3 \nCompleteDigraph 3.5-2 \nCompleteM\ ultipartiteDigraph 3.5-4 \nCycleDigraph 3.5-5 \nDegreeMatrix 5.2-16 \nDigraph 3.1-7 \n    for a group, list, function, and function \ 3.1-7 \n  &nb\ sp; for a list and function \ 3.1-7 \nDigraph6String 9.2-3 \nDigraphAd\ dAllLoops 3.3-38 \nDigraphAddAllLoopsAttr 3.3-38 \nDigraphAddEdge<\ /code>, for a digraph and an edge 3.3-16 \n    for a digraph, source, and range \ 3.3-16 \nDigraphAddEdgeOrbit 3.3-17 \nDigraphAddEdges 3.3-18 \nDig\ raphAddVertex 3.3-14 \ \nDigraphAddVertices, for a digraph and a li\ st of labels 3.3-15 \n\     for a digraph and an integer 3.3-15 \nDigraphAdjacencyF\ unction 5.2-4 \ \nDigraphAllSimpleCircuits 5.3-24 \nDigraphBicompon\ ents 5.3-13 \n<\ code class=\"func\">DigraphByAdjacencyMatrix 3.1-8 \nDigraphByEdges 3.1-9 \nDigraphByInNeighbors 3.1-11 \nDigraphByInNeighbours 3.1-11 \nDigraphByOutNeighboursType 3.1-6 \nDigraphCartesianProduct, \ for a list of digraphs 3.3-30\ \n    for a positive number of digraphs 3.3-30 \nD\ igraphCartesianProductProjections 3.3-32 \nDigraphClique 8.1-3 \nD\ igraphCliques 8.1-4 <\ br />\nDigraphCliquesReps 8.1-4 \nDigraphClosure 3.3-40 \nDigraphColouring, for a digraph and a number of colours <\ a href=\"chap7_mj.html#X86D0A74B83D6B9A7\">7.3-14 \nDigraphConnectedComponent 5.3-10 \nDigraphConnectedComponents 5.3-9 \nDigraphCopy 3.3-\ 1 \nDigraphCopySameMutability 3.3-1 \nDi\ graphCore 7.3-18 \nDigraphDegeneracy 5.3-27 \nDigraphDegeneracyOrd\ ering 5.3-28 \n\ DigraphDiameter 5.3-1 \nDigraphDijkstra, for\ a source 5.3-31 \n&nb\ sp;   for a source and target 5.3-31 \nDigraphDirectProduct, for a list of digraphs 3.3-31\ \n    for a positive number of digraphs 3.3-31 \nDigraphDirectProductProjections 3.3-33 \nDigraphDisjointUnion,\ for a list of digraphs 3.3-27 \n    for an\n arbitrary number of digraphs <\ a href=\"chap3_mj.html#X814F1DFC83DB273F\">3.3-27 \nDigraphDistanceSet, for a digraph, a pos int, and a list 5.3-5 \n   &nbs\ p;for a digraph, a pos int, and an int 5.3-5 \nDigraphDual 3.3-10 \nDigra\ phDualAttr 3.3-10 \nDigraphEdgeLabel 5.1-11 \nDigraphEdgeLabels 5.1-12 \nDigraphEdges \ 5.1-3 \nDigraphEdgeUnion, for a list of\ digraphs 3.3-28 \n&nb\ sp;   for a positive number of digraphs 3.3-28 \nDigraphEmbeddin\ g 7.3-8 \nDigraphEpimorphism 7.3-6 \nDigraphFamily 3.1-6 \nD\ igraphFile 9.2-4 \nDigraphFloydWarshall 5.3-18 \nDigraphFromDigrap\ h6String 9.2-2 \ \nDigraphFromDiSparse6String 9.2-2 \nDigraphFromGra\ ph6String 9.2-2 \nDigraphFromPlainTextString 9.2-12 \nDigraphFromS\ parse6String 9.2-2 \nDigraphGirth 5.3-6 \nDigraphGreedyColouring, for a digraph 7.3-15 \ \n    for a digraph and vertex order 7.3-15 \n    for \ a digraph and vertex order function 7.3-15 \nDigraphGroup 7.2-10 \nDigrap\ hHasLoops 6.1-1 \nDigraphHomomorphism 7.3-2 \nDigraphImmutableCopy\ 3.3-1 \nDigraphImmutableCopyIfImmutable 3.3-2 \nDigraphImmutableC\ opyIfMutable 3.3-2 \nDigraphIndependentSet 8.2-2 \nDigraphIndepend\ entSets 8.2-3 \ \nDigraphIndependentSetsReps 8.2-3 \nDigraphInEdges\ 5.1-13 \nDigraphJoin, for a list of digraphs 3.3-29 \n    for a po\ sitive number of digraphs 3.3-29<\ /a> \nDigraphLayers 5.3-26 \nDigraphLongestD\ istanceFromVertex 5.3-4 \nDigraphLongestSimpleCircuit 5.3-25 \nDig\ raphLoops 5.2-14 \nDigraphMaximalClique 8.1-3 \nDigraphMaximalCliq\ ues 8.1-4 \nDigraphMaximalCliquesAttr 8.1-4 \nDigraphMaximalClique\ sReps 8.1-4 \n<\ code class=\"func\">DigraphMaximalCliquesRepsAttr 8.1-4
\nDigraphMaximal\ IndependentSet 8.2-2 \
\nDigraphMaximalIndependentSets 8.2-3
\nDigra\ phMaximalIndependentSetsAttr 8.2-3
\nDigraphMaximalIndependentSetsReps 8.2-3 \nDigraphMaximalIndependentSetsRepsAttr 8.2-3 \nDigraphMaxima\ lMatching 5.1-17 \nDigraphMaximumMatching 5.1-18 \nDigraphMonomorp\ hism 7.3-4 \nDigraphMutableCopy 3.3-1
\nDigraphMutableCopyIfImmuta\ ble 3.3-2
\nDigraphMutableCopyIfMutable 3.3-2
\nDigraphMycielskian\ 3.3-41
\nDigraphMycielskianAttr 3.3-41
\nDigraphNrConnectedCompon\ ents 5.3-9
\nDigraphNrEdges 5.1-4
\nDigraphNrLoops 5.1-5
\nD\ igraphNrStronglyConnectedComponents 5.3-11
\nDigraphNrVertices <\ a href=\"chap5_mj.html#X7C6F19B57CB2E882\">5.1-2
\nDigraphOddGirth 5.3-7\
\nDigraphOrbitReps 7.2-12
\nDigraphOrbi\ ts 7.2-11
\nDigraphOutEdges 5.1-14
\nDigraphPath 5.3-21
\nD\ igraphPeriod 5.3-17 <\ br />\nDigraphPlainTextLineDecoder 9.2-8
\nDigraphP\ lainTextLineEncoder 9.2-8<\ /a>
\nDigraphRange 5.2-5
\nDigraphReflexiveT\ ransitiveClosure 3.3-12
\nDigraphReflexiveTransitiveClosureAttr \ 3.3-12
\nDigraphReflexiveTransitiveReduction 3.3-13
\nDigraphReflexiveTran\ sitiveReductionAttr 3.3-13\
\nDigraphRemoveAllMultipleEdges 3.3-25
\nDigraphRemoveAllMultipleEdgesAttr 3.3-25
\nDigraphRemoveEdge, \ for a digraph and an edge 3.3-21<\ /a>
\n    for a digraph, source, and range 3.3-21
\n\ DigraphRemoveEdgeOrbit 3.3\ -22
\nDigraphRemoveEdges 3.3-23
\nDigrap\ hRemoveLoops 3.3-24 <\ br />\nDigraphRemoveLoopsAttr 3.3-24
\nDigraphRemov\ eVertex 3.3-19
\ \nDigraphRemoveVertices 3.3-20
\nDigraphReverse3.3-9 \nDigraphReverseAttr 3.3-9 \nDigraphReverseEdge, for a d\ igraph and an edge 3.3-26 \n    for a digraph, source, and range 3.3-26 \nDigraph\ ReverseEdges, for a digraph and a list of edges 3.3-26 \nDigraphs package overview 1. \nDigraphSchreierVector 7.2-13 \nDigraphShortestD\ istance, for a digraph and a list 5.3-2 \n    for a digraph and two verti\ ces 5.3-2 \n &nbs\ p;  for a digraph, a list, and a list 5.3-2 \nDigraphShortestDistance\ s 5.3-3 \nDigraphShortestPath 5.3-22
\nDigraphSinks 5.1-6
\n\ DigraphsMakeDoc 2.4-1 \
\nDigraphSource 5.2-5
\nDigraphSources5.1-7 \nDigraphsRespectsColouring 7.3-12 \nDigraphStabilizer 7.2-14 \nDigraphsTestExtreme 2\ .5-3 \nDigraphsTestInstall 2.5-1 \nDigra\ phsTestStandard 2.5-2 \ \nDigraphStronglyConnectedComponent 5.3-12 \nDigraphStronglyConnectedComponents 5.3-11 \nDigraphsUseBliss 7.2-1 \nDigraphsUseNauty 7.2-1\ \nDigraphSymmetricClosure 3.3-11 \nDigr\ aphSymmetricClosureAttr 3.\ 3-11 \nDigraphTopologicalSort 5.1-8 \nDi\ graphTransitiveClosure 3.3\ -12 \nDigraphTransitiveClosureAttr 3.3-12 \nDigraphTransitiveReduction 3.3-13 \nDigraphTransitiveReductionAttr 3.3-13 \nDigraphUndirectedGirth 5.3-8 \nDigraphVertexLabel \ 5.1-9 \nDigraphVertexLabels \ 5.1-10 \nDigraphVertices 5.1-1 \nDigraph\ WelshPowellOrder 7.3-16 \nDiSparse6String 9.2-3 \nDistanceDigraph<\ /code>, for digraph and int 3.3-3\ 9 \n    for digraph and list 3.3-39 \nDotColored\ Digraph 9.1-2 \ \nDotDigraph 9.1-2 \nDotEdgeColoredDigraph \ 9.1-2 \nDotHighlightedDigraph 9.1-6 \nDotPartialOrderDigraph 9.1-4 \nDotPreorderDigraph 9.1-\ 5 \nDotQuasiorderDigraph 9.1-5 \nDotSymm\ etricColoredDigraph 9.1-3<\ /a> \nDotSymmetricDigraph 9.1-3 \nDotSymmetr\ icEdgeColoredDigraph 9.1-3\ \nDotSymmetricVertexColoredDigraph 9.1-3 \nDotVertexColoredDigraph 9.1-2 \nDotVertexLabelledDigraph 9.1-2 \nDoubleDigraph 3.3-36 \nEdgeDigraph 3.3-34 \nEdgeOrbitsDigraph<\ /code> 3.1-10 \nEdgeUndirectedDigraph 3.3-35 \nEmbeddingsDigraphs \ 7.3-9 \nEmbeddingsDigraphsRepresentatives 7.3-9 \nEmptyDigraph 3.5-6 \nEpimorphismsDigraphs 7\ .3-7 \nEpimorphismsDigraphsRepresentatives7.3-7 \nGeneralisedPetersenGraph 3.5-9 \nGeneratorsOfCayleyDigraph 5.4-2 \nGeneratorsOfEndomorphismMonoid 7.3-13 \nGeneratorsOfEndomorp\ hismMonoidAttr 7.3-13 \ \nGraph 3.2-3 \nGraph6String 9.2-3 \nG\ roupOfCayleyDigraph 5.4-1<\ /a> \nHamiltonianPath 5.3-29 \nHomomorphismD\ igraphsFinder 7.3-1 <\ br />\nHomomorphismsDigraphs 7.3-3 \nHomomorphismsD\ igraphsRepresentatives 7.3\ -3 \nInDegreeOfVertex 5.2-12 \nInDegrees\ 5.2-9 \nInDegreeSequence 5.2-9 \nInDegreeSet 5.2-9 \nInduce\ dSubdigraph 3.3-3 \nInNeighbors 5.2-7 \nInNeighborsMutableCopy 5.2-7 \nInNeighborsOfVertex 5.2-13 \nInNeighbours 5.2-7 \nInNei\ ghboursMutableCopy 5.2-7 \nInNeighboursOfVertex 5.2-13 \nIsAcyclic\ Digraph 6.3-1 \ \nIsAntisymmetricDigraph 6.1-2 \nIsAntiSymmetricDig\ raph 6.1-2 \nIsAperiodicDigraph 6.3-7 \nIsBiconnectedDigraph 6.3-4 \nIsBipartiteDigraph 6.1-3 \nIsBridgelessDigraph 6.3-5 \nI\ sCayleyDigraph 3.1-4 \ \nIsChainDigraph 6.3-2 \nIsClique 8.1-1 \nIsCompleteBipartiteDigraph 6.1-4 \nIsCompleteDigraph 6.1-5 \nIs\ CompleteMultipartiteDigraph 6.1-6 \nIsConnectedDigraph 6.3-3 \nIs\ CycleDigraph 6.3-12 <\ br />\nIsDigraph 3.1-1 \nIsDigraphAutomorphism 7.2-20 \n &nbs\ p;  for a digraph and a transformation or permutation 7.2-20 \nIsDigr\ aphColouring 7.2-21 <\ br />\n    for a transformation 7.2-21 \nIsDigraphCore 6.5-1 \nIsDigraphEdge, for digraph and list 5.1-15 \n    for digraph an\ d two pos ints 5.1-15 \ \nIsDigraphEmbedding 7.3-11 \n    for digraphs and a\ permutation or transformation 7.\ 3-11 \nIsDigraphEndomorphism 7.3-10 \n   &nbs\ p;for digraphs and a permutation or transformation 7.3-10 \nIsDigraphEpimorphis\ m 7.3-10 \n&nbs\ p;   for digraphs and a permutation or transformation 7.3-10 \nI\ sDigraphHomomorphism 7.3-1\ 0 \n    for digraphs and a permutation or trans\ formation 7.3-10 \nIsDigraphIsomorphism 7.2-20 \n    for digraphs and tran\ sformation or permutation 7.2-20<\ /a> \nIsDigraphMonomorphism 7.3-10 \n    for \ digraphs and a permutation or transformation 7.3-10 \nIsDigraphWithAdjacencyFun\ ction 3.1-5 \n<\ code class=\"func\">IsDirectedTree 6.3-8 \nIsDistanceRegularDigraph 6.2-4 \nIsEdgeTransitive 6.5-2 \nIsEmptyDigraph 6.1-7 \nIsEquiva\ lenceDigraph 6.1-8 \nIsEulerianDigraph 6.3-10 \nIsFunctionalDigrap\ h 6.1-9 \nIsHamiltonianDigraph 6.3-11 \nIsImmutableDigraph \ 3.1-3 \nIsIndependentSet 8.\ 2-1 \nIsInRegularDigraph 6.2-1 \nIsIsomo\ rphicDigraph, for digraphs 7.2-15 \n    for digraphs and homogeneous list\ s 7.2-16 \nIsJoinSemilatticeDigraph 6.1-17 \nIsLatticeDigraph 6.1-17 \nIsMatching 5.1-16 \ \nIsMaximalClique 8.1-1 \nIsMaximalIndepende\ ntSet 8.2-1 \n<\ code class=\"func\">IsMaximalMatching 5.1-16 \nIsMaximumMatching \ 5.1-16 \nIsMeetSemilatticeDigraph 6.1-17 \nIsMultiDigraph 6.1-10 \nIsMutableDigraph 3.1-2 \nIsNullDigraph 6.1-7 \nIsomorphismDigraph\ s, for digraphs 7.2-17\ \n    for digraphs and homogeneous lists 7.2-18 \nI\ sOuterPlanarDigraph 6.4-2<\ /a> \nIsOutRegularDigraph 6.2-2 \nIsPartialO\ rderDigraph 6.1-16 \nIsPerfectMatching 5.1-16 \nIsPlanarDigraph 6.4-1 \nIsPreorderDigraph 6.1-15 \nIsQuasiorderDigraph 6.1-15 \nIsReachable 5.3-19 \ \nIsReflexiveDigraph 6.1-11 \nIsRegularDigra\ ph 6.2-3 \nIsStronglyConnectedDigraph 6.3-6 \nIsSubdigraph \ 4.1-1 \nIsSymmetricDigraph \ 6.1-12 \nIsTournament 6.1-13 \nIsTransit\ iveDigraph 6.1-14 \nIsUndirectedForest 6.3-9 \nIsUndirectedSpannin\ gForest 4.1-2 \ \nIsUndirectedSpanningTree 4.1-2 \nIsUndirectedTree\ 6.3-9 \nIsVertexTransitive 6.5-3 \nIteratorFromDigraphFile\ 9.2-7 \nIteratorOfPaths 5\ .3-23 \nJohnsonDigraph 3.5-7 \nKuratowsk\ iOuterPlanarSubdigraph 5.6\ -2 \nKuratowskiPlanarSubdigraph 5.6-1 \n\ LaplacianMatrix 5.2-17\ \nLineDigraph 3.3-34 \nLineUndirectedDigrap\ h 3.3-35 \nMaximalAntiSymmetricSubdigraph 3.3-6 \nMaximalAntiSymme\ tricSubdigraphAttr 3.3-6 \nMaximalSymmetricSubdigraph 3.3-5 \nMaxi\ malSymmetricSubdigraphAttr 3.3-5 \nMaximalSymmetricSubdigraphWithoutLoop\ s 3.3-5 \nMaximalSymmetricSubdigraphWithoutLoopsAttr 3.3-5 \nMonom\ orphismsDigraphs 7.3-5\ \nMonomorphismsDigraphsRepresentatives 7.3-5 \nNautyAutomorphismGroup 7.2-4 \nNautyCanonicalDigraph 7.2-9 \nN\ autyCanonicalLabelling, for a digraph 7.2-7 \n    for a digraph and a lis\ t 7.2-8 \nNrSpanningTrees 5\ .3-30 \nNullDigraph 3.5-6 \nOnDigraphs, for a digraph and a perm 7\ .1-1 \n    for a digraph and a transformation \ 7.1-1 \nOnMultiDigraphs 7.1-\ 2 \n    for a digraph, perm, and perm 7.1-2 \nOu\ tDegreeOfVertex 5.2-10\ \nOutDegrees 5.2-8 \nOutDegreeSequence 5.2-8 \nOutDegreeSet 5.2\ -8 \nOuterPlanarEmbedding 5.6-4 \nOutNei\ ghbors 5.2-6 \n\ OutNeighborsMutableCopy 5.2-6 \nOutNeighborsOfVerte\ x 5.2-11 \nOutNeighbours 5.2-6 \nOutNeighboursMutableCopy \ 5.2-6 \nOutNeighboursOfVertex 5.2-11 \nPartialOrderDigraphJoinOfVertices<\ /code> 5.2-15 \nPartialOrderDigraphMeetOfVertices 5.2-15 \nPetersenGraph<\ /code> 3.5-8 \nPlainTextString 9.2-12 \nPlanarEmbedding 5.6-3 \nPr\ intString 9.2-1 \nQuotientDigraph 3.3-8 \nRandomDigraph 3.4-1 \nRandomLattice 3.4-4 \nRandomMultiDigraph 3.4-2 \nRandomTournam\ ent 3.4-3 \nReadDigraphs 9.2-5 \nReadDIMACSDigraph 9.2-14 \n\ ReadPlainTextDigraph 9.2-1\ 3 \nReducedDigraph 3.3-4 \nReducedDigrap\ hAttr 3.3-4 \n<\ code class=\"func\">RepresentativeOutNeighbours 7.2-19 \nSemigroupOfCayl\ eyDigraph 5.4-1 \nSetDigraphEdgeLabel 5.1-11 \nSetDigraphEdgeLabel\ s, for a digraph and a function 5.1-12 \n    for a digraph and a list of \ lists 5.1-12 \nSetDigraphVertexLabel 5.1-9 \nSetDigraphVertexLabels 5.1-10 \nSparse6String 9.\ 2-3 \nSplash 9.1-1 \nString 9.2-1 \nStrongOrientation 5.3-1\ 6 \nStrongOrientationAttr 5.3-16 \nSubdi\ graphHomeomorphicToK23 5.6\ -5 \nSubdigraphHomeomorphicToK33 5.6-5 \nSubdigraphHomeomorphicToK4 5.6-5 \nTCodeDecoder 9.2-11 \nTournam\ entLineDecoder 9.2-9 \ \nUndirectedSpanningForest 3.3-7 \nUndirected\ SpanningForestAttr 3.3-7 \nUndirectedSpanningTree 3.3-7 \nUndirect\ edSpanningTreeAttr 3.3-7 \nVerticesReachableFrom 5.3-20 \nWriteDig\ raphs 9.2-6 \n<\ code class=\"func\">WriteDIMACSDigraph 9.2-14 \nWritePlainTextDigraph 9.2-13 \n", input := "Digraphs\">\n http://www.maths.qmul.ac.uk/~leonard/grape/Grape\">\n http://www.tcs.tkk.fi/Software/bliss/bliss\">\n https://github.com/graph-algorithms/edge-addition-planarity-suite<\ /URL>edge-addition-planarity-suite\">\n h\ ttp://pallini.di.uniroma1.it/nauty\">\n https://github.com/gap-packages/NautyTracesInte\ rfaceNautyTracesInterface<\ /Alt>\">\n \n http://gap-\ packages.github.io/io/\nIO\">\n \n http://http:\ //gap-packages.github.io/orb/\n Orb<\ /Package>\">\n https://github.com/gap-packages/datastructuresdatastructures\">\n \n\n\n\n\n\n\n\n\n\n ]>\n\n\n\n &Digraphs\ ;\n Version &VERSION;\n\n Jan De Beule\n \ jdebeule@cage.ugent.be\n http://homepages.vub.a\ c.be/~jdbeule\n \n\n Julius Jonušas\n julius.jonusas@tuwien.ac.at\n http://julius.jonusas.\ work\n \n\n James D. Mitchell\n jdm3@s\ t-and.ac.uk\n https://jdbm.me\n \n\n\ Michael Torpey\n mct25@st-andrews.ac.uk\n https://mtorpey.github.io\n \n\n Wilf A. Wi\ lson\n gap@wilf-wilson.net\n http://wilf.me\n \n\n Stuart Burrell\n\n Reinis \ Cirpons\n\n Luke Elliott\n\n Max Horn\n\n Christopher Jefferson\n\n Markus Pfeiffer<\ /Author>\n\n Chris Russell\n\n Finn Smith\ \n\n Murray White\n\n ©right; ©RIGHTYEAR\ S; by\n Jan De Beule, Julius Jonušas, James D. Mitchell, Michael\n \ Torpey, Wilf A. Wilson et al.\n\n &Digraphs; is free software; you can \ redistribute it and/or modify it\n under the terms of the \n http://www.fsf.org/licenses/gpl.html as pu\ blished by the Free Software\n Foundation; either version 3 of the License,\ or (at your option) any later\n version.\n \n\n \n\ The &Digraphs; package is a &GAP; package containing methods for graphs,\n\ digraphs, and multidigraphs.\n \n\n \n W\ e would like to thank Christopher Jefferson for his help in including \n &b\ liss; in &Digraphs;.\n\n This package's methods for computing digraph homom\ orphisms are based on work\n by Max Neunhöffer, and independently Artu\ r Schäfer.\n \n\n\n\n\n\ \n\n\n \n \n The &\ Digraphs; package\n \n &Digraphs; package \ overview\n\n Introduction\n\n This is the \ manual for version &VERSION; of the &Digraphs; package.\n This package was de\ veloped at the University of St Andrews by:\n \n Jan De Beule,\ \n Julius Jonušas,\n James D. Mitchell,\n Michael C. Torpey, and\n Wilf A. Wilson.\n \n Additional contributions were made by:\n \n S\ tuart Burrell,\n Reinis Cirpons,\n Luke Elliot\ t,\n Max Horn,\n Christopher Jefferson,\ \n Markus Pfeiffer,\n Chris Russell,\n Finn Smith, and\n Maria Tsalakou,\n Murray W\ hyte.\n \n\n The &Digraphs; package contains a variety of meth\ ods for efficiently creating\n and storing mutable and immutable digraphs and\ computing information about\n them. Full explanations of all the function\ s contained in the package are\n provided below. \n\n If the &Grape; p\ ackage is available, it will be loaded automatically.\n Digraphs created with\ the &Digraphs; package can be converted to &Grape;\n graphs with , and conversely &Grape; graphs can be\n converted to &Digraphs; o\ bjects with . &Grape; is not\n required for &Digraphs\ ; to run. \n\n The &bliss; tool is included in this \ package. It\n is an open-source tool for computing automorphism groups and c\ anonical forms\n of graphs, written by Tommi Junttila and Petteri Kaski. Seve\ ral of the methods\n in the &Digraphs; package rely on &bliss;. If the &Nauty\ TracesInterface;\n package for GAP is available then it is also possible to u\ se &nauty; for computing automorphism groups and can\ onical forms\n in &Digraphs;. See Section \n for more details. \n \n\n From version 1.0.0 of \ this package, digraphs can be either mutable or immutable.\n Mutable digraphs\ can be changed in-place by many of the methods in the\n package, which avoid\ s unnecessary copying. Immutable digraphs cannot be\n changed in-place, but t\ heir advantage is that the value of an attribute of\n an immutable digraph is\ only ever computed once. Mutable digraphs can be\n converted into immutable \ digraphs in-place using . \ One of the motivations for introducing mutable\n digraphs in version 1.0.0 wa\ s that in practice the authors often wanted\n to create a digraph and immedia\ tely modify it (removing certain edges,\n loops, and so on). Before version 1\ .0.0, this involved copying the digraph\n several times, with each copy being\ discarded almost immediately.\n From version 1.0.0, this unnecessary copying\ can be eliminated by\n first creating a mutable digraph, then changing it in\ -place, and finally\n converting the mutable digraph to an immutable one in-\ place\n (if desirable).\n\n Defin\ itions\n\n For the purposes of this package and its documentation\ , the following\n definitions apply: \n\n A digraph E=(E^0\ ,E^1,r,s), also known as a directed\n graph, consists of a set\ of vertices E^0 and a set of edges\n E^1 together with funct\ ions s, r: E^1 \\to E^0, called the\n source and range\ , respectively.\n\n The source and range of an edge is respectively the val\ ues of s, r at\n that edge.\n\n An edge is called a loop i\ f its source and range are the same.\n\n A digraph is called a multidigr\ aph if there exist two or more edges\n with the same source and the sam\ e range. \n\n A directed walk on a digraph is a sequence of alte\ rnating vertices\n and edges (v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)\ such that each edge\n e_i has source v_i and range v_{i+1\ }. A directed\n path is a directed walk where no vertex (and h\ ence no edge) is\n repeated. A directed circuit is a directed walk \ where v_1 =\n v_n, and a directed cycle is a directed circu\ it where where no\n vertex is repeated, except for v_1 = v_n. \ \n\n The length of a directed walk (v_1, e_1, v_2, e_2, ..., e_{n\ -1},\n v_n) is equal to n-1, the number of edges it contains. \ A\n directed walk (or path) (v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)\ is\n sometimes called a directed walk (or path) from vertex v_1 \ to\n vertex v_n. A directed walk of zero length, i.e. a seque\ nce\n (v) for some vertex v, is called trivial. A triv\ ial\n directed walk is considered to be both a circuit and a cycle, as is t\ he\n empty directed walk (). A simple circuit is another name\ for a\n non-trivial and non-empty directed cycle. \n\n \n\ \n\n \n \n Installing &Digraphs;\n\n \n For those in a hurry\n In this s\ ection we give a brief description of how to start using\n &Digraphs;.\ \n\n It is assumed that you have a working copy of &GAP; with version numbe\ r\n &GAPVERS; or higher. The most up-to-date version of &GAP; and instructi\ ons\n on how to install it can be obtained from the main &GAP; webpage\n \ http://www.gap-system.org. \n\n The following is a summary o\ f the steps that should lead to a\n successful installation of &Digraphs;:\ \n\n \n\n \n ensure that the &IO; package version &\ IOVERS; or higher is available.\n &IO; must be compiled before &Digraph\ s; can be loaded.\n \n\n \n ensure that the &Orb\ ; package version &ORBVERS; or higher is available.\n &Orb; has better \ performance when compiled, but although compilation is\n recommended, i\ t is not required to be compiled for &Digraphs; to be\n loaded.\n \ \n\n ensure that the &datastructures; package version\n \ &DATASTRUCTURESVERS; or higher is available. \n \n\n \ \n This step is optional: certain functions in &Digraphs; requir\ e\n the &Grape; package to be available; see Section\n for full details. To use these\n functions\ make sure\n that the &Grape; package version &GRAPEVERS; or higher is \ available.\n If &Grape; is not available, then &Digraphs; can be used a\ s normal with\n the exception that the functions listed in Subsection\n\ will not work.\n \n \ \n \n This step is optional: certain functions in\ &Digraphs; require\n the &NautyTracesInterface; package to be availa\ ble.\n \n If you want to make use of the\ se functions, please ensure that the\n &NautyTracesInterface; package v\ ersion &NAUTYTRACESINTERFACEVERS; or\n higher is available. If &NautyT\ racesInterface; is not available,\n then &Digraphs; can be used as norm\ al with the exception that \n functions whose names contain Nauty<\ /Q> will not work.\n \n\n \n download the packag\ e archive &ARCHIVENAME;.tar.gz from\n http://digraphs.github.io/Digraphs/.\n \n\n \ \n unzip and untar the file, this should create a directory \ called\n &ARCHIVENAME;.\n \n\n \n \ locate the pkg directory of your &GAP; directory, which contains\n \ the directories lib, doc and so on. Move the directory\n \ &ARCHIVENAME; into the pkg directory.\n \n\n \ \n it is necessary to compile the &Digraphs; package.\n I\ nside the pkg/&ARCHIVENAME; directory, type\n \n\n Further information about this step \ can be found in Section\n .\ \n \n\n \n start &GAP; in the usual way (i.e. ty\ pe gap at the command line).\n \n\n \n ty\ pe LoadPackage(\"digraphs\");\n \n \n\n If you\ want to check that the package is working correctly, you\n should run some\ of the tests described in Section\n .\n \n\n \n \n\n \n Optional package dependencies\n\n \n\n The &Di\ graphs; package is written in &GAP; and C code and requires the\n &IO; pack\ age. The &IO; package is used to read and write transformations,\n partial\ permutations, and bipartitions to a file. \n\n \n\n \n The Grape package\n\n Th\ e &Grape; package must be available for the following operations to be\n \ available:\n\n \n with a digraph\ argument\n with a digraph argumen\ t\n with a &Grape; graph argument<\ /Item>\n \n\n If &Grape; is not available, then &Digraphs; ca\ n be used as normal with\n the exception that the functions above will no\ t work.\n \n \n\n \n \n\n \n Compiling the kernel module\n \ The &Digraphs; package has a &GAP; kernel component in\n C which should \ be compiled. This component contains certain low-level\n functions required\ by &Digraphs;.\n \n\n It is not possible to use the &Digraphs; pack\ age without compiling it. \n\n To compile the kernel component inside th\ e pkg/&ARCHIVENAME;\n directory, type\n \n\n \n\n If you installed the package in anoth\ er 'pkg' directory than the standard 'pkg'\n directory in your &GAP; instal\ lation, then you have to do two things. Firstly\n during compilation you ha\ ve to use the option '--with-gaproot=PATH' of the\n 'configure' script wher\ e 'PATH' is a path to the main GAP root directory (if\n not given the defau\ lt '../..' is assumed). \n\n If you installed &GAP; on several architect\ ures, you must execute the\n configure/make step for each of the architectu\ res. You can either do this\n immediately after configuring and compiling G\ AP itself on this architecture, or\n alternatively set the environment\n \ variable 'CONFIGNAME' to the name of the configuration you used when compilin\ g\n GAP before running './configure'. Note however that your compiler choi\ ce and\n flags (environment variables 'CC' and 'CFLAGS') need to be chosen \ to match the\n setup of the original GAP compilation. For example you have \ to specify 32-bit\n or 64-bit mode correctly!\n \n\n \n \n\n \n Rebuilding the do\ cumentation\n The &Digraphs; package comes complete with pdf, htm\ l, and text versions of\n the documentation. However, you might find it nec\ essary, at some point, to\n rebuild the documentation. To rebuild the docum\ entation, please use the\n function .\n\n \ \n \n Nothing\ \n \n This function should be called with no argume\ nt to compile the &Digraphs;\n documentation.\n \n\n\n \n\n \n \n\n \ \n Testing your installation\n\n In this section we d\ escribe how to test that &Digraphs; is working as\n intended. To test that\ &Digraphs; is installed correctly use\n or for more extensive tests use\n . <\ P/>\n\n If something goes wrong, then please review the instructions in Sec\ tion and ensure that &Digraphs; ha\ s been\n properly installed. If you continue having problems, please use t\ he https://github.com/digraphs/Digraphs/iss\ ues\n to report the issues you are having.\n\n \n \n true or false\ .\n \n This function can be called without argu\ ments to test your installation of\n &Digraphs; is working correctly. These\ tests should take no more than a\n few seconds to complete. To test more \ comprehensively that\n &Digraphs; is working correctly, use .\n \n\n\n \n \ \n true or \ false.\n \n This function can be called without\ arguments to test all of the methods\n included in &Digraphs;.\n These \ tests should take less than a minute to complete. \n\n To quickly test \ that &Digraphs; is installed correctly use . For a more thorough test, use \ .\n \n\n\n \n \n Nothing.\n \n \ This function should be called with no argument. It executes a series of\n \ very demanding tests, which measure the performance of a variety of\n func\ tions on large examples. These tests take a long time to\n complete, at le\ ast several minutes. \n\n For these tests to complete, the digraphs lib\ rary digraphs-lib must\n be downloaded and placed in the digraphs\ directory in a subfolder\n named digraphs-lib.\n This librar\ y can be found on the &Digraphs; website.\n \n\n\n\ \n \n\n \n \n\n\n\n \n Creating digraphs\n\n In this chapt\ er we describe how to create digraphs.\n\n Creating dig\ raphs\n \n \n \n Every digraph in &Digraphs; belongs to the categ\ ory\n IsDigraph. Some basic attributes and operations for digraphs\ \n are , , and\n\ .\n \n\n\n <\ ManSection>\n \n \n IsMutableDigraph is a synonym for a\ nd . A mutable digraph may be\ changed\n in-place by methods in the &Digraphs; package, and is not attrib\ ute-storing\n – see .\n \n\n A mutable digraph may be converted into an immutable attr\ ibute-storing\n digraph by calling on the\n digraph.\n \n\n\n \n \n \n\ IsImmutableDigraph is a subcategory of .\ \n Digraphs that lie in IsImmutableDigraph are immutable and\n at\ tribute-storing. In particular, they lie in\n . \n\n A mutable digraph may be converted to \ an immutable\n digraph that lies in the category IsImmutableDigraph \ by calling\n on the digraph.\ \n\n The operation can be used\n \ to construct a mutable copy of an immutable digraph. It is however not\n po\ ssible to convert an immutable digraph into a mutable digraph in-place.\n \n\n\n \n \n \n IsCayleyDigraph is a subcate\ gory of IsDigraph. Digraphs that\n are Cayley digraphs of a group an\ d that are constructed by the operation\n are\ constructed in this category, and are\n always immutable.\n \n\n\n \n \n \n IsDigraphWithAdjacencyFun\ ction is a subcategory of IsDigraph.\n Digraphs that are crea\ ted using an adjacency function are constructed\n in this category.\n \ \n\n\n \n \n \n \n The t\ ype of all digraphs is DigraphByOutNeighboursType.\n The family of a\ ll digraphs is DigraphFamily.\n \n\n\n <\ ManSection>\n \n \n \n A digraph.\n \n If the optional first argume\ nt filt is present, then this should\n specify the category or repre\ sentation the digraph being created will\n belong to. For example, if fi\ lt is ,\n then the digraph being create\ d will be mutable, if filt is , \ then the digraph will be immutable. \n If the optional first argument fi\ lt is not present, then is used by\ default.\n \n for a list (i.e. an adjacency list)\ \n \n if obj is a list of lists of positive integers \ in\n the range from 1 to Length(obj), then this fu\ nction\n returns the digraph with vertices\n E ^ 0 = [\ 1 .. Length(obj)],\n and edges corresponding to the entries \ of obj. \n\n More precisely, there is an edge from vertex i to j if and\n only if j is in obj[i]\ ; the source of this edge is\n i and the range is j. If <\ C>j occurs in\n obj[i] with multiplicity k, th\ en there are k\n edges from i to j.\n \ \n\n for three lists\n \n if obj is\ a duplicate-free list, and source and\n range are lists \ of equal length consisting of positive integers\n in the list [1 .. \ Length(obj)], then this function\n returns a digraph with ve\ rtices\n E ^ 0 = [1 .. Length(obj)], and\n \ Length(source) edges. For each i in\n [1 .. Le\ ngth(source)] there exists an edge with source\n vertex s\ ource[i] and range vertex range[i].\n See and .\n \n\n The ver\ tices of the digraph will be labelled by the elements of\n obj.\ \n \n\n for an integer, and two lists\n \n if obj is an integer, and source and\n ra\ nge are lists of equal length consisting of positive integers\n in \ the list [1 .. obj], then this function\n returns a digra\ ph with vertices\n E ^ 0 = [1 .. obj], and\n \ Length(source) edges. For each i in\n [1 .. \ Length(source)] there exists an edge with source\n vertex source[i] and range vertex range[i].\n See and .\n \n\n \ for a list and a function\n \n if list is a li\ st and func is a function taking 2\n arguments that are elements\ of list, and func returns\n true or false,\ then this operation creates a digraph with\n vertices [1 .. Length(\ list)] and an edge from vertex\n i to vertex j\ if and only if\n func(list[i], list[j]) re\ turns true.\n \n\n for a group, a list, and two \ functions\n \n The arguments will be G, list, act,\ adj. \n\n Let G be a group acting on the objects in \ list via\n the action act, and let adj be a function \ taking\n two objects from list as arguments and returning tru\ e or\n false. The function adj will describe the adja\ cency\n between objects from list, which is invariant under the \ action of\n G. This variant of the constructor returns a digraph\ with\n vertices the objects of list and directed edges [x, y\ ]\n when f(x, y) is true. \n\n The action \ of the group G on the objects in list is stored\n in the \ attribute , and is used to speed up\n opera\ tions like .\n \n\n for \ a Grape package graph\n \n if obj is a &Grape;\ package graph (i.e. a record for which the\n function IsGraph r\ eturns true), then this function returns\n a digraph isomorphic \ to obj.\n \n\n for a binary relation\n \ \n if obj is a binary relation on the points [1 .. n\ ] for\n some posititve integer n, then this function returns\ the digraph\n defined by obj. Specifically, this function retur\ ns a digraph\n which has n vertices, and which has an edge with \ source i\n and range j if and only if [i,j] is a p\ air in\n the binary relation obj.\n \n \n\ \n gr := Digraph([\n> [2, 5, 8, 10], [2, 3, 4, 2, \ 5, 6, 8, 9, 10], [1],\n> [3, 5, 7, 8, 10], [2, 5, 7], [3, 6, 7, 9, 10], [1, 4]\ ,\n> [1, 5, 9], [1, 2, 7, 8], [3, 5]]);\n\ngap> gr := Digraph([\"a\", \"b\", \"c\"], [\"a\"], [\"b\"]);\n\ \ngap> gr := Digraph(5, [1, 2, 2, 4\ , 1, 1], [2, 3, 5, 5, 1, 1]);\n\ngap> Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function(x,\ y) return Intersection(x, y) = []; end);;\ngap> Digraph(Petersen);\n\ngap> gr := Digraph([1 .. 10], ReturnTru\ e);\n]]>\n\n The\ next example illustrates the uses of the fourth and fifth variants of\n \ this constructor. The resulting digraph is a strongly regular graph, and\n \ it is actually the point graph of the van Lint-Schrijver partial\n geom\ etry, . The algebraic description\n is taken from t\ he seminal paper of Calderbank and Kantor .\n f := GF(3 ^ 4);\nGF(3^4)\ngap> gamma := First(f, x -> Ord\ er(x) = 5);\nZ(3^4)^64\ngap> L := Union([Zero(f)], List(Group(gamma)));\n[ 0*Z\ (3), Z(3)^0, Z(3^4)^16, Z(3^4)^32, Z(3^4)^48, Z(3^4)^64 ]\ngap> omega := Union\ (List(L, x -> List(Difference(L, [x]), y -> x - y)));\n[ Z(3)^0, Z(3), Z(3^4)^\ 5, Z(3^4)^7, Z(3^4)^8, Z(3^4)^13, Z(3^4)^15, \n Z(3^4)^16, Z(3^4)^21, Z(3^4)^\ 23, Z(3^4)^24, Z(3^4)^29, Z(3^4)^31, \n Z(3^4)^32, Z(3^4)^37, Z(3^4)^39, Z(3^\ 4)^45, Z(3^4)^47, Z(3^4)^48, \n Z(3^4)^53, Z(3^4)^55, Z(3^4)^56, Z(3^4)^61, Z\ (3^4)^63, Z(3^4)^64, \n Z(3^4)^69, Z(3^4)^71, Z(3^4)^72, Z(3^4)^77, Z(3^4)^79\ ]\ngap> adj := function(x, y)\n> return x - y in omega;\n> end;\nfunction( \ x, y ) ... end\ngap> digraph := Digraph(AsList(f), adj);\n\ngap> group := Group(Z(3));;\ngap> act := \\*;\n<\ Operation \"*\">\ngap> digraph := Digraph(group, List(f), act, adj);\n\n]]>\n \n\n\n \n \n A digraph.\n \n If t\ he optional first argument filt is present, then this should\n speci\ fy the category or representation the digraph being created will\n belong t\ o. For example, if filt is ,\n then \ the digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the optio\ nal first argument filt is not present, then is used by default. \n\n If list is the adjacency m\ atrix of a digraph in the sense of\n , then \ this operation returns the digraph\n which is defined by list. \ \n\n Alternatively, if list is a square boolean matrix, then this\n \ operation returns the digraph with Length(list)\n v\ ertices which has the edge [i,j] if and only if\n list[i][\ j] is true. \n\n DigraphByAdjacency\ Matrix([\n> [0, 1, 0, 2, 0],\n> [1, 1, 1, 0, 1],\n> [0, 3, 2, 1, 1],\n> [0, 0,\ 1, 0, 1],\n> [2, 0, 0, 0, 0]]);\n\ngap> D := DigraphByAdjacencyMatrix([\n> [true, false, true],\n> [false\ , false, true],\n> [false, true, false]]);\n\ngap> OutNeighbours(D);\n[ [ 1, 3 ], [ 3 ], [ 2 ] ]\ngap> D := Digr\ aphByAdjacencyMatrix(IsMutableDigraph, \n> [[true, false, true],\n> [false, f\ alse, true],\n> [false, true, false]]);\n\n]]>\n \n\n\n \n \ \n A digrap\ h.\n \n If the optional first argument filt \ is present, then this should\n specify the category or representation the d\ igraph being created will\n belong to. For example, if filt is ,\n then the digraph being created will be mutab\ le, if filt is , then the digrap\ h will be immutable. \n If the optional first argument filt is not p\ resent, then is used by default. \n\ \n If list is list of pairs of positive integers, then this function\ \n returns the digraph with the minimum number of vertices m such th\ at\n its list equal list. \n\n If the optional second argument\ n is a positive integer with\n n >= m (with m \ defined as above),\n then this function returns the digraph with n v\ ertices and list\n list. \n\n See .\n\n DigraphByEdges(\n> [[1, 3], [2, 1], [2, 3],\ [2, 5], [3, 6],\n> [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]]);\n\ngap> DigraphByEdges(\n> [[1, 3], [2, 1], [2, \ 3], [2, 5], [3, 6],\n> [4, 6], [5, 2], [5, 4], [5, 6], [6, 6]], 12);\n\ngap> DigraphByEdges(IsMutableDigraph,\ \n> [[1, 3], [2, 1], [2, 3], [2, 5], [3, 6],\n> [4, 6], [5, 2], [5, 4], [5, \ 6], [6, 6]], 12);\n\n]]>\ \n \n\n\n \n \n \n An immutable digraph.\n \ \n \n If G is a permutation group, edges\ is an edge or list of\n edges, and n is a non-negative integer \ such that G fixes\n [1 .. n] setwise, then\n this o\ peration returns an immutable digraph with n vertices and the union\n \ of the orbits of the edges in edges under the action of the\n pe\ rmutation group G. An edge in this context is simply a pair of\n po\ sitive integers.\n \n\n If the optional third argument n is n\ ot present, then the largest\n moved point of the permutation group G is used by default. \n\n digraph := EdgeOrbit\ sDigraph(Group((1, 3), (1, 2)(3, 4)),\n> [[1, \ 2], [4, 5]], 5);\n\ngap> OutNeigh\ bours(digraph);\n[ [ 2, 4, 5 ], [ 1, 3, 5 ], [ 2, 4, 5 ], [ 1, 3, 5 ], [ ] ]\ \ngap> RepresentativeOutNeighbours(digraph);\n[ [ 2, 4, 5 ], [ ] ]\n]]>\n \n\n\n \n \n \n A digraph.\n \n\ If the optional first argument filt is present, then this should\n s\ pecify the category or representation the digraph being created will\n belong\ to. For example, if filt is ,\n then \ the digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the optional \ first argument filt is not present, then is used by default. \n\n If list is a list of lists of posi\ tive integers list the range\n [1 .. Length(list)], then this f\ unction returns the\n digraph with vertices E^0=[1 .. Length(lis\ t)], and\n edges corresponding to the entries of list. More pr\ ecisely, there is an\n edge with source vertex i and range vertex j\ if i is list\n list[j]. \n\n If i occur\ s list list[j] with multiplicity k,\n then there are \ k multiple edges from i to j. \n\n See .\n\n D := DigraphByInNeighbours([\n> \ [2, 5, 8, 10], [2, 3, 4, 5, 6, 8, 9, 10],\n> [1], [3, 5, 7, 8, 10], [2, 5, 7],\ [3, 6, 7, 9, 10], [1, 4],\n> [1, 5, 9], [1, 2, 7, 8], [3, 5]]);\n\ngap> D := DigraphByInNeighbours([[2, 3, 2]\ , [1], [1, 2, 3]]);\n\ngap> D\ := DigraphByInNeighbours(IsMutableDigraph, \n> \ [[2, 3, 2], [1], [1, 2, 3]]);\n\ \n]]>\n \n\n\n \n \n An immutable digraph.\n \n Let G be any group and let gens b\ e a list of elements of\n G. This operation returns an immutable dig\ raph that corresponds to\n the Cayley graph of G with respect\n <\ A>gens. The vertices are the elements of G. There exists an edge\n \ from the vertex u to the vertex v if and only if there exists\ \n a generator g in gens such that x * g = y. \n\n\ \n\ \n If the optional second argument gens is not present, then the\n \ generators of G are used by default. \n\n The digraph created b\ y this operation belongs to the category , the group G can be recovered from the\n digraph using , and the generators\n gens can be obtaine\ d using . \n\n Note that this f\ unction can only return an immutable digraph.\n\n \ G := DihedralGroup(8);\n\ngap> CayleyDig\ raph(G);\n\ngap> G := DihedralGro\ up(IsPermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> CayleyDigraph(G);\n\ngap> digraph := CayleyDigraph(G, [(\ )]);\n\ngap> GroupOfCayleyDigraph(\ digraph) = G;\ntrue\ngap> GeneratorsOfCayleyDigraph(digraph);\n[ () ]]]>\n \n\n\n \n\n C\ hanging representations\n \n \n A binary relation.\n \n If digraph is a digraph with a positive number of vertices\ \n n, and no multiple edges, then this operation returns a binary\n \ relation on the points [1..n].\n The pair [i,j] is in the \ binary relation if and only if\n [i,j] is an edge in digraph.\ \n \n D := Digraph([[3, 2], [1, 2], [2], [3, 4]\ ]);\n\ngap> AsBinaryRelation(D);\n\ Binary Relation on 4 points\n]]>\n \n\n\ \n \n \n \ A digraph, or fail.\n \n If the op\ tional first argument filt is present, then this should\n specify th\ e category or representation the digraph being created will\n belong to. Fo\ r example, if filt is ,\n then the d\ igraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the optional f\ irst argument filt is not present, then is used by default. \n\n If trans is a transformation, a\ nd n is a non-negative integer\n such that the restriction of tra\ ns to [1 .. n] defines\n a transformation of [1 .. \ n], then AsDigraph\n returns the functional digraph with \ n vertices defined by\n trans. See .\n \n\n Specifically, the digraph returned by AsDigraph has n edges:\n for each vertex x in [1 .. n]\ , there is a unique edge\n with source x; this edge has range x^<\ A>trans.\n \n\n If the optional second argument n is \ not supplied, then the degree of\n the transformation trans is used \ by default. If the restriction of\n trans to [1 .. n]\ does not define a transformation of\n [1 .. n], then AsDi\ graph(trans, n)\n returns fail.\n\n f := Transformation([4, 3, 3, 1, 7, 9, 10, 4, 2, 3]);\nTransform\ ation( [ 4, 3, 3, 1, 7, 9, 10, 4, 2, 3 ] )\ngap> AsDigraph(f);\n\ngap> AsDigraph(f, 4);\n\ngap> AsDigraph(f, 5);\nfail\ngap> b := BinaryRelat\ ionOnPoints(\n> [[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\nBinary Relation\ on 5 points\ngap> D := AsDigraph(b);\n\n]]>\n \n\n\n \n \ \n A &Grape; package graph.\n \n If digraph is a mutable or immutable digr\ aph without multiple edges, then this operation\n returns a &Grape; package\ graph that is isomorphic to digraph. \n\n If digraph is \ a multidigraph, then since &Grape; does not support\n multiple edges, the m\ ultiple edges will be reduced to a single edge in the\n result. In order wo\ rds, for a multidigraph this operation will return the\n same as Graph(D\ igraphRemoveAllMultipleEdges(digraph)).\n\n Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n> function\ (x, y) return Intersection(x, y) = []; end);;\ngap> Display(Petersen);\nrec(\n\ adjacencies := [ [ 3, 5, 8 ] ],\n group := \n Group( [ ( 1, 2, 3, 5, 7)( \ 4, 6, 8, 9,10), ( 2, 4)( 6, 9)( 7,10) \n ] ),\n isGraph := true,\n names\ := [ [ 1, 2 ], [ 2, 3 ], [ 3, 4 ], [ 1, 3 ], [ 4, 5 ], \n [ 2, 4 ], [ 1,\ 5 ], [ 3, 5 ], [ 1, 4 ], [ 2, 5 ] ],\n order := 10,\n representatives := [ \ 1 ],\n schreierVector := [ -1, 1, 1, 2, 1, 1, 1, 1, 2, 2 ] )\ngap> Digraph(Pe\ tersen);\n\ngap> Graph(last) = P\ etersen;\ntrue]]>\n \n\n\n \n \n A &Grape; package g\ raph.\n \n If digraph is a digraph, then thi\ s method returns the same as\n , except that if dig\ raph is immutable, then the\n result will be stored as a mutable attrib\ ute of digraph.\n In this latter case, when AsGraph(digrap\ h) is called\n subsequently, the same &GAP; object will be retur\ ned as before.\n\n D := Digraph([[1, 2], [3], []])\ ;\n\ngap> G := AsGraph(D);\nrec( a\ djacencies := [ [ 1, 2 ], [ 3 ], [ ] ], group := Group(()), \n isGraph := tr\ ue, names := [ 1 .. 3 ], order := 3, \n representatives := [ 1, 2, 3 ], schre\ ierVector := [ -1, -2, -3 ] )\n]]>\n \n\ \n\n \n \n <\ Returns>A transformation, or fail\n \n If digraph is a functional digraph, then AsTransformation\n return\ s the transformation which is defined by digraph. See . Otherwise,\n AsTransformation(digraph\ ) returns fail. \n\n If digraph is a function\ al digraph with n vertices, then\n AsTransformation(digrap\ h) will return the\n transformation f of degree at most <\ M>n where for each 1\n \\leq i \\leq n, i ^ f is equal \ to the unique out-neighbour of\n vertex i in digraph.\n\n \ D := Digraph([[1], [3], [2]]);\n\ngap> AsTransformation(D);\nTransformation( [ 1, 3, 2 \ ] )\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> AsTransformation(D);\nTransformation( [ 2, 3, 1 ] )\ngap\ > AsPermutation(last);\n(1,2,3)\ngap> D := Digraph([[2, 3], [], []]);\n\ngap> AsTransformation(D);\nfail]]>\n \n\n\n \n\n \ New digraphs from old\n \n \n \n \n \n A digraph.\n \n Each of these operations returns a new copy of digraph,\ \n of the appropriate mutability, retaining none of the attributes or\n \ properties of digraph. \n\n DigraphCopy is a synonym for \ DigraphCopySameMutability.\n D := CycleDigraph(\ 10);\n\ngap> DigraphCopy(D) = D;\ntr\ ue\ngap> IsIdenticalObj(DigraphCopy(D), D);\nfalse\ngap> DigraphMutableCopy(D)\ ;\n\n]]>\n \n\n\n \n \n \n \n \n A digraph.\n \n Each of these operations returns \ either the original argument\n digraph, or a new copy of digraph<\ /A> of the appropriate\n mutability, according to the mutability of digr\ aph.\n C := CycleDigraph(10);\n\ngap> D := DigraphImmutableCopyIfImmutable(C);\n\ngap> IsIdenticalObj(C, D);\nfalse\ng\ ap> C = D;\ntrue\ngap> D := DigraphImmutableCopyIfMutable(C);\n\ngap> IsIdenticalObj(C, D);\ntrue\ngap> C = D;\ntr\ ue\ngap> D := DigraphMutableCopyIfMutable(C);\n\ngap> IsMutableDigraph(D);\nfalse\ngap> D := DigraphMutableCopyIfI\ mmutable(C);\n\ngap> IsMutableDigr\ aph(D);\ntrue\ngap> C := CycleDigraph(IsMutableDigraph, 10);\n\ngap> D := DigraphImmutableCopyIfImmutable(C);\n<\ mutable digraph with 10 vertices, 10 edges>\ngap> IsIdenticalObj(C, D);\ntrue\ \ngap> C = D;\ntrue\ngap> D := DigraphImmutableCopyIfMutable(C);\n\ngap> IsIdenticalObj(C, D);\nfalse\ngap> C \ = D;\ntrue\ngap> D := DigraphMutableCopyIfMutable(C);\n\ngap> IsMutableDigraph(D);\ntrue\ngap> D := DigraphMutab\ leCopyIfImmutable(C);\n\ngap> IsId\ enticalObj(C, D);\ntrue\ngap> IsMutableDigraph(D);\ntrue]]>\n \n\n\n \n \n A digraph.\n \ \n If digraph is a digraph, and verts is a subset of the\n \ vertices of digraph, then this operation returns a digraph\n constr\ ucted from digraph by retaining precisely those vertices in\n ver\ ts, and those edges whose source and range vertices are both\n containe\ d in verts. \n\n The vertices of the induced subdigraph are\n \ [1..Length(verts)] but the original vertex labels can be\n accessed\ via . \n\n If digraph belong\ s to , then\n digraph is modified in\ place. If digraph belongs to \ , a new immutable digraph containing the\n appropriate\n vertices and ed\ ges is returned. \n\n D := Digraph([[1, 1, 2, 3\ , 4, 4], [1, 3, 4], [3, 1], [1, 1]]);\n\ngap> InducedSubdigraph(D, [1, 3, 4]);\n\ngap> DigraphVertices(last);\n[ 1 .. 3 ]\ngap> D := Di\ graphMutableCopy(D);\n\ngap> n\ ew := InducedSubdigraph(D, [1, 3, 4]);\n\ngap> D = new;\ntrue\n]]>\n \n\ \n\n \n \n \n A digraph.\n \n This function returns a digraph isomorphic to the su\ bdigraph of\n digraph induced by the set of non-isolated vertices,\n\ i.e. the set of those vertices of digraph which are the source or\n\ range of some edge in digraph.\n See .\n \n\n The ordering of the remaining vertices of digraph<\ /A> is preserved, as are\n the labels of the remaining vertices and edges;\ \n see and .\n This can allow one to match a vertex in the reduced digraph to the\ \n corresponding vertex in digraph.\n \n\n If digraph is immutable, then a new immutable digraph is returned.\n Otherwise, the\ isolated vertices of the mutable digraph digraph are\n removed in-p\ lace.\n D := Digraph([[1, 2], [], [], [1, 4], []])\ ;\n\ngap> R := ReducedDigraph(D);\ \n\ngap> OutNeighbours(R);\n[ [ 1,\ 2 ], [ ], [ 1, 3 ] ]\ngap> DigraphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 4, 1 ],\ [ 4, 4 ] ]\ngap> DigraphEdges(R);\n[ [ 1, 1 ], [ 1, 2 ], [ 3, 1 ], [ 3, 3 ] ]\ \ngap> DigraphVertexLabel(R, 3);\n4\ngap> DigraphVertexLabel(R, 2);\n2\ngap> D\ := Digraph(IsMutableDigraph, [[], [3], [2]]);\n\ngap> ReducedDigraph(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n \n \n \n A digraph.\n \n If <\ A>digraph is a digraph, then MaximalSymmetricSubdigraph\n return\ s a symmetric digraph without multiple edges which has the same\n vertex se\ t as digraph, and whose edge list is formed from\n digraph by\ ignoring the multiplicity of edges, and by ignoring\n edges [u,v] f\ or which there does not exist an edge [v,u].\n \n\n The digra\ ph returned by MaximalSymmetricSubdigraphWithoutLoops is\n the same,\ except that loops are removed. \n \n If digraph is immutable,\ then a new immutable digraph is returned.\n Otherwise, the mutable digraph\ digraph is changed in-place into such\n a digraph described above.<\ P/>\n\n See , , and\n for more information.\n D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsSymmetricDigraph(D) and IsM\ ultiDigraph(D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ \ 3, 1 ] ]\ngap> S := MaximalSymmetricSubdigraph(D);\n\ngap> IsSymmetricDigraph(S) and not IsMultiDigrap\ h(S);\ntrue\ngap> OutNeighbours(S);\n[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ]\ngap> D :=\ CycleDigraph(IsMutableDigraph, 3);\n\ngap> MaximalSymmetricSubdigraph(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n \n A digraph.\n \n \ If digraph is a digraph, then MaximalAntiSymmetricSubdigraph\n \ returns an anti-symmetric subdigraph of digraph formed by\n retai\ ning the vertices of digraph, discarding any duplicate edges,\n and \ discarding any edge [i,j] of digraph where i > j\n and\ the reverse edge [j,i] is an edge of digraph.\n In other wor\ ds, for every\n symmetric pair of edges [i,j] and [j,i] in digraph, where \n i and j are distinct, it discards the \ the edge\n [\\max(i,j),\\min(i,j)].\n \n\n If digraph is immutable, then a new immutable digraph is returned.\n Otherwise, the\ mutable digraph digraph is changed in-place. \n\n See for more information.\n D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);\n\ngap> not IsAntiSymmetricDigraph(D) and IsMultiDigraph(D\ );\ntrue\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]\ngap\ > D := MaximalAntiSymmetricSubdigraph(D);\n\ngap> IsAntiSymmetricDigraph(D) and not IsMultiDigraph\ (D);\ntrue\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 4 ], [ 1 ] ]\ngap> D := \ Digraph(IsMutableDigraph, [[2], [1]]);\n\ngap> MaximalAntiSymmetricSubdigraph(D);\n\ngap> D;\n\n]]>\ \n \n\n\n \n \n \n \n \n A digraph, or fail.\n \n If digraph\ is a digraph with at least one vertex, then\n UndirectedSpanningForest<\ /C> returns an undirected spanning forest of\n digraph, otherwise th\ is attribute returns fail. See for the definition of an undirected\n spanning forest. \n\n \ If digraph is a digraph with at least one vertex and whose is connected (see ), then UndirectedSpanningTree returns\n an un\ directed spanning tree of digraph, otherwise this attribute\n return\ s fail. See for the\n def\ inition of an undirected spanning tree. \n\n If digraph is immuta\ ble, then an immutable digraph is returned.\n Otherwise, the mutable digrap\ h digraph is changed in-place into an\n undirected spanning tree of \ digraph. \n\n Note that for an immutable digraph that has known u\ ndirected spanning tree,\n the attribute UndirectedSpanningTree retu\ rns the same digraph as\n the attribute UndirectedSpanningForest.\n\ \n D := Digraph([[1, 2, 1, 3], [1], [4], [3, 4, 3]\ ]);\n\ngap> UndirectedSpannin\ gTree(D);\nfail\ngap> forest := UndirectedSpanningForest(D);\n\ngap> OutNeighbours(forest);\n[ [ 2 ], \ [ 1 ], [ 4 ], [ 3 ] ]\ngap> IsUndirectedSpanningForest(D, forest);\ntrue\ngap>\ DigraphConnectedComponents(forest).comps;\n[ [ 1, 2 ], [ 3, 4 ] ]\ngap> Digra\ phConnectedComponents(MaximalSymmetricSubdigraph(D)).comps;\n[ [ 1, 2 ], [ 3, \ 4 ] ]\ngap> UndirectedSpanningForest(MaximalSymmetricSubdigraph(D))\n> = fores\ t;\ntrue\ngap> D := CompleteDigraph(4);\n\ngap> tree := UndirectedSpanningTree(D);\n\ngap> IsUndirectedSpanningTree(D, tree);\ntrue\ngap\ > tree = UndirectedSpanningForest(D);\ntrue\ngap> UndirectedSpanningForest(Emp\ tyDigraph(0));\nfail\ngap> D := PetersenGraph(IsMutableDigraph);\n\ngap> UndirectedSpanningTree(D);\n\ngap> D;\n\n]]>\n \n\n\n \ \n \n A digraph.\ \n \n If digraph is a digraph, and p \ is a partition of the vertices\n of digraph, then this operation ret\ urns a digraph constructed by\n amalgamating all vertices of digraph\ which lie in the same part of\n p.\n \n\n A partition of \ the vertices of digraph is a list of non-empty\n disjoint lists, suc\ h that the union of all the sub-lists is equal\n to vertex set of digrap\ h. In particular, each vertex must appear\n in precisely one sub-list.\ \n \n\n The vertices of digraph in part i of p \ will become\n vertex i in the quotient, and there exists some edge i\ n digraph\n with source in part i and range in part j \ if and only if there\n is an edge from i to j in the quotient\ .\n In particular, this means that the quotient of a digraph has no multipl\ e edges.\n which was a change introduced in version 1.0.0 of the &Digraphs;\ package.\n \n\n If digraph belongs to , then\n digraph is modified in place. If digraph b\ elongs to , a new immutable digraph w\ ith the above\n properties is returned. \n\n \ D := Digraph([[2, 1], [4], [1], [1, 3, 4]]);\n\ngap> DigraphVertices(D);\n[ 1 .. 4 ]\ngap> DigraphEdges(D);\n[ \ [ 1, 2 ], [ 1, 1 ], [ 2, 4 ], [ 3, 1 ], [ 4, 1 ], [ 4, 3 ], \n [ 4, 4 ] ]\nga\ p> p := [[1], [2, 4], [3]];\n[ [ 1 ], [ 2, 4 ], [ 3 ] ]\ngap> quo := QuotientD\ igraph(D, p);\n\ngap> DigraphVerti\ ces(quo);\n[ 1 .. 3 ]\ngap> DigraphEdges(quo);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 1 ]\ , [ 2, 2 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> QuotientDigraph(EmptyDigraph(0), []);\n\ \n]]>\n \n\n\n \n \n \n A digr\ aph.\n \n The reverse of a digraph is the digraph f\ ormed by reversing the orientation\n of each of its edges, i.e. for every e\ dge [i, j] of a digraph, the\n reverse contains the corresponding ed\ ge [j, i]. \n\n DigraphReverse returns the reverse of the \ digraph digraph. If\n digraph is immutable, then a new immuta\ ble digraph is returned. Otherwise,\n the mutable digraph digraph is\ changed in-place into its reverse.\n D := Digraph\ ([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> DigraphReverse(D);\n\ngap> OutNeighbours(last);\n[ [ 2, 3, 4 ], [ 4, 5 ], [ 1, 2, 5 ], [ \ 4 ], [ 2, 5 ] ]\ngap> D := Digraph([[2, 4], [1], [4], [3, 4]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 4 ], \ [ 2, 1 ], [ 3, 4 ], [ 4, 3 ], [ 4, 4 ] ]\ngap> DigraphEdges(DigraphReverse(D))\ ;\n[ [ 1, 2 ], [ 2, 1 ], [ 3, 4 ], [ 4, 1 ], [ 4, 3 ], [ 4, 4 ] ]\ngap> D := C\ ycleDigraph(IsMutableDigraph, 3);\n\ \ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> DigraphReverse(D);\n\ngap> OutNeighbours(D);\n[ [ 3 ], [ 1\ ], [ 2 ] ]\n]]>\n \n\n\n \n \n \n A digraph.\n \n \ The dual of digraph has the same vertices as digraph,\n \ and there is an edge in the dual from i to j\n whenever th\ ere is no edge from i to j in digraph.\n The dual is sometimes called the complement. \n\n DigraphDual re\ turns the dual of the digraph digraph. If\n digraph is an imm\ utable digraph, then a new immutable digraph is\n returned. Otherwise, the \ mutable digraph digraph is changed in-place\n into its dual.\n D := Digraph([[2, 3], [], [4, 6], [5], [],\n> [7, 8, 9]\ , [], [], []]);\n\ngap> DigraphDua\ l(D);\n\ngap> D := CycleDigraph(I\ sMutableDigraph, 3);\n\ngap> Digraph\ Dual(D);\n\ngap> D;\n\n]]>\n \n\n\ \n \n \ \n \n A\ digraph.\n \n If digraph is a digraph, then\ this attribute gives the minimal\n symmetric digraph which has the same ve\ rtices and contains all the edges of\n digraph. \n \n A dig\ raph is symmetric if its adjacency matrix \n is symmetric. For a digraph with multiple\n edges this means that \ there are the same number of edges from a vertex\n u to a vertex \ v as there are from v to u; \n see . \n\n If digraph is immutable, then a new immutable \ digraph is returned.\n Otherwise, the mutable digraph digraph is cha\ nged in-place into its\n symmetric closure.\n D := \ Digraph([[1, 2, 3], [2, 4], [1], [3, 4]]);\n\ngap> D := DigraphSymmetricClosure(D);\n\ngap> IsSymmetricDigraph(D);\ntrue\ngap> List(Out\ Neighbours(D), AsSet);\n[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 4 ], [ 2, 3, 4 ] ]\ng\ ap> D := Digraph([[2, 2], [1]]);\n\ngap> D := DigraphSymmetricClosure(D);\n\ngap> OutNeighbours(D);\n[ [ 2, 2 ], [ 1, 1 ] ]\ng\ ap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSymmetricClosure(D);\n\ngap> D;\n\n]]>\ \n \n\n\n \n \n \n \n \n A digraph.\n \n If digra\ ph is a digraph with no multiple edges, then these\n attributes return \ the (reflexive) transitive closure of digraph. \n\n A digraph is \ reflexive if it has a loop at every vertex, and it is\n transitiv\ e if whenever [i,j] and [j,k] are edges of\n digraph, [i,k] is also an edge. The (reflexive)\n transitive closur\ e of a digraph digraph is the least\n (reflexive and) transitive\ digraph containing digraph. \n\n If digraph is immutable,\ then a new immutable digraph is returned.\n Otherwise, the mutable digraph\ digraph is changed in-place into its\n (reflexive) transitive closu\ re. \n\n Let n be the number of vertices of digraph, and l\ et\n m be the number of edges. For an arbitrary digraph, these\n \ attributes will use a version of the Floyd-Warshall algorithm, with\n comp\ lexity O(n^3).\n\n However, for a topologically sortable digraph [se\ e ], these attributes will use me\ thods\n with complexity O(m + n + m \\cdot n) when this is faster.\n\n D := DigraphFromDiSparse6String(\".H`eOWR`U\ l^\");\n\ngap> IsReflexiveDigraph(\ D) or IsTransitiveDigraph(D);\nfalse\ngap> OutNeighbours(D);\n[ [ 4, 6 ], [ 1,\ 3 ], [ ], [ 5 ], [ ], [ 7, 8, 9 ], [ ], [ ], \n [ ] ]\ngap> T := Digrap\ hTransitiveClosure(D);\n\ngap> OutNeighbours(T);\n[ [ 4, 6, 5, 7, 8, 9 ], [ 1, 3, 4, 5, 6, 7, 8, 9 ]\ , [ ], [ 5 ], \n [ ], [ 7, 8, 9 ], [ ], [ ], [ ] ]\ngap> RT := DigraphRe\ flexiveTransitiveClosure(D);\n\ngap> OutNeighbours(RT);\n[ [ 4, 6, 5, 7, 8, 9, 1 ], [ 1, 3, 4, 5, 6, 7\ , 8, 9, 2 ], [ 3 ], \n [ 5, 4 ], [ 5 ], [ 7, 8, 9, 6 ], [ 7 ], [ 8 ], [ 9 ] ]\ \ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphReflexiveTransitiveClosure(D);\n\ngap> D;\n\n]\ ]>\n \n\n\n \n \n \n \n \n A digraph.\n \n If digraph is a topologically sortable digraph\n [see ]\n with no multiple edges, then these op\ erations return the\n (reflexive) transitive reduction of digraph.\n\n The (reflexive) transitive reduction of such a digraph is the unique\ \n least subgraph such that the (reflexive) transitive closure of the\n \ subgraph is equal to the (reflexive) transitive closure of digraph\n \ [see ].\n In order words,\ it is the least subgraph of digraph which\n retains the same reacha\ bility as digraph. \n\n If digraph is immutable, then a ne\ w immutable digraph is returned.\n Otherwise, the mutable digraph digrap\ h is changed in-place into its\n (reflexive) transitive reduction. \ \n\n Let n be the number of vertices of an arbitrary digraph, and le\ t\n m be the number of edges.\n Then these operations\n us\ e methods with complexity O(m + n + m \\cdot n).\n \n D := Digraph([[1, 2, 3], [3], [3]]);;\ngap> DigraphHasLoops(\ D);\ntrue\ngap> D1 := DigraphReflexiveTransitiveReduction(D);\n\ngap> DigraphHasLoops(D1);\nfalse\ngap> OutNeigh\ bours(D1);\n[ [ 2 ], [ 3 ], [ ] ]\ngap> D2 := DigraphTransitiveReduction(D);\ \n\ngap> DigraphHasLoops(D2);\ntru\ e\ngap> OutNeighbours(D2);\n[ [ 2, 1 ], [ 3 ], [ 3 ] ]\ngap> DigraphReflexiveT\ ransitiveClosure(D)\n> = DigraphReflexiveTransitiveClosure(D1);\ntrue\ngap> D\ igraphTransitiveClosure(D)\n> = DigraphTransitiveClosure(D2);\ntrue\ngap> D :\ = Digraph(IsMutableDigraph, [[1], [1], [1, 2, 3]]);\n\ngap> DigraphReflexiveTransitiveReduction(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n A digrap\ h.\n \n The operation returns a digraph constructed\ from digraph by adding a\n single new vertex, and no new edges. \n\n If the optional second argument label is a &GAP; object,\n \ then the new vertex will be labelled label.\n \n\n If digr\ aph belongs to , then the\n vertex is a\ dded directly to digraph. If digraph belongs to\n , an immutable copy of digraph with\n the ad\ ditional vertex is returned. \n\n D := Complete\ Digraph(3);\n\ngap> new := Digraph\ AddVertex(D);\n\ngap> D = new;\nfa\ lse\ngap> DigraphVertices(new);\n[ 1 .. 4 ]\ngap> new := DigraphAddVertex(D, G\ roup([(1, 2)]));\n\ngap> DigraphVe\ rtexLabels(new);\n[ 1, 2, 3, Group([ (1,2) ]) ]\ngap> D := CompleteBipartiteDi\ graph(IsMutableDigraph, 2, 3);\n\ng\ ap> new := DigraphAddVertex(D);\n\n\ gap> D = new;\ntrue\n]]>\n \n\n\n \n \n \n \n A digraph.\n \n For a non-negative integer m, this operati\ on returns a digraph\n constructed from digraph by adding m n\ ew vertices.\n \n\n Otherwise, if labels is a list\n consi\ sting of k &GAP; objects, then this operation returns a\n digraph co\ nstructed from digraph by adding k new vertices,\n which are \ labelled according to this list.\n \n\n If digraph belongs to\ , then the\n vertices are added directly t\ o digraph, which is changed in-place.\n If digraph belongs to\ , then\n digraph itself is return\ ed if no vertices are added\n (i.e. m=0 or labels is e\ mpty), otherwise\n the result is a new immutable digraph.\n \n\n \ D := CompleteDigraph(3);\n\ngap> new := DigraphAddVertices(D, 3);\n\ngap> DigraphVertices(new);\n[ 1 .. 6 ]\ngap> new := \ DigraphAddVertices(D, [Group([(1, 2)]), \"d\"]);\n\ngap> DigraphVertexLabels(new);\n[ 1, 2, 3, Group([ (1,2) ]),\ \"d\" ]\ngap> DigraphAddVertices(D, 0) = D;\ntrue\ngap> D := CompleteBipartit\ eDigraph(IsMutableDigraph, 2, 3);\n\ \ngap> new := DigraphAddVertices(D, 4);\n\ngap> D = new;\ntrue\n]]>\n \n\n\ \n \n \n \n A digraph.\n \n If edge is a pair of ver\ tices of digraph, or src and\n ran are vertices of \ digraph,\n then this operation returns a digraph constructed from di\ graph\n by adding a new edge with source edge[1] [src<\ /A>] and range\n edge[2] [ran].\n \n \n I\ f digraph belongs to , then the edge\n \ is added directly to digraph. If digraph belongs to , then an immutable copy of digraph wit\ h the\n additional edge is returned. \n\n \ D1 := Digraph([[2], [3], []]);\n\n\ gap> DigraphEdges(D1);\n[ [ 1, 2 ], [ 2, 3 ] ]\ngap> D2 := DigraphAddEdge(D1, \ [3, 1]);\n\ngap> DigraphEdges(D2);\ \n[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D3 := DigraphAddEdge(D2, [2, 3]);\n\ngap> DigraphEdges(D3);\n[ [ 1\ , 2 ], [ 2, 3 ], [ 2, 3 ], [ 3, 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph\ , 4);\n\ngap> new := DigraphAddEdge(\ D, [1, 3]);\n\ngap> DigraphEdges(new\ );\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 3, 4 ], [ 4, 1 ] ]\ngap> D = new;\ntrue\ \n]]>\n \n\n\n \n \n \n A new d\ igraph.\n \n \n This operation returns a new digra\ ph with the same vertices and edges as\n digraph and with additional\ edges consisting of the orbit of the\n edge edge under the action o\ f the of\n digraph. If edge is \ already an edge in digraph, then\n digraph is returned unchan\ ged. The argument digraph must be\n an immutable digraph.\n \ \n\n An edge is simply a pair of vertices of digraph.\n\n gr1 := CayleyDigraph(DihedralGroup(8));\n\ngap> gr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);\n\ngap> DigraphEdges(gr1);\n[ [ 1, 2 \ ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], [ 2, 5 ], [ 2, 6 ], \n [ 3, 8 ], [ 3, 4 ], [\ 3, 7 ], [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], \n [ 5, 7 ], [ 5, 6 ], [ 5, 8 ], [ 6, \ 4 ], [ 6, 8 ], [ 6, 2 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 ], [ 8, 3 ], [ 8, 2 ],\ [ 8, 5 ] ]\ngap> DigraphEdges(gr2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ]\ , [ 2, 1 ], [ 2, 5 ], \n [ 2, 6 ], [ 2, 7 ], [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ \ 3, 6 ], \n [ 4, 6 ], [ 4, 7 ], [ 4, 1 ], [ 4, 5 ], [ 5, 7 ], [ 5, 6 ], \n [ \ 5, 8 ], [ 5, 4 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ], [ 6, 3 ], \n [ 7, 5 ], [ 7, 1\ ], [ 7, 3 ], [ 7, 2 ], [ 8, 3 ], [ 8, 2 ], \n [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3\ := DigraphRemoveEdgeOrbit(gr2, [1, 8]);\n\ngap> gr3 = gr1;\ntrue\n]]>\n \n\n\n \n \n A digraph.\n \n If edges is\ a (possibly empty) list of pairs of vertices of\n digraph, then thi\ s operation returns a digraph constructed from\n digraph by adding t\ he edges specified by edges. More\n precisely, for every edge in edges, a new edge\n will be added with source edge[1] an\ d range edges[2].\n \n\n If an edge is included in edges with multiplicity k,\n then it will be added k times.\n\n \ If digraph belongs to , then the edg\ es\n are added directly to digraph. If digraph belongs to , then the result is returned as an immu\ table\n digraph.\n \n\n func := function\ (n)\n> local source, range, i;\n> source := [];\n> range := [];\n> for i \ in [1 .. n - 2] do\n> Add(source, i);\n> Add(range, i + 1);\n> od;\n> \ return Digraph(n, source, range);\n> end;;\ngap> D := func(1024);\n\ngap> new := DigraphAddEdges(D,\n> [[1\ 023, 1024], [1, 1024], [1023, 1024], [1024, 1]]);\n\ngap> D = new;\nfalse\ngap> D2 := DigraphMutableC\ opy(func(1024));\n\ngap> new :\ = DigraphAddEdges(D2,\n> [[1023, 1024], [1, 1024], [1023, 1024], [1024, 1]]);\ \n\ngap> D2 = new;\ntrue\ \n]]>\n \n\n\n \n \n A digraph.\n \n If v is a vertex of digraph, then thi\ s operation returns a\n digraph constructed from digraph by removing\ vertex v,\n along with any edge whose source or range vertex is \ v. \n\n If digraph has n vertices, then the vertices o\ f the returned\n digraph are [1..n-1], but the original labels can b\ e\n accessed via . \n\n If dig\ raph belongs to , then the\n vertex is \ removed directly from digraph. If digraph belongs\n to , an immutable copy of digraph\n without\ the vertex is returned. \n\n D := Digraph([\"a\ \", \"b\", \"c\"],\n> [\"a\", \"a\", \"b\", \"c\", \"c\"],\n>\ [\"b\", \"c\", \"a\", \"a\", \"c\"]);\n\ngap> DigraphVertexLabels(D);\n[ \"a\", \"b\", \"c\" ]\ \ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 3, 1 ], [ 3, 3 ] ]\ \ngap> new := DigraphRemoveVertex(D, 2);\n\ngap> DigraphVertexLabels(new);\n[ \"a\", \"c\" ]\ngap> D := CycleDig\ raph(IsMutableDigraph, 5);\n\ngap> n\ ew := DigraphRemoveVertex(D, 1);\n\n\ gap> DigraphVertexLabels(D);\n[ 2, 3, 4, 5 ]\ngap> D = new;\ntrue\n]]>\n \n\n\n \n \n A digraph.\n \ \n If verts is a (possibly empty) duplicate-free list \ of vertices of\n digraph, then this operation returns a digraph cons\ tructed from\n digraph by removing every vertex in verts, alo\ ng with any edge\n whose source or range vertex is in verts. \n\n\ If digraph has n vertices, then the vertices of the new\n \ digraph are [1 .. n-Length(verts)], but the\n original label\ s can be accessed via . \n\n If d\ igraph belongs to , then the\n vertices\ are removed directly from digraph. If digraph belongs\n to <\ Ref Filt=\"IsImmutableDigraph\"/>, an immutable copy of digraph\n wi\ thout the vertices is returned. \n\n D := Digra\ ph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);\n\ngap> SetDigraphVertexLabels(D, [\"a\", \"b\", \"c\", \"d\",\ \"e\"]);\ngap> new := DigraphRemoveVertices(D, [2, 4]);\n\ngap> DigraphVertexLabels(new);\n[ \"a\", \"c\", \"e\ \" ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\n\ngap> new := DigraphRemoveVertices(D, [1, 3]);\n\ngap> DigraphVertexLabels(D);\n[ 2, 4, 5 ]\ngap> \ D = new;\ntrue\n]]>\n \n\n\n \n \n \n A di\ graph.\n \n If digraph is a digraph with no \ multiple edges and edge is a\n pair of vertices of digraph, o\ r src and ran are\n vertices of digraph, then this ope\ ration returns a digraph\n constructed from digraph by removing the \ edge specified by\n edge or [src, ran]. \n\n If d\ igraph belongs to , then the edge i\ s removed directly from\n digraph. If digraph belongs to , an immutable copy of digraph wit\ hout\n the edge is returned.\n \n\n Note that if digraph b\ elongs to ,\n then a new copy of digra\ ph will be returned even if edge or \n [src, ran] does no\ t define an edge of digraph. \n\n D := C\ ycleDigraph(250000);\n\ngap> D :\ = DigraphRemoveEdge(D, [250000, 1]);\n\ngap> new := DigraphRemoveEdge(D, [25000, 2]);;\ngap> new = D;\ \ntrue\ngap> IsIdenticalObj(new, D);\nfalse\ngap> D := DigraphMutableCopy(D);;\ \ngap> new := DigraphRemoveEdge(D, 2500, 2);;\ngap> IsIdenticalObj(new, D);\nt\ rue]]>\n\n \n\n\n \n \n \n A\ new digraph.\n \n \n This operation returns a new\ digraph with the same vertices as\n digraph and with the orbit of t\ he edge edge (under the action\n of the \ of digraph) removed.\n If edge is not an edge in digraph<\ /A>, then digraph is\n returned unchanged. The argument digraph must be an immutable\n digraph.\n \n\n An edge is simply a pai\ r of vertices of digraph.\n gr1 := CayleyDi\ graph(DihedralGroup(8));\n\ngap> \ gr2 := DigraphAddEdgeOrbit(gr1, [1, 8]);\n\ngap> DigraphEdges(gr1);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 1 ], \ [ 2, 5 ], [ 2, 6 ], \n [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 4, 6 ], [ 4, 7 ], [ 4,\ 1 ], \n [ 5, 7 ], [ 5, 6 ], [ 5, 8 ], [ 6, 4 ], [ 6, 8 ], [ 6, 2 ], \n [ 7,\ 5 ], [ 7, 1 ], [ 7, 3 ], [ 8, 3 ], [ 8, 2 ], [ 8, 5 ] ]\ngap> DigraphEdges(gr\ 2);\n[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 8 ], [ 2, 1 ], [ 2, 5 ], \n [ 2, 6 \ ], [ 2, 7 ], [ 3, 8 ], [ 3, 4 ], [ 3, 7 ], [ 3, 6 ], \n [ 4, 6 ], [ 4, 7 ], [\ 4, 1 ], [ 4, 5 ], [ 5, 7 ], [ 5, 6 ], \n [ 5, 8 ], [ 5, 4 ], [ 6, 4 ], [ 6, \ 8 ], [ 6, 2 ], [ 6, 3 ], \n [ 7, 5 ], [ 7, 1 ], [ 7, 3 ], [ 7, 2 ], [ 8, 3 ],\ [ 8, 2 ], \n [ 8, 5 ], [ 8, 1 ] ]\ngap> gr3 := DigraphRemoveEdgeOrbit(gr2, [\ 1, 8]);\n\ngap> gr3 = gr1;\ntrue\ \n]]>\n \n\n\n \n \n A digraph.\n \n If one of the following holds:\n \n \ \n digraph is a digraph with no multiple edges, and\n \ edges is a list of pairs of vertices of digraph, or\n \ \n digraph is a digraph and edges is an empty\ list \n \n\n then this operation returns a digraph constr\ ucted from digraph\n by removing all of the edges specified by ed\ ges\n (see ).\n \n\n If digraph belongs to , then the edge\n is removed directly from digraph. If <\ A>digraph belongs to , the edge i\ s removed from an immutable copy of\n digraph and this new digraph i\ s returned. \n\n Note that if edges is empty, then this operation\ \n will always return digraph rather than a copy. Also, if any eleme\ nt\n of edges is invalid (i.e. does not define an edge of digraph\ )\n then that element will simply be ignored.\n\n D := CycleDigraph(250000);\n\ngap> D := DigraphRemoveEdges(D, [[250000, 1]]);\n\ngap> D := DigraphMutableCopy(D);\n\ngap> new := DigraphRemoveEdges(D, [[\ 1, 2], [2, 3], [3, 100]]);\n\ngap> new = D;\ntrue\n]]>\n \n\n\n \ \n \n \n A digraph.\n \n If digraph is a digraph, then this operation\ returns a\n digraph constructed from digraph by removing every loo\ p. A loop\n is an edge with equal source and range. \n\n If digrap\ h is immutable, then a new immutable digraph is returned.\n Otherwise, \ the loops are removed from the mutable digraph digraph\n in-place.\n\ D := Digraph([[1, 2, 4], [1, 4], [3, 4], [1, 4, 5\ ], [1, 5]]);\n\ngap> DigraphRemov\ eLoops(D);\n\ngap> D := Digraph(Is\ MutableDigraph, [[1, 2], [1]]);\n\ng\ ap> DigraphRemoveLoops(D);\n\ngap> D\ ;\n\n]]>\n \ \n\n\n \n \n \n A digraph.\n \n If d\ igraph is a digraph, then this operation returns a\n digraph construct\ ed from digraph by removing all multiple edges.\n The result is the \ largest subdigraph of digraph which does not\n contain multiple edge\ s. \n\n If digraph is immutable, then a new immutable digraph is \ returned.\n Otherwise, the multiple edges of the mutable digraph digraph\ are\n removed in-place.\n D1 := Digraph([[\ 1, 2, 3, 2], [1, 1, 3], [2, 2, 2]]);\n\ngap> D2 := DigraphRemoveAllMultipleEdges(D1);\n\ngap> OutNeighbours(D2);\n[ [ 1, 2, 3 ], [ 1, 3 ], [\ 2 ] ]\ngap> D := Digraph(IsMutableDigraph, [[2, 2], [1]]);\n\ngap> DigraphRemoveAllMultipleEdges(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n \n \n A digraph.\n \n If digraph is a digraph without multiple edges, and edges is a\n l\ ist of pairs of vertices of digraph (the entries of each pair\n corr\ esponding to the source and the range of an edge, respectively),\n then \ DigraphReverseEdges returns a digraph constructed from\n digraph\ by reversing the orientation of every edge specified by\n edges. If\ only one edge is to be reversed, then\n DigraphReverseEdge can be u\ sed instead. In this case, the second\n argument should just be a single ve\ rtex-pair, or the second and third\n arguments should be the source and ran\ ge of an edge respectively. \n\n Note that even though digraph c\ annot have multiple edges, the\n output may have multiple edges. \n\n \ If digraph belongs to , then the edge\ s\n are reversed in digraph. If digraph belongs to , an immutable copy of digraph with the\ \n specified edges reversed is returned. \n \ D := DigraphFromDiSparse6String(\".Tg?i@s?t_e?_qEsC\");\n\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 7 ], [ 1, 8 \ ], [ 5, 21 ], [ 7, 19 ], [ 9, 1 ], \n [ 11, 2 ], [ 21, 1 ] ]\ngap> new := Dig\ raphReverseEdge(D, [7, 19]);\n\ng\ ap> DigraphEdges(new);\n[ [ 1, 2 ], [ 1, 7 ], [ 1, 8 ], [ 5, 21 ], [ 9, 1 ], [\ 11, 2 ], \n [ 19, 7 ], [ 21, 1 ] ]\ngap> D2 := DigraphMutableCopy(new);;\nga\ p> new := DigraphReverseEdges(D2, [[19, 7]]);;\ngap> D2 = new;\ntrue\ngap> D =\ new;\ntrue\n]]>\n \n\n\n \n \n \n A digraph.\ \n \n In the first form, if D1, D2, etc. are di\ graphs, then\n DigraphDisjointUnion returns their disjoint union.\n\ \n In the second form, if list is a non-empty list of digraphs, then\ \n DigraphDisjointUnion returns the disjoint union of the digraphs\n\ contained in the list. \n\n For a disjoint union of digraphs, the \ vertex set is the disjoint union of\n the vertex sets, and the edge list is\ the disjoint union of the edge lists.\n \n\n More specifically, for\ a collection of digraphs D1, D2,\n ..., the disjoint \ union with have\n DigraphNrVertices(D1) +\n \ DigraphNrVertices(D2) + ... vertices.\n \ The edges of D1 will remain unchanged, whilst the edges of the\n \ ith digraph, D[i], will be changed so that they\n belong \ to the vertices of the disjoint union corresponding to\n D[i]\ . In particular, the edges of D[i] will\n have their source \ and range increased by\n DigraphNrVertices(D1) + ... +\n DigraphNrVertices(D[i-1]).\n\n\ \n\n Note that previously set w\ ill be lost. \n\n If the first digraph D1 [list[1]] belong\ s to , then D1 [list[1]] is\ modified in\n place to contain the appropriate vertices and edges. If d\ igraph belongs\n to , a new immutable\ digraph containing the\n appropriate vertices and edges is returned.\n D1 := CycleDigraph(3);\n\ngap> OutNeighbours(D1);\n[ [ 2 ], [ 3 ], [ 1 ] ]\ngap> D2 := Complet\ eDigraph(3);\n\ngap> OutNeighbours\ (D2);\n[ [ 2, 3 ], [ 1, 3 ], [ 1, 2 ] ]\ngap> union := DigraphDisjointUnion(D1\ , D2);\n\ngap> OutNeighbours(union\ );\n[ [ 2 ], [ 3 ], [ 1 ], [ 5, 6 ], [ 4, 6 ], [ 4, 5 ] ]\n]]>\n \n\n\n \n \n \ \n A digraph.\n \n In the first form, i\ f D1, D2, etc. are digraphs, then\n DigraphEdgeUnion r\ eturns their edge union.\n\n In the second form, if list is a non-em\ pty list of digraphs, then\n DigraphEdgeUnion returns the edge union\ of the digraphs contained\n in the list. \n\n The vertex set of th\ e edge union of a collection of digraphs is the\n union of the verte\ x sets, whilst the edge list of the edge union is\n the concatenation of the edge lists. The number of vertices of the\n edge union is equal t\ o the maximum number of vertices of one of the\n digraphs, whilst th\ e number of edges of the edge union will equal the\n sum of the numb\ er of edges of each digraph. \n\n Note that previously set will be lost.\n \n\n If the first digraph D1 [list[1]] belongs to , the\ n D1 [list[1]] is modified in\n place to contain the appropri\ ate vertices and edges. If digraph belongs\n to , a new immutable digraph containing the\n appropriate vertic\ es and edges is returned.\n D := CycleDigraph(10);\n\ngap> DigraphEdgeUnion(D, D);\n\ngap> D1 := Digraph([[2], [1]]);\ \n\ngap> D2 := Digraph([[2, 3], [2\ ], [1]]);\n\ngap> union := Digraph\ EdgeUnion(D1, D2);\n\ngap> Ou\ tNeighbours(union);\n[ [ 2, 2, 3 ], [ 1, 2 ], [ 1 ] ]\ngap> union = DigraphByE\ dges(\n> Concatenation(DigraphEdges(D1), DigraphEdges(D2)));\ntrue\n]]>\n \n\n\n \n \n \n A digraph.\n \n In the first form, if\ D1, D2, etc. are digraphs, then\n DigraphJoin returns\ their join.\n\n In the second form, if list is a non-empty list of \ digraphs,\n then DigraphJoin returns the join of the digraphs contai\ ned in the\n list. \n\n The join of a collection of digraphs D1, D2, ... is\n formed by first taking the of the collection.\ \n\n In the disjoint union, if i \\neq j then there are no edges bet\ ween\n vertices corresponding to digraphs D[i] and\n D<\ /A>[j] in the collection; the join is created by including\n all suc\ h edges. \n\n For example, the join of two empty digraphs is a complete \ bipartite\n digraph. \n\n Note that previously set will be lost.\n \n\n If the first digraph D1 [list[1]] belongs to , then D\ 1 [list[1]] is modified in\n place to contain the appropriate ve\ rtices and edges. If digraph belongs\n to , a new immutable digraph containing the\n appropriate vertices and\ edges is returned.\n D := CompleteDigraph(3);\n\ngap> IsCompleteDigraph(DigraphJoin(D, \ D));\ntrue\ngap> D2 := CycleDigraph(3);\n\ngap> DigraphJoin(D, D2);\n\ \n]]>\n \n\n\n \n \n \n A digraph.\ \n \n In the first form, if gr1, gr2, etc. are \ digraphs, then\n DigraphCartesianProduct returns a digraph isomorphi\ c to their\n Cartesian product.\n \n\n In the second form, if \ list is a non-empty list of digraphs,\n then DigraphCartesianProduct\ returns a digraph isomorphic to the \n Cartesian product of the digrap\ hs contained in the list.\n \n \n Mathematically, the Cartesian prod\ uct of two digraphs G, H\n is a digraph with vertex set Ca\ rtesian(DigraphVertices(G),\n DigraphVertices(H)) such that there is an\ edge from [u, u'] to\n [v, v'] iff u = v and there \ is an edge from u' to\n v' in H or u' = v' and\ there is an edge from\n u to v in G.\n \n\n \ Due to technical reasons, the digraph D returned by\n DigraphCar\ tesianProduct has vertex set\n [1 .. DigraphNrVertices(G)*DigraphNr\ Vertices(H)] instead, and the\n exact\n method of encoding pairs of\ vertices into integers is implementation\n specific. The original vertex \ pair can be somewhat regained by using\n . In addition,\n are pr\ eserved: if vertex pair\n [u,u'] was encoded as i then the ve\ rtex label of i\n will be the pair of vertex labels of u and \ u' i.e.\n DigraphVertexLabel(D,i) = [DigraphVertexLabel(G,u),\n \ DigraphVertexLabel(H,u')].\n \n\n As the Cartesian product i\ s associative, the Cartesian product of a collection \n of digraphs gr1<\ /A>, gr2, ... is computed in the obvious \n fashion. \n \n\n gr := ChainDigraph(4);\n\ngap> gr2 := CycleDigraph(3);\n\ngap> gr3 := DigraphCartesianProduct(gr, gr2);\n\ngap> IsIsomorphicDigraph(gr3, \n> Digraph([[2,\ 5], [3, 6], [4, 7], [8], \n> [6, 9], [7, 10], [8, 11], [12],\n> \ [10, 1], [11, 2], [12, 3], [4]]));\ntrue\n]]>\n \ \n\n\n \n \n \n A digraph.\n \n In the first form\ , if gr1, gr2, etc. are digraphs, then\n DigraphDirectProd\ uct returns a digraph isomorphic to their direct\n product.\n \n\ \n In the second form, if list is a non-empty list of digraphs, then\ \n DigraphDirectProduct returns a digraph isomorphic to the direct\n\ product of the digraphs contained in the list.\n \n\n Mathematic\ ally, the direct product of two digraphs G, H is a\n digraph \ with vertex set Cartesian(DigraphVertices(G),\n DigraphVertices(H)) \ such that there is an edge from [u, u']\n to [v, v'] iff ther\ e is an edge from u to v in\n G and an edge from u'\ to v' in H. \n \n\n Due to technical reasons, th\ e digraph D returned by\n DigraphDirectProduct has vertex set\ \n [1 .. DigraphNrVertices(G)*DigraphNrVertices(H)] instead, and th\ e\n exact method of encoding pairs of vertices into integers is implementat\ ion\n specific. The original vertex pair can be somewhat regained by using\ \n . In addition\n are preserved: if vertex pair\n [u,u']\ was encoded as i then the vertex label of i\n will be the pa\ ir of vertex labels of u and u' i.e.\n DigraphVertexLabel(\ D,i) = [DigraphVertexLabel(G,u),\n DigraphVertexLabel(H,u')].\n \ \n\n As the direct product is associative, the direct product of a collecti\ on \n of digraphs gr1, gr2, ... is computed in the obv\ ious \n fashion. \n \n\n gr := ChainDigraph(\ 4);\n\ngap> gr2 := CycleDigraph(3);\n\ \ngap> gr3 := DigraphDirectProduct(gr\ , gr2);\n\ngap> IsIsomorphicDigra\ ph(gr3, \n> Digraph([[6], [7], [8], [], \n> [10], [11], [12], [],\n> \ [2], [3], [4], []]));\ntrue\n]]>\n \n\n\n \n \n A list of transformations.\n \n If digraph is a Cartesian product digraph, \n digra\ ph = DigraphCartesianProduct(gr_1, gr_2, ... ), then\n DigraphCartes\ ianProductProjections returns a list proj\n such that proj[i]\ is the projection onto the i-th\n coordinate of the product.\n \ \n\n A projection is an idempotent endomorphism of digraph. I\ f\n gr1, gr2, ... are all loopless digraphs, then the induced \n \ subdigraph of digraph on the image of proj[i] is \n isomorphi\ c to gr_i.\n \n\n Currently this attribute is only set upon c\ reating an immutable \n digraph via DigraphCartesianProduct and ther\ e is no way \n of calculating it for other digraphs.\n \n\n For m\ ore information see\n \n\n D := DigraphCartesianProd\ uct(ChainDigraph(3), CycleDigraph(4),\n> Digraph([[2], [2]]));;\ngap> HasDigra\ phCartesianProductProjections(D);\ntrue\ngap> proj := DigraphCartesianProductP\ rojections(D);; Length(proj);\n3\ngap> IsIdempotent(proj[2]);\ntrue\ngap> Rank\ OfTransformation(proj[3]);\n2\ngap> S := ImageSetOfTransformation(proj[2]);;\n\ gap> IsIsomorphicDigraph(CycleDigraph(4), InducedSubdigraph(D, S));\ntrue\n]]>\ \n \n\n\n \n \n A list of tr\ ansformations.\n \n If digraph is a direct p\ roduct digraph, \n digraph = DigraphDirectProduct(gr_1, gr_2, ... ),\ then\n DigraphDirectProductProjections returns a list proj\n\ such that proj[i] is the projection onto the i-th\n coord\ inate of the product.\n \n\n A projection is an idempotent endomorph\ ism of digraph. If\n gr1, gr2, ... are all loopless digraphs,\ then the image \n of digraph under proj[i] is isomorphic to \ gr_i.\n \n\n Currently this attribute is only set upon creati\ ng an immutable\n digraph via DigraphDirectProduct and there is no w\ ay of \n calculating it for other digraphs.\n \n\n For more infor\ mation, see\n \n\n D := DigraphDirectProduct(ChainDigra\ ph(3), CycleDigraph(4),\n> Digraph([[2], [2]]));;\ngap> HasDigraphDirectProduc\ tProjections(D);\ntrue\ngap> proj := DigraphDirectProductProjections(D);; Leng\ th(proj);\n3\ngap> IsIdempotent(proj[2]);\ntrue\ngap> RankOfTransformation(pro\ j[3]);\n2\ngap> P := DigraphRemoveAllMultipleEdges(\n> ReducedDigraph(OnDigrap\ hs(D, proj[2])));; \ngap> IsIsomorphicDigraph(CycleDigraph(4), P);\ntrue\n]]><\ /Example>\n \n\n\n \n \n \n A digraph.\n \n Given a digraph digraph, the operation returns the digraph\n obtained by associating a\ vertex with each edge of digraph, and\n creating an edge from a ver\ tex v to a vertex u if and only\n if the terminal vertex of t\ he edge associated with v is the start\n vertex of the edge associat\ ed with u. \n\n Note that the returned digraph is always a new im\ mutable digraph, and the\n argument digraph is never modified.\n \ LineDigraph(CompleteDigraph(3));\n\ngap> LineDigraph(ChainDigraph(3));\n\n]]>\n \n\ \n\n \n \ \n \n A digra\ ph.\n \n Given a symmetric digraph digraph, \ the operation returns the\n symmetric digraph obtained by associating a ver\ tex with each edge of\n digraph, ignoring directions and multiplicit\ es, and adding an edge\n between two vertices if and only if the correspond\ ing edges have a vertex in\n common. \n\n Note that the returned digr\ aph is always a new immutable digraph, and the\n argument digraph is\ never modified.\n LineUndirectedDigraph(CompleteD\ igraph(3));\n\ngap> LineUndirected\ Digraph(DigraphSymmetricClosure(ChainDigraph(3)));\n\n]]>\n \n\n\n \n \n A digra\ ph.\n \n Let digraph be a digraph with verte\ x set V. This function\n returns the double digraph of digraph. The vertex set of the double\n digraph is the orginal vertex set togethe\ r with a duplicate. The edges are\n [u_1, v_2] and [u_2, v_1]\ if and only if [u, v] is an\n edge in digraph, together with\ the original edges and their\n duplicates. \n\n If digraph is\ mutable, then digraph is modified in-place. If\n digraph is \ immutable, then a new immutable digraph constructed as\n described above is\ returned.\n gamma := Digraph([[2], [3], [1]]);\n<\ immutable digraph with 3 vertices, 3 edges>\ngap> DoubleDigraph(gamma);\n\n]]>\n \n\n\n \n \n A digraph.\n \n Let digr\ aph be a digraph with vertex set V. This function\n returns the \ bipartite double digraph of digraph. The vertex set of\n the double \ digraph is the original vertex set together with a duplicate. The\n edges a\ re [u_1, v_2] and [u_2, v_1] if and only if [u,\n v]\ is an edge in digraph. The resulting graph is bipartite,\n since th\ e orignal edges are not included in the resulting digraph. \n\n If di\ graph is mutable, then digraph is modified in-place. If\n dig\ raph is immutable, then a new immutable digraph constructed as\n descri\ bed above is returned.\n gamma := Digraph([[2], [3\ ], [1]]);\n\ngap> BipartiteDoubleD\ igraph(gamma);\n\n]]>\n \ \n\n\n \n \n \n A digraph.\n \n For a digraph \ digraph this operation returns a new digraph constructed\n from d\ igraph, such that a loop is added for every vertex which did not\n have\ a loop in digraph. \n\n If digraph is immutable, then a n\ ew immutable digraph is returned.\n Otherwise, the loops are added to the l\ oopless vertices of the mutable digraph\n digraph in-place.\n D := EmptyDigraph(13);\n\ngap> D := DigraphAddAllLoops(D);\n\ngap> OutNeighbours(D);\n[ [ 1 ], [ 2 ], [ 3 ], [ 4 ]\ , [ 5 ], [ 6 ], [ 7 ], [ 8 ], [ 9 ], \n [ 10 ], [ 11 ], [ 12 ], [ 13 ] ]\ngap\ > D := Digraph([[1, 2, 3], [1, 3], [1]]);\n\ngap> D := DigraphAddAllLoops(D);\n\ngap> OutNeighbours(D);\n[ [ 1, 2, 3 ], [ 1, 3, 2 ], [ 1\ , 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphAddAllLoops(D);\n\ngap> D;\n\n]]>\n \n\n\n \n \n <\ Oper Name=\"DistanceDigraph\" Arg=\"digraph, list\"\n Label=\"for digra\ ph and list\"/>\n A digraph.\n \n The fir\ st argument is a digraph, the second argument is a non-negative\n integer o\ r a list of positive integers. This operation returns a digraph on\n the sa\ me set of vertices as digraph, with two vertices being adjacent\n if\ and only if the distance between them in digraph equals i\n \ or is a number in list. See . \n\n If digraph is \ mutable, then digraph is modified in-place. If\n digraph is i\ mmutable, then a new immutable digraph constructed as\n described above is \ returned.\n digraph := DigraphFromSparse6String(\n\ > \":]n?AL`BC_DEbEF`GIaGHdIJeGKcKL_@McDHfILaBJfHMjKM\");\n\ngap> DistanceDigraph(digraph, 1);\n\ngap> DistanceDigraph(digraph, [1, 2]);\n\n]]>\n \n<\ /ManSection>\n\n \n \n A digraph.\n \n Given a symmet\ ric loopless digraph with no multiple edges digraph,\n the k-closure of digraph is defined to be the unique\n smallest sym\ metric loopless digraph C with no multiple edges on the\n vertices o\ f digraph that contains all the edges of digraph\n and satsif\ ies the property that the sum of the degrees of every two\n non-adjacenct v\ ertices in C is less than k. See , , \ , and . \n\n The operation DigraphC\ losure returns the k-closure of\n digraph.\n \ D := CompleteDigraph(6);\n\ngap> D := DigraphRemoveEdges(D, [[1, 2], [2, 1]]);\n\ngap> closure := DigraphClosure(D, 6);\n\ngap> IsCompleteDigraph(closure);\ntrue\ \n]]>\n \n\n\n \n \n \n A digraph.\n \n\ If digraph is a symmetric digraph, then\n DigraphMycielskian<\ /C> returns the Mycielskian of digraph. \n \n The Mycielskian \ of a symmetric digraph is a larger symmetric digraph\n constructed from it,\ which has a larger chromatic number. For further\n information, see h\ ttps://en.wikipedia.org/wiki/Mycielskian. \n\n If digraph i\ s immutable, then a new immutable digraph is returned.\n Otherwise, the mut\ able digraph digraph is changed in-place into\n its Mycielskian. \ \n\n D := CycleDigraph(2);\n\ngap> ChromaticNumber(D);\n2\ngap> D := DigraphMycielskian\ (D);\n\ngap> ChromaticNumber(D);\ \n3\ngap> D := DigraphMycielskian(D);\n\ngap> ChromaticNumber(D);\n4\ngap> D := CompleteBipartiteDigraph(IsMut\ able, 2, 3);\n\ngap> DigraphMyciels\ kian(D);\n\ngap> D;\n\n]]>\n \n\n\n \n\n Random digraphs\n \n \n A \ digraph.\n \n If the optional first argument fil\ t is present, then this should\n specify the category or representation\ the digraph being created will\n belong to. For example, if filt is\ ,\n then the digraph being created will be\ mutable, if filt is , then the \ digraph will be immutable. \n If the optional first argument filt is\ not present, then is used by defaul\ t.\n\n If n is a positive integer, then this function returns a \ random\n digraph with n vertices and without multiple edges. The res\ ult\n may or may not have loops. \n\n If the optional second argument\ p is a float with value\n 0 \\leq p \\leq 1, then an edge will exist between each\n pair of vertices with probability\ approximately p.\n If p is not specified, then a random prob\ ability will be assumed\n (chosen with uniform probability).\n RandomDigraph(1000);\n\ngap> RandomDigraph(10000, 0.023);\n\ngap> RandomDigraph(IsMutableDigraph, 1000, 1 / 2);\n\n]]>\n \n\ \n\n \n \n A digraph.\n \n If n i\ s a positive integer, then this function returns a random\n digraph with n vertices. If the optional second argument\n m is a positive i\ nteger, then the digraph will have m\n edges. If m is not spe\ cified, then the number of edges will be\n chosen randomly (with uniform pr\ obability) from the range\n [1 .. {n \\choose 2}].\n \ \n\n The method used by this function chooses each edge from the set \ of\n all possible edges with uniform probability. No effort is made to avoi\ d\n creating multiple edges, so it is possible (but not guaranteed) that th\ e\n result will have multiple edges. The result may or may not have loops.<\ P/>\n RandomMultiDigraph(1000);\n\ngap> RandomMultiDigraph(1000, 950);\n\n]]>\n \n\n\n \n \n A digraph.\n \n If the \ optional first argument filt is present, then this should\n specify \ the category or representation the digraph being created will\n belong to. \ For example, if filt is ,\n then the\ digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the optional\ first argument filt is not present, then is used by default. \n\n If n is a non-negative inte\ ger, this function returns a random\n tournament with n vertices. Se\ e . \n RandomTournamen\ t(10);\n\ngap> RandomTournament(IsMutab\ leDigraph, 10);\n\n]]>\ \n \n\n\n \n \n A digraph.\n \n If \ the optional first argument filt is present, then this should\n spec\ ify the category or representation the digraph being created will\n belong \ to. For example, if filt is ,\n then\ the digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the opt\ ional first argument filt is not present, then is used by default. \n\n If n is a positive int\ eger, this function return a random lattice\n with m vertices, where\ it is guaranteed that m is between\n n and 2 * n. See . \n Rando\ mLattice(10);\n\ngap> Ra\ ndomLattice(IsMutableDigraph, 10);\n\n]]>\n \n\n\n \n\n <\ Heading>Standard examples\n \n \n A digraph.\n \n\ If n is a positive integer, this function returns a chain\n with\ n vertices and n - 1 edges.\n Specifically,\n for \ each vertex i (with i < n), there is a directed\n e\ dge with source i and range i + 1. \n\n If the optional f\ irst argument filt is present, then this should\n specify the catego\ ry or representation the digraph being created will\n belong to. For exampl\ e, if filt is ,\n then the digraph b\ eing created will be mutable, if filt is , then the digraph will be immutable. \n If the optional first arg\ ument filt is not present, then is used by default. \n\n The of a chain represents a\n total order.\n ChainDigraph(42);\n\ngap> ChainD\ igraph(IsMutableDigraph, 10);\n\n]]\ >\n \n\n\n \n \n A digraph.\n <\ Description>\n If n is a non-negative integer, this function returns\ the complete\n digraph with n vertices. See . \n\n If the optional first argument filt is present,\ then this should\n specify the category or representation the digraph bein\ g created will\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if f\ ilt is , then the digraph will be i\ mmutable. \n If the optional first argument filt is not present, the\ n is used by default.\n \ CompleteDigraph(20);\n\ngap> CompleteDigraph(IsMutableDigraph, 10);\n\n]]>\n \n\n\n \n \n \ A digraph.\n \n A complete bipartite digra\ ph is a digraph whose vertices\n can be partitioned into two non-empty vert\ ex sets, such there exists\n a unique edge with source i and range <\ C>j if and only if\n i and j lie in different vertex sets\ . \n\n If m and n are positive integers, this function\n \ returns the complete bipartite digraph with vertex sets of sizes\n m<\ /A> (containing the vertices [1 .. m]) and\n n (containing th\ e vertices [m + 1 .. m + n]).\n \n\n If the optional first ar\ gument filt is present, then this should\n specify the category or r\ epresentation the digraph being created will\n belong to. For example, if <\ A>filt is ,\n then the digraph being cr\ eated will be mutable, if filt is , then the digraph will be immutable. \n If the optional first argument <\ A>filt is not present, then is u\ sed by default.\n\n CompleteBipartiteDigraph(2, 3)\ ;\n\ngap>\ CompleteBipartiteDigraph(IsMutableDigraph, 3, 2);\n\n]]>\n \n\n\n \n \n A digraph.\n \n For a list\ orders of n positive integers,\n this function returns the\ digraph containing n independent sets\n of vertices of orders \ [l[1] .. l[n]]. Moreover,\n each vertex is adjacent to \ every other not contained in the\n same independent set.\n \n\n \ If the optional first argument filt is present, then this should\n\ specify the category or representation the digraph being created will\n \ belong to. For example, if filt is ,\n then the digraph being created will be mutable, if filt is , then the digraph will be immutable. \ \n If the optional first argument filt is not present, then is used by default.\n CompleteMultipartiteDigraph([5, 4, 2]);\n\ngap> CompleteMultipartiteDigraph(\ IsMutableDigraph, [5, 4, 2]);\n\n]\ ]>\n \n\n\n \n \n A digraph.\n \n If n is a positive integer, this function returns a \ cycle\n digraph with n vertices and n edges. Specifically\ ,\n for each vertex i (with i < n), there is a dire\ cted\n edge with source i and range i + 1. In addition, there\ is\n an edge with source n and range 1. \n\n If the o\ ptional first argument filt is present, then this should\n specify t\ he category or representation the digraph being created will\n belong to. F\ or example, if filt is ,\n then the \ digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the optional \ first argument filt is not present, then is used by default.\n CycleDigraph(1);\n\ngap> CycleDigraph(123);\n\ngap> CycleDigraph(IsMutableDigraph, 10);\n\n]]>\n \n\n\n \n \n \n A digraph.<\ /Returns>\n \n If n is a non-negative integer, this fu\ nction returns the\n empty or null digraph with n vert\ ices. An empty\n digraph is one with no edges.\n \n \n If th\ e optional first argument filt is present, then this should\n specif\ y the category or representation the digraph being created will\n belong to\ . For example, if filt is ,\n then t\ he digraph being created will be mutable, if filt is , then the digraph will be immutable. \n If the option\ al first argument filt is not present, then is used by default. \n\n NullDigraph is a synonym \ for EmptyDigraph.\n\n EmptyDigraph(20);\n\ngap> NullDigraph(10);\n\ngap> EmptyDigraph(IsMutableDigraph, 10);\n]]>\n \n\n\n \n \n A digraph.\n \n If n and k<\ /A> are non-negative integers, then this operation\n returns a symmetric di\ graph which corresponds to the undirected Johnson\n graph J(n, k)\ . \n\n If the optional first argument filt is present, then \ this should\n specify the category or representation the digraph being crea\ ted will\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if filt is , then the digraph will be immutab\ le. \n If the optional first argument filt is not present, then is used by default. \n\n The Jo\ hnson graph J(n, k) has vertices given by all the\n k-sub\ sets of the range [1 .. n], and two vertices are\n connecte\ d by an edge iff their intersection has size k - 1.\n\n gr := JohnsonDigraph(3, 1);\n\ngap> OutNeighbours(gr);\n[ [ 2, 3 ], [ 1, 3 ], [ 1,\ 2 ] ]\ngap> gr := JohnsonDigraph(4, 2);\n\ngap> OutNeighbours(gr);\n[ [ 2, 3, 4, 5 ], [ 1, 3, 4, 6 ]\ , [ 1, 2, 5, 6 ], [ 1, 2, 5, 6 ], \n [ 1, 3, 4, 6 ], [ 2, 3, 4, 5 ] ]\ngap> J\ ohnsonDigraph(1, 0);\n\ngap> JohnsonDig\ raph(IsMutableDigraph, 1, 0);\n\n]]>\n \n\n\n \n \n A digraph.\n \n\n From https://en.wikipedia.org/wiki/Petersen_graph: \n\n\ The Petersen graph is an undirected graph with 10 vertices and 15 edges\ .\n It is a small graph that serves as a useful example and counterexample \ for\n many problems in graph theory. The Petersen graph is named after Juli\ us\n Petersen, who in 1898 constructed it to be the smallest bridgeless cub\ ic\n graph with no three-edge-coloring. \n \n If the optional \ first argument filt is present, then this should\n specify the categ\ ory or representation the digraph being created will\n belong to. For examp\ le, if filt is ,\n then the digraph \ being created will be mutable, if filt is , then the digraph will be immutable. \n If the optional first ar\ gument filt is not present, then is used by default. \n\n See also .\n\n ChromaticNumber(PetersenGraph());\n3\ngap\ > PetersenGraph(IsMutableDigraph);\n\n]]>\n \n\n\n \n \n A digra\ ph.\n \n If n is a positive integer and k\ is a non-negative\n integer less than n / 2, then this o\ peration returns\n the generalised Petersen graph GPG(n, k\ ). \n\n From\n https://en.wikipedia.org/wiki/Generalized_Pe\ tersen_graph:\n \n\n The generalized Petersen graphs are a \ family of cubic\n graphs formed by connecting the vertices of a regular pol\ ygon to the\n corresponding vertices of a star polygon. They include the Pe\ tersen graph\n and generalize one of the ways of constructing the Petersen \ graph. The\n generalized Petersen graph family was introduced in 1950 by H.\ S. M.\n Coxeter and was given its name in 1969 by Mark Watkins.\n <\ P/>\n\n If the optional first argument filt is present, then this sh\ ould\n specify the category or representation the digraph being created wil\ l\n belong to. For example, if filt is ,\n then the digraph being created will be mutable, if filt is <\ Ref\n Filt=\"IsImmutableDigraph\"/>, then the digraph will be immutable. \n\ If the optional first argument filt is not present, then is used by default. \n\n See also .\n\n GeneralisedPetersenGraph(\ 7, 2);\n\ngap> General\ isedPetersenGraph(40, 1);\n\ngap> D := GeneralisedPetersenGraph(5, 2);\n\ngap> IsIsomorphicDigraph(D, PetersenGraph());\n\ true\ngap> GeneralisedPetersenGraph(IsMutableDigraph, 9, 4);\n\n]]>\n \n\n\ \n \n\n\n \n Operators\n\n Operators for digraphs<\ /Heading>\n \n digraph1 = digraph2\n \n = (for digraphs)\n \ returns true if digraph1 and digraph2 have the same\n\ vertices, and DigraphEdges(digraph1) =\n DigraphEdges\ (digraph2), up to some re-ordering of the edge\n lists. \ \n\n Note that this operator does not compare the vertex labels\n \ of digraph1 and digraph2.\n \n\n digr\ aph1 < digraph2\n \n < (for digraphs)\n\n This operator returns true\ if one of the following holds:\n\n \n \n \ The number n_1 of vertices in digraph1 is less than\n \ the number n_2 of vertices in digraph2;\n \n \ \n n_1 = n_2, and the number m_1 of edge\ s in\n digraph1 is less than the number m_2 of edges i\ n\n digraph2;\n \n \n \ n_1 = n_2, m_1 = m_2, and\n DigraphEdges(digraph\ 1) is less than\n DigraphEdges(digraph2) after\ having both of these sets\n have been sorted with respect to the le\ xicographical order.\n \n \n \n \n\n \n \n <\ Returns>true or false.\n \n If sup\ er and sub are digraphs, then this operation returns\n true if sub is a subdigraph of super, and false\n if it \ is not. \n\n A digraph sub is a subdigraph of a digraph <\ A>super if\n sub and super share the same number of verti\ ces, and the\n collection of edges of super (including repeats) cont\ ains the\n collection of edges of sub (including repeats). \n\n\ In other words, sub is a subdigraph of super if and only if\ \n DigraphNrVertices(sub) = DigraphNrVertices(super), \ and\n for each pair of vertices i and j, there are at least a\ s many\n edges of the form [i, j] in super as there are in sub.\n \n\n g := Digraph([[2, 3], [1],\ [2, 3]]);\n\ngap> h := Digraph([[\ 2, 3], [], [2]]);\n\ngap> IsSubdig\ raph(g, h);\ntrue\ngap> IsSubdigraph(h, g);\nfalse\ngap> IsSubdigraph(Complete\ Digraph(4), CycleDigraph(4));\ntrue\ngap> IsSubdigraph(CycleDigraph(4), ChainD\ igraph(4));\ntrue\ngap> g := Digraph([[2, 2], [1]]);\n\ngap> h := Digraph([[2], [1]]);\n\ngap> IsSubdigraph(g, h);\ntrue\ngap> IsSubdigraph(h\ , g);\nfalse]]>\n \n\n\n \ \n \n \n true o\ r false.\n \n The operation IsUndirectedS\ panningTree returns true if the\n digraph sub is an undir\ ected spanning tree of the digraph\n super, and the operation IsU\ ndirectedSpanningForest returns\n true if the digraph sub\ is an undirected spanning forest of\n the digraph super. \n\n \ An undirected spanning tree of a digraph super is a subdigraph\ \n of super that is an undirected tree (see \n and ). Note that a digraph whose <\ Ref\n Oper=\"MaximalSymmetricSubdigraph\"/> is not connected has no undir\ ected\n spanning trees (see ). \n\n \ An undirected spanning forest of a digraph super is a\n sub\ digraph of super that is an undirected forest (see and ), and is not\n conta\ ined in any larger such subdigraph of super. Equivalently, an\n und\ irected spanning forest is a subdigraph of super whose connected\n c\ omponents coincide with those of the of super (see ). \n\n Note that an undirected spanning tree is an undirected spannin\ g forest that\n is connected.\n\n D := Complete\ Digraph(4);\n\ngap> tree := Digrap\ h([[3], [4], [1, 4], [2, 3]]);\n\n\ gap> IsSubdigraph(D, tree) and IsUndirectedTree(tree);\ntrue\ngap> IsUndirecte\ dSpanningTree(D, tree);\ntrue\ngap> forest := EmptyDigraph(4);\n\ngap> IsSubdigraph(D, forest) and IsUndirectedFore\ st(forest);\ntrue\ngap> IsUndirectedSpanningForest(D, forest);\nfalse\ngap> Is\ Subdigraph(tree, forest);\ntrue\ngap> D := DigraphDisjointUnion(CycleDigraph(2\ ), CycleDigraph(2));\n\ngap> IsUnd\ irectedTree(D);\nfalse\ngap> IsUndirectedForest(D) and IsUndirectedSpanningFor\ est(D, D);\ntrue]]>\n \n\n\n \ \n\n \n Attributes and operations\n Ve\ rtices and edges\n \n \n A list of integers.\n \ \n Returns the vertices of the digraph digraph. \n\n Note tha\ t the vertices of a digraph are always a range of\n positive integers from \ 1 to the number of vertices of the graph.\n \ gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \"b\"\ ],\n> [\"b\", \"c\", \"a\"]);\n\ngap> DigraphVertices(gr);\n[ 1 .. 3 ]\ngap> gr := Digraph([1, \ 2, 3, 4, 5, 7],\n> [1, 2, 2, 4, 4],\n> [2, 7\ , 5, 3, 7]);\n\ngap> DigraphVertic\ es(gr);\n[ 1 .. 6 ]\ngap> DigraphVertices(RandomDigraph(100));\n[ 1 .. 100 ]\n\ gap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphVertices(D);\n[ 1 .. 3 ]\n]]>\n \n\n\n \n \n An integer.\n \n Returns\ the number of vertices of the digraph digraph. \n\n gr := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\",\ \"b\", \"b\"],\n> [\"b\", \"c\", \"a\"]);\n\ngap> DigraphNrVertices(gr);\n3\ngap> gr := Digrap\ h([1, 2, 3, 4, 5, 7],\n> [1, 2, 2, 4, 4],\n> \ [2, 7, 5, 3, 7]);\n\ngap> Digraph\ NrVertices(gr);\n6\ngap> DigraphNrVertices(RandomDigraph(100));\n100\ngap> D :\ = CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphNrVertices(D);\n3\n]]>\n \n\n\n \n \n A list of lists.\n \n DigraphEdges re\ turns a list of edges of the digraph digraph,\n where each edge is a\ pair of elements of of\n the form [sour\ ce,range].\n \n\n The entries of DigraphEdges(digraph<\ /A>) are in one-to-one\n corresponence with the edges of digraph<\ /A>. Hence\n DigraphEdges(digraph) is duplicate-free \ if and only if\n digraph contains no multiple edges. \n\n The\ entries of DigraphEdges are guaranteed to be sorted by their\n firs\ t component (i.e. by the source of each edge), but they are not\n necessari\ ly then sorted by the second component.\n gr := Di\ graphFromDiSparse6String(\".DaXbOe?EAM@G~\");\n\ngap> edges := ShallowCopy(DigraphEdges(gr));; Sort(edges)\ ;\ngap> edges;\n[ [ 1, 1 ], [ 1, 3 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 1 ], \ \n [ 2, 2 ], [ 2, 3 ], [ 2, 5 ], [ 3, 2 ], [ 3, 4 ], [ 3, 5 ], \n [ 4, 2 ], \ [ 4, 4 ], [ 4, 5 ], [ 5, 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ \ngap> DigraphEdges(D);\n[ [ 1, 2 ],\ [ 2, 3 ], [ 3, 1 ] ]\n]]>\n \n\n\n <\ ManSection>\n \n An \ integer.\n \n This function returns the number of e\ dges of the digraph digraph.\n gr := Digrap\ h([\n> [1, 3, 4, 5], [1, 2, 3, 5], [2, 4, 5], [2, 4, 5], [1]]);;\ngap> Digraph\ NrEdges(gr);\n15\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> \ [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\", \"a\"]);\n\ngap> DigraphNrEdges(gr);\n3\ngap> D\ := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphNrEdges(D);\n3\n]]>\n \n\n\n \n \n <\ Returns>An integer.\n \n This function returns the \ number of loops of the digraph digraph. See . \n D := Digraph([[2, 3], [1, 4], [3, 3, \ 5], [], [2, 5]]);\n\ngap> Dig\ raphNrLoops(D);\n3\ngap> D := EmptyDigraph(5);\n\ngap> DigraphNrLoops(D);\n0\ngap> D := CompleteDigraph(5);\n\ngap> DigraphNrLoops(D);\n0\n]]>\n \n\n\n \n \n A list of vertices.\n \n This function returns a list of the sinks of the digraph\n digraph.\n A sink of a digraph is a vertex with out-degree zero.\n \ See .\n gr := Dig\ raph([[3, 5, 2, 2], [3], [], [5, 2, 5, 3], []]);\n\ngap> DigraphSinks(gr);\n[ 3, 5 ]\ngap> D := CycleDigrap\ h(IsMutableDigraph, 3);\n\ngap> Digr\ aphSinks(D);\n[ ]\n]]>\n \n\n\n \n \n A list\ of vertices.\n \n This function returns an immutab\ le list of the sources of the digraph\n digraph.\n A source of a \ digraph is a vertex with in-degree zero.\n See .\n gr := Digraph([[3, 5, 2, 2], [3], [], [5, \ 2, 5, 3], []]);\n\ngap> Digra\ phSources(gr);\n[ 1, 4 ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphSources(D);\n[ ]\n]]>\n \n\n\n \n \n A list of positive integers\ , or fail.\n \n If digraph is a digra\ ph whose only directed cycles are loops, then\n DigraphTopologicalSort returns the vertices of digraph ordered\n so that every edge's so\ urce appears no earlier in the list than its range.\n If the digraph dig\ raph contains directed cycles of length greater\n than 1, then t\ his operation returns fail.\n \n\n See Section for the definition\n of a directed cycl\ e, and the definition of a loop.\n\n \n\n The method used for this a\ ttribute has complexity O(m+n) where\n m is the number of edg\ es (counting multiple edges as one) and\n n is the number of vertice\ s in the digraph. \n D := Digraph([\n> [2, 3],\ [], [4, 6], [5], [], [7, 8, 9], [], [], []]);\n\ngap> DigraphTopologicalSort(D);\n[ 2, 5, 4, 7, 8, 9, 6, 3, 1 ]\ \ngap> D := Digraph(IsMutableDigraph, [[2, 3], [3], [4], []]);\n\ngap> DigraphTopologicalSort(D);\n[ 4, 3, 2, 1 ]\ \n]]>\n \n\n\n \n \n \n \n If digraph is a digraph, then the first operation returns the label\n of the vertex \ i. The second operation can be used to set the label\n of the vertex\ i in digraph to the arbitrary &GAP; object\n obj. \n\n The label of a vertex can be changed an arbitrary number of times. I\ f no\n label has been set for the vertex i, then the default value i\ s\n i. \n\n If digraph is a digraph created from a rec\ ord with a component\n vertices, then the labels of the vertices are\ set to the value of\n this component. \n\n Induced subdigraphs, and \ some other operations which create new digraphs from\n old ones, inherit th\ eir labels from their parents.\n\n D := DigraphFro\ mDigraph6String(\"&DHUEe_\");\n\n\ gap> DigraphVertexLabel(D, 3);\n3\ngap> D := Digraph([\"a\", \"b\", \"c\"], []\ , []);\n\ngap> DigraphVertexLabel(D, \ 2);\n\"b\"\ngap> SetDigraphVertexLabel(D, 2, \"d\");\ngap> DigraphVertexLabel(\ D, 2);\n\"d\"\ngap> D := InducedSubdigraph(D, [1, 2]);\n\ngap> DigraphVertexLabel(D, 2);\n\"d\"\ngap> D := Digraph(\ IsMutableDigraph, [\"e\", \"f\", \"g\"], [], []);\n\ngap> DigraphVertexLabel(D, 1);\n\"e\"\ngap> SetDigraphVertexLabe\ l(D, 1, \"h\");\ngap> DigraphVertexLabel(D, 1);\n\"h\"\ngap> InducedSubdigraph\ (D, [1, 2]);\n\ngap> DigraphVertexLabel\ (D, 1);\n\"h\"\n]]>\n \n\n\n \n \n \n \n If di\ graph is a digraph, then DigraphVertexLabels returns a\n copy of\ the labels of the vertices in digraph.\n SetDigraphVertexLabels<\ /C> can be used to set the labels of the vertices\n in digraph to th\ e list of\n arbitrary &GAP; objects list. \n\n The label of a\ vertex can be changed an arbitrary number of times. If no\n label has been\ set for the vertex i, then the default value is\n i. \n\ \n If digraph is a digraph created from a record with a component\n \ vertices, then the labels of the vertices are set to the value of\n \ this component. \n\n Induced subdigraphs, and other operations which \ create new digraphs from\n old ones, inherit their labels from their parent\ s.\n\n D := DigraphFromDigraph6String(\"&DHUEe_\")\ ;\n\ngap> DigraphVertexLabels(D);\ \n[ 1 .. 5 ]\ngap> D := Digraph([\"a\", \"b\", \"c\"], [], []);\n\ngap> DigraphVertexLabels(D);\n[ \"a\", \"b\", \"\ c\" ]\ngap> SetDigraphVertexLabel(D, 2, \"d\");\ngap> DigraphVertexLabels(D);\ \n[ \"a\", \"d\", \"c\" ]\ngap> D := InducedSubdigraph(D, [1, 3]);\n\ngap> DigraphVertexLabels(D);\n[ \"a\", \"c\" \ ]\ngap> D := Digraph(IsMutableDigraph, [\"e\", \"f\", \"g\"], [], []);\n\ngap> SetDigraphVertexLabels(D, [\"h\", \"i\ \", \"j\"]);\ngap> DigraphVertexLabels(D);\n[ \"h\", \"i\", \"j\" ]\ngap> Indu\ cedSubdigraph(D, [1, 3]);\n\ngap> Digra\ phVertexLabels(D);\n[ \"h\", \"j\" ]\n]]>\n \n\n\n \n \n \ \n \n If digraph is a digraph without multiple edges, \ then the first\n operation returns the label of the edge from vertex i to vertex\n j. The second operation can be used to set the label\ of the edge\n between vertex i and vertex j to the arbitrary\ &GAP; object\n obj. \n\n The label of an edge can be changed\ an arbitrary number of times. If no\n label has been set for the edge, the\ n the default value is 1. \n\n Induced subdigraphs, and some oth\ er operations which create new digraphs from\n old ones, inherit their edge\ labels from their parents.\n\n See also .\ \n\n D := DigraphFromDigraph6String(\"&DHUEe_\");\ \n\ngap> DigraphEdgeLabel(D, 3, 1\ );\n1\ngap> SetDigraphEdgeLabel(D, 2, 5, [42]);\ngap> DigraphEdgeLabel(D, 2, 5\ );\n[ 42 ]\ngap> D := InducedSubdigraph(D, [2, 5]);\n\ngap> DigraphEdgeLabel(D, 1, 2);\n[ 42 ]\ngap> D := ChainD\ igraph(IsMutableDigraph, 5);\n\ngap>\ DigraphEdgeLabel(D, 2, 3);\n1\ngap> SetDigraphEdgeLabel(D, 4, 5, [1729]);\nga\ p> DigraphEdgeLabel(D, 4, 5);\n[ 1729 ]\ngap> InducedSubdigraph(D, [4, 5]);\n<\ mutable digraph with 2 vertices, 1 edge>\ngap> DigraphEdgeLabel(D, 1, 2);\n[ 1\ 729 ]\n]]>\n \n\n\n \n <\ Oper Name=\"DigraphEdgeLabels\" Arg=\"digraph\"/>\n \n \n \n If digraph i\ s a digraph without multiple edges, then\n DigraphEdgeLabels returns\ a copy of the labels of the edges\n in digraph as a list of lists <\ C>labels such that\n labels[i][j] is the label on the edge from \ vertex i\n to vertex OutNeighbours(digraph)[i][j].\n\n \ SetDigraphEdgeLabels can be used to set the labels of the edges in\n digraph without multiple edges to the list labels of lists of\n \ arbitrary &GAP; objects such that list[i][j] is the label on the edge\ \n from vertex i to the vertex OutNeighbours(digraph>[i][j].\ \n\n Alternatively SetDigraphEdgeLabels can be called with binary fu\ nction\n func that as its second argument that when passed two verti\ ces i\n and j returns the label for the edge between vertex <\ C>i and vertex\n j. \n\n The label of an edge can be chan\ ged an arbitrary number of times. If no\n label has been set for an edge, t\ hen the default value is 1. \n\n Induced subdigraphs, and some o\ ther operations which create new digraphs\n from old ones, inherit their la\ bels from their parents.\n\n D := DigraphFromDigra\ ph6String(\"&DHUEe_\");\n\ngap> D\ igraphEdgeLabels(D);\n[ [ 1 ], [ 1, 1, 1 ], [ 1 ], [ 1, 1, 1 ], [ 1, 1, 1 ] ]\ \ngap> SetDigraphEdgeLabel(D, 2, 1, \"d\");\ngap> DigraphEdgeLabels(D);\n[ [ 1\ ], [ \"d\", 1, 1 ], [ 1 ], [ 1, 1, 1 ], [ 1, 1, 1 ] ]\ngap> D := InducedSubdi\ graph(D, [1, 2, 3]);\n\ngap> Digra\ phEdgeLabels(D);\n[ [ 1 ], [ \"d\", 1 ], [ 1 ] ]\ngap> OutNeighbours(D);\n[ [ \ 3 ], [ 1, 3 ], [ 1 ] ]\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 2\ , 3);\n\ngap> DigraphEdgeLabels(D);\ \n[ [ 1, 1, 1 ], [ 1, 1, 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ] ]\ngap> SetDigraphE\ dgeLabel(D, 2, 4, \"a\");\ngap> DigraphEdgeLabels(D);\n[ [ 1, 1, 1 ], [ 1, \"a\ \", 1 ], [ 1, 1 ], [ 1, 1 ], [ 1, 1 ] ]\ngap> InducedSubdigraph(D, [1, 2, 3, 4\ ]);\n\ngap> DigraphEdgeLabels(D);\n[\ [ 1, 1 ], [ 1, \"a\" ], [ 1, 1 ], [ 1, 1 ] ]\ngap> OutNeighbors(D);\n[ [ 3, 4\ ], [ 3, 4 ], [ 1, 2 ], [ 1, 2 ] ]\n]]>\n \n\n\n \n \n A list of edges.\n \n DigraphI\ nEdges returns the list of all edges of digraph\n which have \ vertex as their range.\n D := Digraph([[2, 2],\ [3, 3], [4, 4], [1, 1]]);\n\ \ngap> DigraphInEdges(D, 2);\n[ [ 1, 2 ], [ 1, 2 ] ]\n]]>\n \n\n\n \n \n A list of edges.\n \n DigraphOutEdges returns the list of all edges of digraph\ \n which have vertex as their source.\n \ D := Digraph([[2, 2], [3, 3], [4, 4], [1, 1]]);\n\ngap> DigraphOutEdges(D, 2);\n[ [ 2, 3 ], [ 2, 3 ] ]\n]]>\ \n \n\n\n \n \n \n true or false.\n \n In the first form, this function returns true\ if and only if the list\n list specifies an edge in the digraph \ digraph. Specifically,\n this operation returns true if list\ is a pair of positive\n integers where list[1] is the so\ urce and list[2]\n is the range of an edge in digraph,\ and false otherwise. \n\n The second form simply returns tru\ e if [u, v] is\n an edge in digraph, and false otherwise.\n\n D := Digraph([[2, 2], [6], [\ ], [3], [], [1]]);\n\ngap> Is\ DigraphEdge(D, [1, 1]);\nfalse\ngap> IsDigraphEdge(D, [1, 2]);\ntrue\ngap> IsD\ igraphEdge(D, [1, 8]);\nfalse\n]]>\n \n\ \n\n \n \n <\ Oper Name=\"IsMaximalMatching\" Arg=\"digraph, list\"/>\n \n \n true or false.\n \n If digraph is a digraph and list is a list of pai\ rs of\n vertices of digraph, then IsMatching returns true<\ /K> if\n list is a matching of digraph. The operation\n <\ C>IsMaximalMatching returns true if list is a maximal \n \ matching, IsMaximumMatching returns true if list is a\n \ maximum matching and IsPerfectMatching returns true if \n <\ A>list is a perfect, matching of digraph, respectively. \n Othe\ rwise, each of these operations return false.\n \n\n A mat\ ching M of a digraph digraph is a subset of the\n edges o\ f digraph, i.e. DigraphEdges(digraph), such\n that no \ pair of distinct edges in M are incident to the same vertex\n of \ digraph. Note that this definition allows a matching to contain\n loop\ s. See . The matching M is\n ma\ ximal if it is contained in no larger matching of the digraph,\n is \ maximum if it has the greatest cardinality among all matchings and\n is\ perfect if every vertex of the digraph is incident to an edge in\n \ the matching. Every maximum or perfect matching is maximal. Note, however,\n \ that not every perfect matching of digraphs with loops is maximum.\n\n <\ Example> D := Digraph([[1, 2], [1, 2], [2, 3, 4], [3, 5], [1]])\ ;\n\ngap> IsMatching(D, [[2, 1], \ [3, 2]]);\nfalse\ngap> edges := [[3, 2]];;\ngap> IsMatching(D, edges);\ntrue\n\ gap> IsMaximalMatching(D, edges);\nfalse\ngap> edges := [[2, 1], [3, 4]];;\nga\ p> IsMaximalMatching(D, edges);\ntrue\ngap> IsPerfectMatching(D, edges);\nfals\ e\ngap> edges := [[1, 2], [3, 3], [4, 5]];;\ngap> IsPerfectMatching(D, edges);\ \ntrue\ngap> IsMaximumMatching(D, edges);\nfalse\ngap> edges := [[1, 1], [2, 2\ ], [3, 3], [4, 5]];;\ngap> IsMaximumMatching(D, edges);\ntrue\n]]>\n\ \n\n\n \n \n A list of pairs of vertices.\n \n This function returns a maximal matching of the di\ graph digraph.\n \n\n For the definition of a maximal matchin\ g, see . \n\n D :\ = DigraphFromDiSparse6String(\".IeAoXCJU@|SHAe?d\");\n\ngap> M := DigraphMaximalMatching(D);; IsMaximalMatchin\ g(D, M);\ntrue\ngap> D := RandomDigraph(100);;\ngap> IsMaximalMatching(D, Digr\ aphMaximalMatching(D));\ntrue\ngap> D := GeneralisedPetersenGraph(IsMutableDig\ raph, 9, 2);\n\ngap> IsMaximalMatc\ hing(D, DigraphMaximalMatching(D));\ntrue\n]]>\n \n\n\n \n \n A list of pairs of vertices.\n \n This function returns a maximum matching of the digraph digraph.\n \n\n For the definition of a maximum matching, see .\n If digraph is bipartite (see ), then \n the algorithm used has complexity O(m*sqr\ t(n)). Otherwise for general \n graphs the complexity is O(m*n*log(n\ )). Here n is the number of vertices \n and m is the numb\ er of edges.\n\n D := DigraphFromDigraph6String(\"\ &I@EA_A?AdDp[_c??OO\");\n\ngap> \ M := DigraphMaximumMatching(D);; IsMaximalMatching(D, M);\ntrue\ngap> Length(M\ );\n5\ngap> D := Digraph([[5, 6, 7, 8], [6, 7, 8], [7, 8], [8], \n> \ [], [], [], []]);;\ngap> M := DigraphMaximumMatching(D);\n[ [ 1, 5 ], [\ 2, 6 ], [ 3, 7 ], [ 4, 8 ] ]\ngap> D := GeneralisedPetersenGraph(IsMutableDig\ raph, 9, 2);\n\ngap> M := DigraphM\ aximumMatching(D);;\ngap> IsMaximalMatching(D, M);\ntrue\ngap> Length(M);\n9\n\ ]]>\n \n\n\n \n\n <\ Heading>Neighbours and degree\n \n \n \n A square matrix of non-negative integers.\n \n This function returns the adjacency matrix mat\ of the digraph\n digraph.\n The value of the matrix entry mat\ [i][j] is the number of edges\n in digraph with source i \ and range j. If digraph\n has no vertices, then the empty lis\ t is returned. \n\n The function AdjacencyMatrix returns an immu\ table list of\n lists, whereas the function AdjacencyMatrixMutableCopy returns a copy\n of AdjacencyMatrix that is a mutable list of mut\ able lists. \n\n gr := Digraph([\n> [2, 2, 2],\ [1, 3, 6, 8, 9, 10], [4, 6, 8],\n> [1, 2, 3, 9], [3, 3], [3, 5, 6, 10], [1, 2\ , 7],\n> [1, 2, 3, 10, 5, 6, 10], [1, 3, 4, 5, 8, 10],\n> [2, 3, 4, 6, 7, 10]]\ );\n\ngap> mat := Adjacency\ Matrix(gr);;\ngap> Display(mat);\n[ [ 0, 3, 0, 0, 0, 0, 0, 0, 0, 0 ]\ ,\n [ 1, 0, 1, 0, 0, 1, 0, 1, 1, 1 ],\n [ 0, 0, 0, 1, 0, 1, \ 0, 1, 0, 0 ],\n [ 1, 1, 1, 0, 0, 0, 0, 0, 1, 0 ],\n [ 0, 0, \ 2, 0, 0, 0, 0, 0, 0, 0 ],\n [ 0, 0, 1, 0, 1, 1, 0, 0, 0, 1 \ ],\n [ 1, 1, 0, 0, 0, 0, 1, 0, 0, 0 ],\n [ 1, 1, 1, 0, 1, 1,\ 0, 0, 0, 2 ],\n [ 1, 0, 1, 1, 1, 0, 0, 1, 0, 1 ],\n [ 0, 1,\ 1, 1, 0, 1, 1, 0, 0, 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, \ 3);\n\ngap> Display(AdjacencyMatrix(\ D));\n[ [ 0, 1, 0 ],\n [ 0, 0, 1 ],\n [ 1, 0, 0 ] ]\n]]>\ \n \n\n\n \n \n A polynomial with integer coe\ fficients.\n \n This function returns the character\ istic polynomial of the digraph\n digraph. That is it returns the c\ haracteristic polynomial\n of the adjacency matrix of the digraph digr\ aph\n D := Digraph([\n> [2, 2, 2], [1, 3, 6, 8\ , 9, 10], [4, 6, 8],\n> [1, 2, 3, 9], [3, 3], [3, 5, 6, 10], [1, 2, 7],\n> [1,\ 2, 3, 10, 5, 6, 10], [1, 3, 4, 5, 8, 10],\n> [2, 3, 4, 6, 7, 10]]);\n\ngap> CharacteristicPolynomial(D);\ \nx_1^10-3*x_1^9-7*x_1^8-x_1^7+14*x_1^6+x_1^5-26*x_1^4+51*x_1^3-10*x_1^2\\\n+1\ 8*x_1-30\ngap> D := CompleteDigraph(5);\n\ngap> CharacteristicPolynomial(D);\nx_1^5-10*x_1^3-20*x_1^2-15*x_1-4\n\ gap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> CharacteristicPolynomial(D);\nx_1^3-1\n]]>\n \n\n\n \n \n \n A square matrix of booleans.\n \n If digraph is a digraph with a positive number of vertices\ \n n, then BooleanAdjacencyMatrix(digraph)\n \ returns the boolean adjacency matrix mat of digraph. The\n \ value of the matrix entry mat[j][i] is true if and only if\n \ there exists an edge in digraph with source j and range\n \ i. If digraph has no vertices, then the empty list is\n returne\ d. \n\n Note that the boolean adjacency matrix loses information about \ multiple\n edges. \n\n The attribute BooleanAdjacencyMatrix \ returns an immutable list of\n immutable lists, whereas the function\n <\ C>BooleanAdjacencyMatrixMutableCopy returns a copy of the\n BooleanA\ djacencyMatrix that is a mutable list of mutable lists. \n gr := Digraph([[3, 4], [2, 3], [1, 2, 4], [4]]);\n\ngap> PrintArray(BooleanAdjacencyMatrix(gr))\ ;\n[ [ false, false, true, true ],\n [ false, true, true, false ]\ ,\n [ true, true, false, true ],\n [ false, false, false, true ]\ ]\ngap> gr := CycleDigraph(4);;\ngap> PrintArray(BooleanAdjacencyMatrix(gr));\ \n[ [ false, true, false, false ],\n [ false, false, true, false ],\ \n [ false, false, false, true ],\n [ true, false, false, false ] \ ]\ngap> BooleanAdjacencyMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDigraph\ (IsMutableDigraph, 3);\n\ngap> Print\ Array(BooleanAdjacencyMatrix(D));\n[ [ false, true, false ],\n [ false, \ false, true ],\n [ true, false, false ] ]\n]]>\n \n\n\n \n \n A function.\n \n \ If digraph is a digraph, then DigraphAdjacencyFunction returns\ \n a function which takes two integer parameters x, y and returns\n \ true if there exists an edge from vertex x to vertex y\ \n in digraph and false if not.\n\n digraph := Digraph([[1, 2], [3], []]);\n\ngap> foo := DigraphAdjacencyFunction(digraph);\nfunction( u, v ) ..\ . end\ngap> foo(1, 1);\ntrue\ngap> foo(1, 2);\ntrue\ngap> foo(1, 3);\nfalse\ng\ ap> foo(3, 1);\nfalse\ngap> gr := Digraph([\"a\", \"b\", \"c\"],\n> \ [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\", \"a\"]);\n\ngap> foo := DigraphAdjacencyFu\ nction(gr);\nfunction( u, v ) ... end\ngap> foo(1, 2);\ntrue\ngap> foo(3, 2);\ \nfalse\ngap> foo(3, 1);\nfalse\ngap> D := CycleDigraph(IsMutableDigraph, 3);\ \n\ngap> foo := DigraphAdjacencyFunc\ tion(D);\nfunction( u, v ) ... end\ngap> foo(1, 2);\ntrue\ngap> foo(2, 1);\nfa\ lse\n]]>\n \n\n\n \n \n \n A list of positive integers.\n \n DigraphRange and DigraphSource return the range and so\ urce of\n the digraph digraph. More precisely, position i in\ \n DigraphRange(digraph) is the range of the ith edge \ of\n digraph.\n\n gr := Digraph([\n> [2,\ 1, 3, 5], [1, 3, 4], [2, 3], [2], [1, 2, 3, 4]]);\n\ngap> DigraphRange(gr);\n[ 2, 1, 3, 5, 1, 3, 4, 2, 3, 2, 1\ , 2, 3, 4 ]\ngap> DigraphSource(gr);\n[ 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 5,\ 5 ]\ngap> DigraphEdges(gr);\n[ [ 1, 2 ], [ 1, 1 ], [ 1, 3 ], [ 1, 5 ], [ 2, 1\ ], [ 2, 3 ], \n [ 2, 4 ], [ 3, 2 ], [ 3, 3 ], [ 4, 2 ], [ 5, 1 ], [ 5, 2 ], \ \n [ 5, 3 ], [ 5, 4 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphRange(D);\n[ 2, 3, 1 ]\ngap>\ DigraphSource(D);\n[ 1, 2, 3 ]\n]]>\n \n\n\n \n \n \n \n \n The adjacencies of a digraph.\n \n\ This function returns the list out of out-neighbours of each vertex\ \n of the digraph digraph.\n More specifically, a vertex j\ appears in out[i] each time\n there exists an edge with source i\ and range j in\n digraph. \n\n The function Ou\ tNeighbours returns an immutable list of\n lists, whereas the function \ OutNeighboursMutableCopy returns a copy\n of OutNeighbours wh\ ich is a mutable list of mutable lists. \n\n g\ r := Digraph([\"a\", \"b\", \"c\"],\n> [\"a\", \"b\", \"b\"],\ \n> [\"b\", \"a\", \"c\"]);\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ 1, 3 ], [ ] ]\ngap> gr := D\ igraph([[1, 2, 3], [2, 1], [3]]);\n\ngap> OutNeighbours(gr);\n[ [ 1, 2, 3 ], [ 2, 1 ], [ 3 ] ]\ngap> gr := Digra\ phByAdjacencyMatrix([\n> [1, 2, 1],\n> [1, 1, 0],\n> [0, 0, 1]]);\n\ngap> OutNeighbours(gr);\n[ [ 1, 2, \ 2, 3 ], [ 1, 2 ], [ 3 ] ]\ngap> OutNeighboursMutableCopy(gr);\n[ [ 1, 2, 2, 3 \ ], [ 1, 2 ], [ 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> OutNeighbours(D);\n[ [ 2 ], [ 3 ], [ 1 \ ] ]\n]]>\n \n\n\n \n \n \n \n <\ Oper Name=\"InNeighborsMutableCopy\" Arg=\"digraph\"/>\n A list of l\ ists of vertices.\n \n This function returns the li\ st inn of in-neighbours of each vertex\n of the digraph digraph.\n More specifically, a vertex j appears in inn[i] each ti\ me\n there exists an edge with source j and range i in\n <\ A>digraph. \n\n The function InNeighbours returns an immutab\ le list of\n lists, whereas the function InNeighboursMutableCopy ret\ urns a copy\n of InNeighbours which is a mutable list of mutable lis\ ts. \n\n Note that each entry of inn is sorted into ascending or\ der. \n\n gr := Digraph([\"a\", \"b\", \"c\"],\ \n> [\"a\", \"b\", \"b\"],\n> [\"b\", \"a\",\ \"c\"]);\n\ngap> InNeighbours(gr)\ ;\n[ [ 2 ], [ 1 ], [ 2 ] ]\ngap> gr := Digraph([[1, 2, 3], [2, 1], [3]]);\n\ngap> InNeighbours(gr);\n[ [ 1, 2 ],\ [ 1, 2 ], [ 1, 3 ] ]\ngap> gr := DigraphByAdjacencyMatrix([\n> [1, 2, 1],\n>\ [1, 1, 0],\n> [0, 0, 1]]);\n\ngap> InNeighbours(gr);\n[ [ 1, 2 ], [ 1, 1, 2 ], [ 1, 3 ] ]\ngap> InNeighb\ oursMutableCopy(gr);\n[ [ 1, 2 ], [ 1, 1, 2 ], [ 1, 3 ] ]\ngap> D := CycleDigr\ aph(IsMutableDigraph, 3);\n\ngap> In\ Neighbours(D);\n[ [ 3 ], [ 1 ], [ 2 ] ]\n]]>\n \n\n\n \n \n\ \n \n A list of non-negative integers.\ \n \n\n Given a digraph digraph with n vertices\ , the function\n OutDegrees returns an immutable list out of \ length n, such that\n for a vertex i in digraph, the v\ alue of out[i] is the\n out-degree of vertex i.\n See . \n\n The function OutDegreeSequence returns the same list,\n after it has been sorted into non-increasing or\ der. \n\n The function OutDegreeSet returns the same list, sorte\ d into\n increasing order with duplicate entries removed. \n\n D := Digraph([[1, 3, 2, 2], [], [2, 1], []]);\n\ngap> OutDegrees(D);\n[ 4, 0, 2, 0 ]\ng\ ap> OutDegreeSequence(D);\n[ 4, 2, 0, 0 ]\ngap> OutDegreeSet(D);\n[ 0, 2, 4 ]\ \ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> OutDegrees(D);\n[ 1, 1, 1 ]\n]]>\n \n\n\n \n \n \n \n A list of non-negative integers.\n \n\n Given a digraph digraph with n vert\ ices, the function\n InDegrees returns an immutable list inn \ of length n, such that\n for a vertex i in digraph, th\ e value of inn[i] is\n the in-degree of vertex i.\n See . \n\n The function InDegreeSequence returns the same list,\n after it has been sorted into non-increasing or\ der. \n\n The function InDegreeSet returns the same list, sorted\ into\n increasing order with duplicate entries removed. \n\n D := Digraph([[1, 3, 2, 2, 4], [], [2, 1, 4], []]);\n\ngap> InDegrees(D);\n[ 2, 3, 1, 2 \ ]\ngap> InDegreeSequence(D);\n[ 3, 2, 2, 1 ]\ngap> InDegreeSet(D);\n[ 1, 2, 3 \ ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> InDegrees(D);\n[ 1, 1, 1 ]\n]]>\n \n\n\n \n \n The non-negative integer.\n \n This operation returns the out-degree of the vertex vertex in the\n digraph digraph.\n The out-degree of vertex is \ the number of edges in digraph\n whose source is vertex.\n \ \n D := Digraph([\n> [2, 2, 1], [1, 4], [2, 2\ , 4, 2], [1, 1, 2, 2, 1, 2, 2]]);\n\ngap> OutDegreeOfVertex(D, 1);\n3\ngap> OutDegreeOfVertex(D, 2);\n2\ng\ ap> OutDegreeOfVertex(D, 3);\n4\ngap> OutDegreeOfVertex(D, 4);\n7\n]]>\n \n\n\n \n \n \n A list of vertices.\n \n\n This operation returns the list out of vertices of t\ he digraph\n digraph.\n A vertex i appears in the list \ out each time there exists an\n edge with source vertex and ran\ ge i in digraph; in\n particular, this means that out \ may contain duplicates. \n D := Digraph([\n> [2\ , 2, 3], [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2]]);\n\ngap> OutNeighboursOfVertex(D, 1);\n[ 2, 2, 3 ]\ \ngap> OutNeighboursOfVertex(D, 3);\n[ 2, 2, 3 ]\n]]>\n \n\n\n \n \n A non-negative integer.\n \n This operation returns the in-degree of the vertex vertex i\ n the\n digraph digraph.\n The in-degree of vertex is the \ number of edges in digraph\n whose range is vertex.\n D := Digraph([\n> [2, 2, 1], [1, 4], [2, 2, 4, 2], [1, 1,\ 2, 2, 1, 2, 2]]);\n\ngap> I\ nDegreeOfVertex(D, 1);\n5\ngap> InDegreeOfVertex(D, 2);\n9\ngap> InDegreeOfVer\ tex(D, 3);\n0\ngap> InDegreeOfVertex(D, 4);\n2\n]]>\n \n\n\n \n \n \n A list of postitive vertices.\n \ \n\n This operation returns the list inn of vertices of the digraph\ \n digraph.\n A vertex i appears in the list inn ea\ ch time there exists an\n edge with source i and range vertex in digraph; in\n particular, this means that inn may contai\ n duplicates. \n\n D := Digraph([\n> [2, 2, 3]\ , [1, 3, 4], [2, 2, 3], [1, 1, 2, 2, 1, 2, 2]]);\n\ngap> InNeighboursOfVertex(D, 1);\n[ 2, 4, 4, 4 ]\ngap>\ InNeighboursOfVertex(D, 2);\n[ 1, 1, 3, 3, 4, 4, 4, 4 ]\n]]>\n \n\n\n \n \n A list of vertices.\n \n\ If digraph is a digraph, then DigraphLoops returns the list\ \n consisting of the of digraph\n \ at which there is a loop. See . \n\n D := Digraph([[2], [3], []]);\n\ngap> DigraphHasLoops(D);\nfalse\ngap> DigraphLoops(D);\n\ [ ]\ngap> D := Digraph([[3, 5], [1], [2, 4, 3], [4], [2, 1]]);\n\ngap> DigraphLoops(D);\n[ 3, 4 ]\ngap> D := Di\ graph(IsMutableDigraph, [[1], [1]]);\n\ngap> DigraphLoops(D);\n[ 1 ]\n]]>\n \n\n\n \n \n \n A positive integer or fail\n \n If the first argument is a partial order digra\ ph then these operations return the\ meet, or\n the join, of the two input vertices. If the meet (or join) is d\ oes not\n exist then fail is returned. The meet (or join) is guarant\ eed to\n exist when the first argument satisfies\n (or\n ) - see \ the documentation for these\n properties for the definition of the meet (or\ the join).\n\n D := Digraph([[1], [1, 2], [1, 3],\ [1, 2, 3, 4]]);\n\ngap> PartialOr\ derDigraphMeetOfVertices(D, 2, 3);\n4\ngap> PartialOrderDigraphJoinOfVertices(\ D, 2, 3);\n1\ngap> PartialOrderDigraphMeetOfVertices(D, 1, 2);\n2\ngap> Partia\ lOrderDigraphJoinOfVertices(D, 3, 4);\n3\ngap> D := Digraph([[1], [2], [1, 2, \ 3], [1, 2, 4]]);\n\ngap> PartialOr\ derDigraphMeetOfVertices(D, 3, 4);\nfail\ngap> PartialOrderDigraphJoinOfVertic\ es(D, 3, 4);\nfail\ngap> PartialOrderDigraphMeetOfVertices(D, 1, 2);\nfail\nga\ p> PartialOrderDigraphJoinOfVertices(D, 1, 2);\nfail\n]]>\n \n\n\n \n \n A square matrix of non-negative integers.\ \n \n This function returns the outdegree matrix mat o\ f the digraph\n digraph. The value of the ith diagonal matrix\ entry is the\n outdegree of the vertex i in digraph. All off\ -diagonal entries \n are 0.\n If digraph has no vertices, \ then the empty list is returned. \n\n See fo\ r more information.\n D := Digraph([[1, 2, 3], [4]\ , [1, 3, 4], []]);\n\ngap> PrintAr\ ray(DegreeMatrix(D));\n[ [ 3, 0, 0, 0 ],\n [ 0, 1, 0, 0 ],\n [ 0, \ 0, 3, 0 ],\n [ 0, 0, 0, 0 ] ]\ngap> D := CycleDigraph(5);;\ngap> PrintA\ rray(DegreeMatrix(D));\n[ [ 1, 0, 0, 0, 0 ],\n [ 0, 1, 0, 0, 0 ],\n\ [ 0, 0, 1, 0, 0 ],\n [ 0, 0, 0, 1, 0 ],\n [ 0, 0, 0, 0, 1 ]\ ]\ngap> DegreeMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDigraph(IsMutabl\ eDigraph, 3);\n\ngap> Display(Degree\ Matrix(D));\n[ [ 1, 0, 0 ],\n [ 0, 1, 0 ],\n [ 0, 0, 1 ] ]\n]]>\n \n\n\n \n \n A square matrix of integers.\n \n This function returns the outdegree Laplacian mat\ rix mat of the\n digraph digraph. The outdegree Laplacian mat\ rix is defined as\n DegreeMatrix(digraph) - AdjacencyMatrix(digraph)\ . If\n digraph has no vertices, then the empty list is returned. \n\n See and \ for more\n information.\n gr := Digraph([[1, 2\ , 3], [4], [1, 3, 4], []]);\n\ngap\ > PrintArray(LaplacianMatrix(gr));\n[ [ 2, -1, -1, 0 ],\n [ 0, 1, \ 0, -1 ],\n [ -1, 0, 2, -1 ],\n [ 0, 0, 0, 0 ] ]\ngap> Laplac\ ianMatrix(EmptyDigraph(0));\n[ ]\ngap> D := CycleDigraph(IsMutableDigraph, 3)\ ;\n\ngap> Display(LaplacianMatrix(D)\ );\n[ [ 1, -1, 0 ],\n [ 0, 1, -1 ],\n [ -1, 0, 1 ] ]\n]]>\n \n\n\n \n\n Reachability and connectivity\n \n \n An integer or fail.\n \n This function returns the diameter of the digraph digraph.\n \n\n If a digraph digraph is strongly connect\ ed and has at least 1\n vertex, then the diameter is the maximum sho\ rtest distance between\n any pair of distinct vertices. Otherwise then the \ diameter of\n digraph is undefined, and this function returns the va\ lue\n fail. \n\n See \ . \n D := Digraph([[2], [3], [4, 5], [5], [1, \ 2, 3, 4, 5]]);\n\ngap> DigraphDia\ meter(D);\n3\ngap> D := ChainDigraph(2);\n\ngap> DigraphDiameter(D);\nfail\ngap> IsStronglyConnectedDigraph(D);\nfa\ lse\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 6, 2);\n\ngap> DigraphDiameter(D);\n4\n]]>\n\ \n\n\n \n \n \n \n An integer or fail\n \n If there is a directed path in the digraph digraph between verte\ x\n u and vertex v, then this operation returns the length of\ the\n shortest such directed path. If no such directed path exists, then \ this\n operation returns fail. See Section for the definition of a directed path. \n\n\ If the second form is used, then list should be a list of length tw\ o,\n containing two positive integers which correspond to the vertices u\ \n and v. \n\n Note that as usual, a vertex is considered\ to be at distance 0 from\n itself . \n\n If the third form is used, \ then list1 and list2 are both lists\n of vertices. The shorte\ st directed path between list1 and\n list2 is then the length\ of the shortest directed path which starts\n with a vertex in list1\ and terminates at a vertex in list2, if\n such directed path exists\ . If list1 and list2 have non-empty\n intersection, the opera\ tion returns 0. \n D := Digraph([[2], [3\ ], [1, 4], [1, 3], [5]]);\n\ngap> \ DigraphShortestDistance(D, 1, 3);\n2\ngap> DigraphShortestDistance(D, [3, 3]);\ \n0\ngap> DigraphShortestDistance(D, 5, 2);\nfail\ngap> DigraphShortestDistanc\ e(D, [1, 2], [4, 5]);\n2\ngap> DigraphShortestDistance(D, [1, 3], [3, 5]);\n0\ \n]]>\n \n\n\n \n \n A square matri\ x.\n \n If digraph is a digraph with n vertices, then this\n function returns an n \\times n matrix ma\ t, where each entry is\n either a non-negative integer, or fail.\ If n = 0, then an\n empty list is returned. \n\n If there i\ s a directed path from vertex i to vertex j, then\n the value\ of mat[i][j] is the length of the shortest such directed\n path. If\ no such directed path exists, then the value of mat[i][j] is\n f\ ail. We use the convention that the distance from every vertex to\n it\ self is 0, i.e. mat[i][i] = 0 for all vertices i.\n \n\n The method used in this function is a version of the Floyd-Warshall\ \n algorithm, and has complexity O(n^3).\n\n D := Digraph([[1, 2], [3], [1, 2], [4]]);\n\ngap> mat := DigraphShortestDistances(D);;\ngap> PrintArray(mat\ );\n[ [ 0, 1, 2, fail ],\n [ 2, 0, 1, fail ],\n [ \ 1, 1, 0, fail ],\n [ fail, fail, fail, 0 ] ]\ngap> D := C\ ycleDigraph(IsMutableDigraph, 3);\n\ \ngap> DigraphShortestDistances(D);\n[ [ 0, 1, 2 ], [ 2, 0, 1 ], [ 1, 2, 0 ] ]\ \n]]>\n \n\n\n \n \n An \ integer, or infinity.\n \n If digraph\ is a digraph and v is a vertex in digraph,\n then this opera\ tion returns the length of the longest directed walk in\n digraph wh\ ich begins at vertex v. See Section for the definitions of directed\n walk, directed cyc\ le, and loop.\n \n\n \n \n If there exists a\ directed walk starting at vertex v\n which traverses a loop or \ a directed cycle,\n then we consider there to be a walk of infinite len\ gth from v\n (realised by repeatedly traversing the loop/directe\ d cycle),\n and so the result is infinity.\n To disallow \ walks using loops, try using\n : \ \n\n DigraphLongestDistanceFromVertex(DigraphRemoveLoops(digraph<\ /A>,v)).\n \n\n \n Otherwise, if all \ directed walks\n starting at vertex v have finite length,\n \ then the length of the longest such walk is returned.\n \n <\ /List>\n\n Note that the result is 0 if and only if v is a si\ nk of\n digraph. See .\n\n D := Digraph([[2], [3, 4], [], [5], [], [6]]);\n\ngap> DigraphLongestDistanceFromVertex(D, 1);\n3\ng\ ap> DigraphLongestDistanceFromVertex(D, 3);\n0\ngap> 3 in DigraphSinks(D);\ntr\ ue\ngap> DigraphLongestDistanceFromVertex(D, 6);\ninfinity\n]]>\n <\ /Description>\n\n\n \n \n \n A list of \ vertices\n \n This operation returns the list of al\ l vertices in digraph digraph\n such that the shortest distance to a\ vertex vertex is\n distance or is in the list distances. \n\n digraph should be a digraph, vertex should be a \ positive\n integer, distance should be a non-negative integer, and\n\ distances should be a list of non-negative integers.\n\n D := Digraph([[2], [3], [1, 4], [1, 3]]);\n\ngap> DigraphDistanceSet(D, 2, [1, 2]);\n[ 3, 1, 4\ ]\ngap> DigraphDistanceSet(D, 3, 1);\n[ 1, 4 ]\ngap> DigraphDistanceSet(D, 2,\ 0);\n[ 2 ]\n]]>\n \n\n\n \n \n An integer, or \ infinity.\n \n This attribute returns the girth of the digraph digraph.\n The girth of a digraph i\ s the length of its shortest simple circuit.\n See Section for the definitions\n of simple circuit, d\ irected cycle, and loop.\n \n\n If digraph has no directed cy\ cles, then this function will return\n infinity. If digraph \ contains a loop, then this function will\n return 1.\n \n\n \ In the worst case, the method used in this function is a version of the\n \ Floyd-Warshall algorithm, and has complexity O(n ^ 3), where\n \ n is the number of vertices in digraph. If the digraph has\n\ known automorphisms [see ], then the performan\ ce\n is likely to be better.\n \n\n For symmetric digraphs, see a\ lso . \n \ D := Digraph([[1], [1]]);\n\ngap>\ DigraphGirth(D);\n1\ngap> D := Digraph([[2, 3], [3], [4], []]);\n\ngap> DigraphGirth(D);\ninfinity\ngap> D := D\ igraph([[2, 3], [3], [4], [1]]);\n\ \ngap> DigraphGirth(D);\n3\ngap> D := GeneralisedPetersenGraph(IsMutableDigrap\ h, 6, 2);\n\ngap> DigraphGirth(D);\ \n2\n]]>\n \n\n\n \n \n An integer, or in\ finity.\n \n This attribute returns the odd \ girth of the digraph digraph.\n The odd girth of a digrap\ h is the length of its shortest simple\n circuit of odd length. See Sectio\ n for the definitions of\ simple circuit, directed\n cycle, and loop.\n \n\n If digraph\ has no directed cycles of odd length, then this function\n will return\ infinity, even if digraph has a directed cycle\n of even len\ gth. If digraph contains a loop, then this function\n will return <\ C>1.\n \n\n See also . \n D := Digraph([[2], [3, 1], [1]]);\n\ngap> DigraphOddGirth(D);\n3\ngap> D := CycleDigraph(4\ );\n\ngap> DigraphOddGirth(D);\ninfin\ ity\ngap> D := Digraph([[2], [3], [], [3], [4]]);\n\ngap> DigraphOddGirth(D);\ninfinity\ngap> D := CycleDigraph(\ IsMutableDigraph, 4);\n\ngap> Digrap\ hDisjointUnion(D, CycleDigraph(5));\n\ngap> DigraphOddGirth(D);\n5\n]]>\n \n\ \n\n \n \n An integer or infinity.\n \n I\ f digraph is a symmetric digraph, then this function returns the\n g\ irth of digraph when treated as an undirected graph (i.e. each pair\n \ of edges [i, j] and [j, i] is treated as a single edge between\ \n i and j). \n\n The girth of an undirected gr\ aph is the length of its shortest simple\n cycle, i.e. the shortest non-tri\ vial path starting and ending at the same\n vertex and passing through no v\ ertex or edge more than once. \n\n If digraph has no cycles, the\ n this function will return\n infinity. \n D := Digraph([[2, 4], [1, 3], [2, 4], [1, 3]]);\n\ngap> DigraphUndirectedGirth(D);\n4\ngap> D := Digraph(\ [[2], [1, 3], [2]]);\n\ngap> Digra\ phUndirectedGirth(D);\ninfinity\ngap> D := Digraph([[1], [], [4], [3]]);\n\ngap> DigraphUndirectedGirth(D);\n1\n\ gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> DigraphUndirectedGirth(D);\n5\n]]>\ \n \n\n\n \n \n \n A record.\n \n \ This function returns the record wcc corresponding to the weakly\n \ connected components of the digraph digraph. Two vertices of\n \ digraph are in the same weakly connected component whenever they are\n \ equal, or there exists a directed path (ignoring the orientation of edges)\n \ between them. More formally, two vertices are in the same weakly connected\ \n component of digraph if and only if they are in the same strongly\ \n connected component (see )\n of the of digraph. \ \n\n The set of weakly connected components is a partition of\n the \ vertex set of digraph. \n\n The record wcc has 2 componen\ ts: comps and id.\n The component comps is a list of t\ he weakly connected components\n of digraph (each of which is a lis\ t of vertices).\n The component id is a list such that the vertex i is an\n element of the weakly connected component comps[id[i]]. \n\n The method used in this function has complexity O(m+n),\ where\n m is the number of edges and\n n is the number of\ vertices in the digraph. \n\n DigraphNrConnectedComponents(digra\ ph) is simply a shortcut\n for Length(DigraphConnectedComponents\ (digraph).comps),\n and is no more efficient.\n\n gr := Digraph([[2], [3, 1], []]);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps := [ [ 1, 2,\ 3 ] ], id := [ 1, 1, 1 ] )\ngap> gr := Digraph([[1], [1, 2], []]);\n\ngap> DigraphConnectedComponents(gr);\nrec\ ( comps := [ [ 1, 2 ], [ 3 ] ], id := [ 1, 1, 2 ] )\ngap> DigraphNrConnectedCo\ mponents(gr);\n2\ngap> gr := EmptyDigraph(0);\n\ngap> DigraphConnectedComponents(gr);\nrec( comps := [ ], id := [ \ ] )\ngap> D := CycleDigraph(IsMutableDigraph, 2);\n\ngap> G := CycleDigraph(3);\n\ngap> DigraphDisjointUnion(D, G);\n\ngap> DigraphConnectedComponents(D);\nrec( comps := [ [ 1, 2 ], [ 3,\ 4, 5 ] ], id := [ 1, 1, 2, 2, 2 ] )\n]]>\n \n\n\n \n \n A list of vertices.\n \ \n If vertex is a vertex in the digraph digraph, then\n th\ is operation returns the connected component of vertex\n in digra\ ph.\n See for more informati\ on.\n D := Digraph([[3], [2], [1, 2], [4]]);\n\ngap> DigraphConnectedComponent(D, 3)\ ;\n[ 1, 2, 3 ]\ngap> DigraphConnectedComponent(D, 2);\n[ 1, 2, 3 ]\ngap> Digra\ phConnectedComponent(D, 4);\n[ 4 ]\n]]>\n \n\n\n \n \n \n A record.\n \n This funct\ ion returns the record scc corresponding to the strongly\n connected\ components of the digraph digraph. Two vertices of\n digraph are in the same strongly connected component whenever they\n are equal, o\ r there is a directed path from each vertex to the other. The\n set of str\ ongly connected components is a partition of the vertex set of\n digraph\ .\n \n\n The record scc has 2 components: comps an\ d id.\n The component comps is a list of the strongly connect\ ed components\n of digraph (each of which is a list of vertices).\n \ The component id is a list such that the vertex i is an\n \ element of the strongly connected component comps[id[i]]. \n\n T\ he method used in this function is a non-recursive version of Gabow's\n Alg\ orithm and has complexity O(m+n) where\n m<\ /M> is the number of edges (counting multiple edges as one) and\n n \ is the number of vertices in the digraph. \n\n DigraphNrStronglyConn\ ectedComponents(digraph) is simply a\n shortcut for\n Leng\ th(DigraphStronglyConnectedComponents(digraph).comps),\n and is \ no more efficient.\n\n gr := Digraph([[2], [3, 1],\ []]);\n\ngap> DigraphStronglyConn\ ectedComponents(gr);\nrec( comps := [ [ 3 ], [ 1, 2 ] ], id := [ 2, 2, 1 ] )\n\ gap> DigraphNrStronglyConnectedComponents(gr);\n2\ngap> D := DigraphDisjointUn\ ion(CycleDigraph(4), CycleDigraph(5));\n\ngap> DigraphStronglyConnectedComponents(D);\nrec( comps := [ [ 1, 2, 3\ , 4 ], [ 5, 6, 7, 8, 9 ] ], \n id := [ 1, 1, 1, 1, 2, 2, 2, 2, 2 ] )\ngap> Di\ graphNrStronglyConnectedComponents(D);\n2\ngap> D := CycleDigraph(IsMutableDig\ raph, 2);\n\ngap> G := CycleDigraph(\ 3);\n\ngap> DigraphDisjointUnion(D, G\ );\n\ngap> DigraphStronglyConnectedC\ omponents(D);\nrec( comps := [ [ 1, 2 ], [ 3, 4, 5 ] ], id := [ 1, 1, 2, 2, 2 \ ] )\n]]>\n \n\n\n \n \n A list of vertices.\n \n If vertex is \ a vertex in the digraph digraph, then\n this operation returns the s\ trongly connected component of vertex\n in digraph.\n See \ for more information.\n \ D := Digraph([[3], [2], [1, 2], [3]]);\n\ngap> DigraphStronglyConnectedComponent(D, 3);\ \n[ 1, 3 ]\ngap> DigraphStronglyConnectedComponent(D, 2);\n[ 2 ]\ngap> Digraph\ StronglyConnectedComponent(D, 4);\n[ 4 ]\n]]>\n \n\n\n \n \n A pair of lists of vertices, or fail.\n \ \n If digraph is a bipartite digraph, i.e. if it satis\ fies , then DigraphBicomponents returns a\n pair of bicomponents of digraph. Otherwise,\n Di\ graphBicomponents returns fail. \n\n For a bipartite digraph,\ the vertices can be partitioned into two non-empty\n sets such that the so\ urce and range of any edge are in distinct sets. The\n parts of this partit\ ion are called bicomponents of digraph.\n Equivalently, a pai\ r of bicomponents of digraph consists of the\n color-classes of a 2-\ coloring of digraph. \n\n For a bipartite digraph with at least \ 3 vertices, there is a unique pair of\n bicomponents of bipartite if and on\ ly if the digraph is connected. See \ for more information.\n D := CycleDigraph(3);\n\ngap> DigraphBicomponents(D);\nfail\nga\ p> D := ChainDigraph(5);\n\ngap> Digr\ aphBicomponents(D);\n[ [ 1, 3, 5 ], [ 2, 4 ] ]\ngap> D := Digraph([[5], [1, 4]\ , [5], [5], []]);\n\ngap> DigraphB\ icomponents(D);\n[ [ 1, 3, 4 ], [ 2, 5 ] ]\ngap> D := CompleteBipartiteDigraph\ (IsMutableDigraph, 2, 3);\n\ngap> D\ igraphBicomponents(D);\n[ [ 1, 2 ], [ 3, 4, 5 ] ]\n]]>\n \n\n\n \n \n A list of vertices.\n \n \ A connected digraph is biconnected if it is still connected (in the\ \n sense of ) when any vertex is removed.\ \n If the digraph digraph is not biconnected but is connected, then\ any\n vertex v of digraph whose removal makes the resulting \ digraph\n disconnected is called an articulation point. \n\n <\ C>ArticulationPoints returns a list of the articulation points of\n \ digraph, if any, and, in particular, returns the empty list if\n dig\ raph is not connected. \n\n Multiple edges are ignored by this meth\ od. \n\n The method used in this operation has complexity O(m+n)\ where\n m is the number of edges and n is the number of vert\ ices in\n the digraph. \n \n If D has a bridge (see ), then a node incident\n to the bridge is an articulatio\ n point if and only if it has degree at\n least 2. It follows that \ if D has a bridge and at least\n 3 nodes, then at least one o\ f the nodes incident to the bridge is\n an articulation point. The convers\ e does not hold, there are digraphs\n with articulation points, but no brid\ ges.\n \n\n See also and \n \ .\n ArticulationPoi\ nts(CycleDigraph(5));\n[ ]\ngap> D := Digraph([[2, 7], [3, 5], [4], [2], [6],\ [1], []]);;\ngap> ArticulationPoints(D);\n[ 2, 1 ]\ngap> ArticulationPoints(C\ hainDigraph(5));\n[ 4, 3, 2 ]\ngap> ArticulationPoints(NullDigraph(5));\n[ ]\ \ngap> D := ChainDigraph(IsMutableDigraph, 4);\n\ngap> ArticulationPoints(D);\n[ 3, 2 ]\n]]>\n \n\n\n \n \n A (possibly empty) list of edges.\n \n \ A connected digraph is 2-edge-connected if it is still connected (in\ \n the sense of ) when any edge is\n r\ emoved. If the digraph D is not 2-edge-connected but is\n connected\ , then any edge [u, v] of D whose removal\n makes the resulti\ ng digraph disconnected is called a bridge. \n\n Bridges r\ eturns a list of the bridges of D, if any, and, in\n particular, ret\ urns the empty list if D is not connected. \n\n Multiple edges a\ re ignored by this method. \n\n The method used in this operation has c\ omplexity O(m+n) where\n m is the number of edges and n is the number of vertices in\n the digraph.\n \n\n If D \ has a bridge, then a node incident to the bridge is an\n articulation point\ (see ) if and only if\n it has degree at\ least 2. It follows that if D has a\n bridge and at least <\ M>3 nodes, then at least one of the nodes\n incident to the bridge is a\ n articulation point. The converse does not\n hold, there are digraphs with\ articulation points, but no bridges.\n \n\n See also and \n .\n D := Digraph([[2, 5], [1, 3, 4, 5], [2, 4], [2, 3], [1, 2]])\ ;\n\ngap> Bridges(D);\n[ ]\ngap>\ D := Digraph([[2], [3], [4], [2]]);\n\ngap> Bridges(D);\n[ [ 1, 2 ] ]\ngap> Bridges(ChainDigraph(2));\n[ [ 1, 2\ ] ]\ngap> ArticulationPoints(ChainDigraph(2));\n[ ]\n]]>\n \n\n\n \n \n \n A digraph or fail.\n \n A strong orienta\ tion of a connected symmetric digraph D (if\n it exists) is a st\ rongly connected subdigraph C of D such\n that for every edge\ [u, v] of D either [u, v] or\n [v, u] is an ed\ ge of C but not both. Robbin's Theorem states\n that a digraph admi\ ts a strong orientation if and only if it is\n bridgeless (see ). \n \n\n This operation returns a strong or\ ientation of the digraph D if \n D is symmetric and D \ admits a strong orientation. If D\n is symmetric but does not admit \ a strong orientation, then fail is\n returned. If D is not sy\ mmetric, then an error is given. \n \n\n If D is immutable, <\ C>StrongOrientation(D) returns an\n immutable digraph, and if D is mutable, then\n StrongOrientation(D) returns a muta\ ble digraph. \n\n The method used in this operation has complexity O\ (m+n) where\n m is the number of edges and n is the numbe\ r of vertices in\n the digraph.\n StrongOrientation\ (DigraphSymmetricClosure(CycleDigraph(5))) \n> = CycleDigraph(5);\ntrue\ngap> \ D := DigraphSymmetricClosure(Digraph(\n> [[2, 7], [3, 5], [4], [2], [6], [1], \ []]));;\ngap> IsBridgelessDigraph(D);\nfalse\ngap> StrongOrientation(D);\nfail\ \ngap> StrongOrientation(NullDigraph(0));\n\ngap> StrongOrientation(DigraphDisjointUnion(CompleteDigraph(3), \n> \ CompleteDigraph(3)));\nfail\n]]>\n \n\n\n \n \n An integer.\n \n This function returns the period of the digraph digraph. \n\ \n If a digraph digraph has at least one directed cycle, then the pe\ riod\n is the greatest positive integer which divides the lengths of all di\ rected\n cycles of digraph. If digraph has no directed cycle\ s, then\n this function returns 0. See Section for the definition of a directed cycle. \n\n A digraph with a period of 1 is said to be aperiodic. \ See\n . \n D := Digraph([[6], [1], [2], [3], [4, 4], [5]]);\n\ngap> DigraphPeriod(D);\n6\ngap> D := Digraph([[2], [\ 3, 5], [4], [5], [1, 2]]);\n\ngap>\ DigraphPeriod(D);\n1\ngap> D := ChainDigraph(2);\n\ngap> DigraphPeriod(D);\n0\ngap> IsAcyclicDigraph(D);\ntrue\nga\ p> D := GeneralisedPetersenGraph(IsMutableDigraph, 5, 2);\n\ngap> DigraphPeriod(D);\n1\n]]>\n \n\n\n \n \n A matrix.\n <\ Description>\n If digraph is a digraph with n vertices, then\ \n this operation returns an n \\times n matrix mat containin\ g\n the output of a generalised version of the Floyd-Warshall algorithm,\n \ applied to digraph. \n\n The operation DigraphFloydWarshal\ l is customised by the arguments\n func, nopath, and e\ dge.\n The arguments nopath and edge can be arbitrary &GA\ P; objects.\n The argument func must be a function which accepts 4 a\ rguments:\n the matrix mat, followed by 3 postive integers. The func\ tion\n func is where the work to calculate the desired outcome must \ be\n performed. \n\n This method initialises the matrix mat b\ y setting entry\n mat[i][j] to equal edge if there is an edge\ with source\n i and range j, and by setting entry mat[i][\ j] to equal\n nopath otherwise.\n The final part of Digrap\ hFloydWarshall then calls the function\n func inside three neste\ d for loops, over the vertices of\n digraph: \n\n \n\n The matrix mat is then returne\ d as the result. An example of using\n DigraphFloydWarshall to calcu\ late the shortest (non-zero) distances\n between the vertices of a digraph \ is shown below: \n\n D := DigraphFromDigraph6S\ tring(\"&EAHQeDB\");\n\ngap> func\ := function(mat, i, j, k)\n> if mat[i][k] <> -1 and mat[k][j] <> -1 then\n>\ if (mat[i][j] = -1) or (mat[i][j] > mat[i][k] + mat[k][j]) then\n> \ mat[i][j] := mat[i][k] + mat[k][j];\n> fi;\n> fi;\n> end;\nfunction( mat\ , i, j, k ) ... end\ngap> shortest_distances := DigraphFloydWarshall(D, func, \ -1, 1);;\ngap> Display(shortest_distances);\n[ [ 3, -1, -1, 2, 1, 2 \ ],\n [ 4, 2, 1, 3, 2, 1 ],\n [ 3, 1, 2, 2, 1, 2 ],\n\ [ 1, -1, -1, 1, 1, 2 ],\n [ 2, -1, -1, 1, 2, 1 ],\n [ \ 3, -1, -1, 2, 1, 1 ] ]\n]]>\n \n\n\n \n \n\ true or false.\n \n This o\ peration returns true if there exists a non-trivial directed\n walk \ from vertex u to vertex v in the digraph digraph,\n an\ d false if there does not exist such a directed walk. See Section\n \ for the definition of a\n \ non-trivial directed walk.\n \n\n The method for IsReachable\ has worst case complexity of O(m +\n n) where m is the num\ ber of edges and n the number of\n vertices in digraph.\n\n D := Digraph([[2], [3], [2, 3]]);\n\ngap> IsReachable(D, 1, 3);\ntrue\ngap> IsReachable(D\ , 2, 1);\nfalse\ngap> IsReachable(D, 3, 3);\ntrue\ngap> IsReachable(D, 1, 1);\ \nfalse\n]]>\n \n\n\n \n \ \n A li\ st.\n \n This operation returns a list of the verti\ ces v, for which there exists\n a non-trivial directed walk from ve\ rtex root to vertex v in the digraph\n digraph. See Se\ ction \n for the definition\ of a non-trivial directed walk.\n \n\n The method for VerticesRe\ achableFrom has worst case complexity of O(m +\n n) where m\ is the number of edges and n the number of\n vertices in dig\ raph.\n\n D := CompleteDigraph(5);\n\ngap> VerticesReachableFrom(D, 1);\n[ 2, 1, 3, \ 4, 5 ]\ngap> VerticesReachableFrom(D, 3);\n[ 1, 2, 3, 4, 5 ]\ngap> D := EmptyD\ igraph(5);\n\ngap> VerticesReachableF\ rom(D, 1);\n[ ]\ngap> VerticesReachableFrom(D, 3);\n[ ]\ngap> D := CycleDigr\ aph(4);\n\ngap> VerticesReachableFrom\ (D, 1);\n[ 2, 3, 4, 1 ]\ngap> VerticesReachableFrom(D, 3);\n[ 4, 1, 2, 3 ]\nga\ p> D := ChainDigraph(5);\n\ngap> Vert\ icesReachableFrom(D, 1);\n[ 2, 3, 4, 5 ]\ngap> VerticesReachableFrom(D, 3);\n[\ 4, 5 ]\ngap> VerticesReachableFrom(D, 5);\n[ ]\n]]>\n \n\n\n \n \n A pair of lists, or fail.\n \n If there exists a non-trivial directed path (or a non-trivial cy\ cle, in the\n case that u = v) from vertex u to\ vertex\n v in the digraph digraph, then this operation retur\ ns such a\n directed path (or directed cycle). Otherwise, this operation re\ turns\n fail. See Section for the\n definition of a directed path and a directed cycle.\n \n\n A directed path (or directed cycle) of non-zero length n-1,\n \ (v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n),\n is represented by a pa\ ir of lists [v,a] as follows:\n\n \n \n v\ is the list [v_1, v_2, ..., v_n].\n \n \n \ a is the list of positive integers [a_1, a_2, ..., a_{n-1}]\n where for each each i < n, a_i is the position of\ \n v_{i+1} in\n OutNeighboursOfVertex(digraph,v_i)\n corresponding to the edge e_i. T\ his is can be useful if the\n position of a vertex in a list of out-nei\ ghours is significant, for\n example in orbit digraphs.\n \ \n \n\n The method for DigraphPath has worst case complexi\ ty of O(m +\n n) where m is the number of edges and n the number of\n vertices in digraph.\n\n D\ := Digraph([[2], [3], [2, 3]]);\n\ \ngap> DigraphPath(D, 1, 3);\n[ [ 1, 2, 3 ], [ 1, 1 ] ]\ngap> DigraphPath(D, 2\ , 1);\nfail\ngap> DigraphPath(D, 3, 3);\n[ [ 3, 3 ], [ 2 ] ]\ngap> DigraphPath\ (D, 1, 1);\nfail\n]]>\n \n\n\n \n \n A pair of lists, or fail.\n \n Returns th\ e shortest directed path in the digraph digraph from the vertex\n u \ to the vertex v, if such a path exists. If u =\n v<\ /C>, then the shortest non-trivial cycle is returned, again, if it\n exists\ . Otherwise, this operation returns fail. See Section\n for the definition of a directed\n path \ and a directed cycle.\n \n\n See for det\ ails on the output.\n\n The method for DigraphShortestPath has worst\ case complexity of\n O(m + n) where m is the number of edges\ and n the\n number of vertices in digraph.\n\n D := Digraph([[1, 2], [3], [2, 4], [1], [2, 4]]);\n\ngap> DigraphShortestPath(D, 5, 1);\n[ [ 5, 4, 1 \ ], [ 2, 1 ] ]\ngap> DigraphShortestPath(D, 3, 3);\n[ [ 3, 2, 3 ], [ 1, 1 ] ]\n\ gap> DigraphShortestPath(D, 5, 5);\nfail\ngap> DigraphShortestPath(D, 1, 1);\n\ [ [ 1, 1 ], [ 1 ] ]\n]]>\n \n\n\n \n \n An iterator.\n \n If digraph is a digraph \ or a list of adjacencies which defines a\n digraph - see - then this operation returns an\n iterator of the non-trivial \ directed paths (or directed cycles, in the case\n that u = v) in digraph from the vertex u\n to the vertex v\ .\n \n\n See for more information about \ the repesentation\n of a directed path or directed cycle which is used, and\ see for more information ab\ out iterators.\n See Section \ for the definition of\n a directed path and a directed cycle.\n \n\n\ D := Digraph([[1, 4, 4, 2], [3, 5], [2, 3], [1, 2], [\ 4]]);\n\ngap> iter := Iterat\ orOfPaths(D, 1, 4);\n\ngap> NextIterator(iter);\n[ [ 1, 4 ], [ 2 ] ]\ \ngap> NextIterator(iter);\n[ [ 1, 4 ], [ 3 ] ]\ngap> NextIterator(iter);\n[ [\ 1, 2, 5, 4 ], [ 4, 2, 1 ] ]\ngap> IsDoneIterator(iter);\ntrue\ngap> iter := I\ teratorOfPaths(D, 4, 3);\n\ngap> NextIterator(iter);\n[ [ 4, 1, 2, 3\ ], [ 1, 4, 1 ] ]\n]]>\n \n\n\n \n \n A list of lists of vertices.\n \n If digraph<\ /A> is a digraph, then DigraphAllSimpleCircuits\n returns a list of \ the simple circuits in digraph. \n\n See Section for the definition\n of a simple c\ ircuit, and related notions. Note that a loop is a simple\n circuit. \n\ \n For a digraph without multiple edges, a simple circuit is uniquely\n \ determined by its subsequence of vertices. However this is not the case for\n \ a multidigraph. The attribute DigraphAllSimpleCircuits ignores\n \ multiple edges, and identifies a simple circuit using only its subsequence\n \ of vertices. For example, although the simple circuits (v, e, v) and\ \n (v, e', v) (for distinct edges e and e') are\n m\ athematically distinct, DigraphAllSimpleCircuits considers them to\n \ be the same. \n\n With this approach, a directed circuit of length \ n can be determined\n by a list of its first n vertices. Thus a \ simple circuit (v_1,\n e_1, v_2, e_2, ..., e_{n-1}, v_n, e_{n+1}, v_1)\ can be represented as\n the list [v_1, \\ldots, v_n], or any cy\ clic permutation thereof. For\n each simple circuit of digraph,\n \ DigraphAllSimpleCircuits(digraph) includes precisely one such\ \n list to represent the circuit. \n\n D :\ = Digraph([[], [3], [2, 4], [5, 4], [4]]);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 4 ], [ 4, 5 ], [ 2, 3 ] ]\n\ gap> D := ChainDigraph(10);;\ngap> DigraphAllSimpleCircuits(D);\n[ ]\ngap> D \ := Digraph([[3], [1], [1]]);\n\nga\ p> DigraphAllSimpleCircuits(D);\n[ [ 1, 3 ] ]\ngap> D := Digraph([[1, 1]]);\n<\ immutable multidigraph with 1 vertex, 2 edges>\ngap> DigraphAllSimpleCircuits(\ D);\n[ [ 1 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 3);\n\ngap> DigraphAllSimpleCircuits(D);\n[ [ 1, 2, 3 ] ]\ \n]]>\n \n\n\n \n \n A list of v\ ertices, or fail.\n \n If digraph is \ a digraph, then DigraphLongestSimpleCircuit\n returns the longest simple circuit in digraph. See Section\n for the definition of simple\n circuit, and the \ definition of length for a simple circuit. \n\n This attribute computes\ \n DigraphAllSimpleCircuits(digraph) to find all the\n\ simple circuits of digraph, and returns one of maximal length. A\n\ simple circuit is represented as a list of vertices, in the same way as\n \ described in . \n\n If dig\ raph has no simple circuits, then this attribute returns\n fail.\ If digraph has multiple simple circuits of maximal\n length, then \ this attribute returns one of them. \n\n D := D\ igraph([[], [3], [2, 4], [5, 4], [4]]);;\ngap> DigraphLongestSimpleCircuit(D);\ \n[ 4, 5 ]\ngap> D := ChainDigraph(10);;\ngap> DigraphLongestSimpleCircuit(D);\ \nfail\ngap> D := Digraph([[3], [1], [1, 4], [1, 1]]);;\ngap> DigraphLongestSi\ mpleCircuit(D);\n[ 1, 3, 4 ]\ngap> D := GeneralisedPetersenGraph(IsMutableDigr\ aph, 4, 1);\n\ngap> DigraphLongestS\ impleCircuit(D);\n[ 1, 2, 3, 4, 8, 7, 6, 5 ]\n]]>\n \ \n\n\n \n \n A list.\n \n This opera\ tion returns a list list such that list[i] is the\n list of v\ ertices whose minimum distance from the vertex vertex in\n digrap\ h is i - 1. Vertex vertex is\n assumed to be at distance \ 0 from itself.\n D := CompleteDigraph(4);;\ \ngap> DigraphLayers(D, 1);\n[ [ 1 ], [ 2, 3, 4 ] ]\n]]>\n \n\n\n \n \n A non-negative integer, or fail.\n \n If digraph is a symmetric digraph without multi\ ple edges - see\n and - then\n this attribute returns the degeneracy of digraph. \n\n The degeneracy of a digraph is the least integer k such\ \n that every induced of digraph contains a vertex whose number of\n\ neighbours (excluding itself) is at most k. Note that this means\n \ that loops are ignored. \n\n If digraph is not symmetric or ha\ s multiple edges then this\n attribute returns fail.\n <\ ![CDATA[\ngap> D := DigraphSymmetricClosure(ChainDigraph(5));;\ngap> DigraphDe\ generacy(D);\n1\ngap> D := CompleteDigraph(5);;\ngap> DigraphDegeneracy(D);\n4\ \ngap> D := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4], [2], []]);\n\ngap> DigraphDegeneracy(D);\n1\ngap> D := \ GeneralisedPetersenGraph(IsMutableDigraph, 10, 3);\n\ngap> DigraphDegeneracy(D);\n3\n]]>\n \n\n\n \n \n A list of integers, or fail.\n \n If digraph is a digraph for which\n Dig\ raphDegeneracy(digraph) is a non-negative integer\n k<\ /C> - see - then this attribute\n returns\ a degeneracy ordering of the vertices of the vertices of\n digraph.\ \n\n A degeneracy ordering of digraph is a list ordering \ of the\n vertices of digraph ordered such that for any\n position\ i of the list, the vertex ordering[i] has at most\n k\ neighbours in later position of the list. \n\n If DigraphDegeneracy(\ digraph) returns fail,\n then this attribute retur\ ns fail.\n D := DigraphSymmetricClosure(Cha\ inDigraph(5));;\ngap> DigraphDegeneracyOrdering(D);\n[ 5, 4, 3, 2, 1 ]\ngap> D\ := CompleteDigraph(5);;\ngap> DigraphDegeneracyOrdering(D);\n[ 5, 4, 3, 2, 1 \ ]\ngap> D := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4], [2], []]);\n\ngap> DigraphDegeneracyOrdering(D);\n[ 1,\ 6, 5, 2, 4, 3 ]\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 3, 1);\ \n\ngap> DigraphDegeneracyOrdering(\ D);\n[ 6, 5, 4, 1, 3, 2 ]\n]]>\n \n\n\n \ \n \n A list or fail.\n \n Returns a Hamiltonian\ path if one exists, fail if not. \n\n A Hamiltonian path \ of a digraph with n vertices is directed\n cycle of length n.\ If digraph is a digraph that contains a\n Hamiltonian path, then th\ is function returns one, described in the form\n used by . Note if digraph has\n 0 or 1 vertices, then \ HamiltonianPath returns [] or\n [1], respectively. \ \n\n The method used in this attribute has the same worst case complexity a\ s\n .\n\n D \ := Digraph([[]]);\n\ngap> HamiltonianPa\ th(D);\n[ 1 ]\ngap> D := Digraph([[2], [1]]);\n\ngap> HamiltonianPath(D);\n[ 1, 2 ]\ngap> D := Digraph([[3], [],\ [2]]);\n\ngap> HamiltonianPath(D)\ ;\nfail\ngap> D := Digraph([[2], [3], [1]]);\n\ngap> HamiltonianPath(D);\n[ 1, 2, 3 ]\ngap> D := GeneralisedPete\ rsenGraph(IsMutableDigraph, 5, 2);\n\ngap> HamiltonianPath(D);\nfail\n]]>\n \n\n\n \n \n\ An integer.\n \n Returns the number of s\ panning trees of the symmetric digraph\n digraph.\n NrSpanning\ Trees will return an error if digraph is not a\n symmetric digra\ ph. \n\n See and\n for more information. \n\n D := CompleteDigraph(5);\n\ \ngap> NrSpanningTrees(D);\n125\ngap> D := DigraphSymmetricClosure(CycleDigrap\ h(24));;\ngap> NrSpanningTrees(D);\n24\ngap> NrSpanningTrees(EmptyDigraph(0));\ \n0\ngap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);\n\ngap> NrSpanningTrees(D);\n1134225\n]]>\n \n\n\n \n \n \n Two lists.\n \n If digraph is a digraph and source and target are\n vertices of dig\ raph, then DigraphDijkstra calculates the\n length of the shortest\ path from source to target and returns\n two lists. Each eleme\ nt of the first list is the distance of the\n corresponding element from s\ ource. If a vertex was not visited in\n the process of calculating the s\ hortest distance to target or if\n there is no path connecting that ve\ rtex with source, then\n the corresponding distance is infinity\ . Each element of the\n second list gives the previous vertex in the shortes\ t path\n from source to the corresponding vertex. For\n source and for any vertices that remained unvisited this\n will be -1. \n\n If the optional second argument target is not present, then\n \ DigraphDijkstra returns the shortest path from source to\n \ every vertex that is reachable from source.\n\n mat := [[0, 1, 1], [0, 0, 1], [0, 0, 0]];\n[ [ 0, 1, 1 ], [ 0, 0, 1 ], [ 0\ , 0, 0 ] ]\ngap> D := DigraphByAdjacencyMatrix(mat);\n\ngap> DigraphDijkstra(D, 2, 3);\n[ [ infinity, 0, 1 ], [ \ -1, -1, 2 ] ]\ngap> DigraphDijkstra(D, 1, 3);\n[ [ 0, 1, 1 ], [ -1, 1, 1 ] ]\n\ gap> DigraphDijkstra(D, 1, 2);\n[ [ 0, 1, 1 ], [ -1, 1, 1 ] ]\n]]>\n\ \n\n\n \n\n Cayley gr\ aphs of groups\n \n \n \n A group or semigroup.\n \n If d\ igraph is an immutable Cayley graph of a group G and\n digrap\ h belongs to the category , then\n Gr\ oupOfCayleyDigraph returns G.\n \n\n If digraph is\ a Cayley graph of a semigroup S and\n digraph belongs to the\ category , then\n SemigroupOfCayleyDigra\ ph returns S.\n \n\n See also .\n G := DihedralGroup(IsPermGroup, \ 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> digraph := CayleyDigraph(G);\n\ngap> GroupOfCayleyDigraph(digraph) = G;\ \ntrue\n]]>\n \n\n\n \n \ \n A list \ of generators.\n \n If digraph is an immutab\ le Cayley graph of a group or semigroup with\n respect to a set of generato\ rs gens and digraph belongs to\n the category , then\n GeneratorsOfCayleyDigraph return the list o\ f generators gens\n over which digraph is defined.\n \ \n\n See also \n or .\n G := DihedralGroup(Is\ PermGroup, 8);\nGroup([ (1,2,3,4), (2,4) ])\ngap> digraph := CayleyDigraph(G);\ \n\ngap> GeneratorsOfCayleyDigrap\ h(digraph) = GeneratorsOfGroup(G);\ntrue\ngap> digraph := CayleyDigraph(G, [()\ ]);\n\ngap> GeneratorsOfCayleyDigr\ aph(digraph) = [()];\ntrue]]>\n \n\n\n \ \n\n Associated semigroups\n \n \n \n A semilattice of partial perms.\n \n The operation AsSemigroup requires that filt be equal t\ o\n .\n If digraph is a\n or \n then AsSemigroup returns a semigroup of \ partial perms which is\n isomorphic to the semigroup whose elements are t\ he vertices of\n digraph with the binary operation\n .\n If digraph\n sat\ isfies but not\n then\n AsSemigroup returns a semigroup \ of partial perms which is\n isomorphic to the semigroup whose elements ar\ e the vertices of\n digraph with the binary operation\n .\n \n\n The operati\ on AsMonoid behaves similarly to AsSemigroup\n except that \ filt may also be equal to\n ,\n digraph must satisfy , and the\n output satisfies . \n\n The output of both of these operations is guaranteed to be o\ f minimal\n degree (see ).\n Furthermore the of\n the output is guaranteed to be the unique generating \ set of minimal size.\n di := Digraph([[1], [1, 2\ ], [1, 3], [1, 4], [1, 2, 3, 5]]);\n\ngap> S := AsSemigroup(IsPartialPermSemigroup, di);\n\ngap> ForAll(Elements(S), IsIdempotent);\ntrue\ \ngap> IsInverseSemigroup(S);\ntrue\ngap> Size(S);\n5\ngap> di := Digraph([[1]\ , [1, 2], [1, 2, 3]]);\n\ngap> M :\ = AsMonoid(IsPartialPermMonoid, di);\n\ngap> Size(M);\n3]]>\n \n \ \n\n \n \n <\ Returns> A Clifford semigroup of partial perms. \n \ \n The operation AsSemigroup requires that filt be equal to\ \n .\n If \ Y is a or\n ,\n gps is a list of groups corresponding \ to each vertex, and\n homs is a list containing for each edge (\ i, j) in\n the transitive reduction of digraph a triple [i,\ j, hom]\n where hom is a group homomorphism from gps[i] to\n gps[j], and the diagram of homomorphisms commutes, then\n \ AsSemigroup returns a semigroup of partial perms which is\n is\ omorphic to the strong semilattice of groups S[Y; gps; homs].\n G1 := AlternatingGroup(4);;\ngap> G2 := SymmetricGroup(\ 2);;\ngap> G3 := SymmetricGroup(3);;\ngap> gr := Digraph([[1, 3], [2, 3], [3]]\ );;\ngap> sgn := function(x)\n> if SignPerm(x) = 1 then\n> return ();\n> fi;\n\ > return (1, 2);\n> end;;\ngap> hom13 := GroupHomomorphismByFunction(G1, G3, s\ gn);;\ngap> hom23 := GroupHomomorphismByFunction(G2, G3, sgn);;\ngap> T := AsS\ emigroup(IsPartialPermSemigroup,\n> gr,\n> [G1, G2, G3], [[1, 3, hom13], [2, 3\ , hom23]]);;\ngap> Size(T);\n20\ngap> D := GreensDClasses(T);;\ngap> List(D, x\ -> Size(x));\n[ 6, 12, 2 ]\n]]>\n \n \n\n \n\n Planarity\n \ \n \n A \ list or fail.\n \n KuratowskiPlanarSubdig\ raph returns the immutable list of\n lists of out-neighbours of a (not \ necessarily induced) subdigraph of the\n digraph digraph that witnes\ ses the fact that digraph is not\n planar, or fail if digr\ aph is planar. In other words,\n KuratowskiPlanarSubdigraph ret\ urns the out-neighbours of a\n subdigraph of digraph that is homeomo\ rphic to the complete graph\n with 5 vertices, or to the complete bi\ partite graph with vertex sets\n of sizes 3 and 3. \n \ \n The directions and multiplicities of any edges in digraph are\n \ ignored when considering whether or not digraph is planar. \n\n \ See also \n and .\n \n \n This method uses the reference impleme\ ntation in\n &edge-addition-planarity-suite; by John Boyer of the algorithm\ s described\n in .\n\n D := Dig\ raph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4, 7], [6, 8], \ [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiPlanarSubdigraph(D);\nfail\ngap> D := Digraph([[2, 4, 7, \ 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, \ 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, \ 6, 7, 8], [3, 5]]);\n\ngap> IsPl\ anarDigraph(D);\nfalse\ngap> KuratowskiPlanarSubdigraph(D);\n[ [ 2, 9, 7 ], [ \ 3 ], [ 6 ], [ 5, 9 ], [ 6 ], [ ], [ 4 ], \n [ 7, 9, 3 ], [ ], [ ] ]\ngap> \ D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1,\ 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiPlanarSubdigraph(D);\nfail\ngap> D\ := Digraph(IsMutableDigraph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 1\ 0], [2, 5, 8, 9],\n> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4,\ 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> IsPlanarDigraph(D);\nfalse\ngap> Kuratowsk\ iPlanarSubdigraph(D);\n[ [ 2, 9, 7 ], [ 3 ], [ 6 ], [ 5, 9 ], [ 6 ], [ ], [ 4\ ], \n [ 7, 9, 3 ], [ ], [ ] ]\n]]>\n \n\n\n \n \n A list or fail.\n \n \ KuratowskiOuterPlanarSubdigraph returns the immutable list of\n i\ mmutable lists of out-neighbours of a (not necessarily induced)\n subdigrap\ h of the digraph digraph that witnesses the fact\n that digraph is not outer planar, or fail if\n digraph is outer planar.\ In other words,\n KuratowskiOuterPlanarSubdigraph returns the out-\ neighbours of a\n subdigraph of digraph that is homeomorphic to the \ complete graph\n with 4 vertices, or to the complete bipartite graph\ with vertex sets\n of sizes 2 and 3. \n \n The dir\ ections and multiplicities of any edges in digraph are\n ignored whe\ n considering whether or not digraph is outer planar. \n \n\n \ See also \n ,\n , and \n \ .\n \n\n This method uses the reference implementation in\n &edge\ -addition-planarity-suite; by John Boyer of the algorithms described\n in <\ Cite Key=\"BM06\"/>.\n\n D := Digraph([[3, 5, 10], [8,\ 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, \ 2], [5, 9]]);\n\ngap> Kuratowski\ OuterPlanarSubdigraph(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11 ],\ [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph([[2, 4, 7, 9,\ 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, 10\ ], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6,\ 7, 8], [3, 5]]);\n\ngap> IsOute\ rPlanarDigraph(D);\nfalse\ngap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ ], [\ ], [ ], [ 8, 9 ], [ ], [ ], [ 9, 4 ], [ 7, 9 ], [ ], \n [ ] ]\ngap> D \ := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7\ , 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ 3, 5,\ 10 ], [ 9, 8, 10 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ \ ], [ ] ]\ngap> D := Digraph(IsMutableDigraph, [[2, 4, 7, 9, 10],\n> [1, 3, 4,\ 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, \ 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\ \n\ngap> IsOuterPlanarDigraph(D);\ \nfalse\ngap> KuratowskiOuterPlanarSubdigraph(D);\n[ [ ], [ ], [ ], [ 8, 9 \ ], [ ], [ ], [ 9, 4 ], [ 7, 9 ], [ ], \n [ ] ]]]>\n \n\n\n \n \n A list or fail.\n \n \ If digraph is a planar digraph, then \n PlanarEmbedding re\ turns the immutable list of lists of\n out-neighbours of a subdigraph of digraph such that each vertex's\n neighbours are given in clockwise or\ der. If digraph is not planar,\n then fail is returned. \ \n \n The directions and multiplicities of any edges in digraph a\ re\n ignored by PlanarEmbedding.\n \n\n See also \n .\n \n\n This method uses the reference \ implementation in\n &edge-addition-planarity-suite; by John Boyer of the al\ gorithms described\n in .\n\n D\ := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [4, 7], [\ 6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> PlanarEmbedding(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ \ 6 ], [ 11, 7 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digrap\ h([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3,\ 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9\ ], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> PlanarEmbedding(D);\nfail\ngap> D := Digraph(IsMutableDigraph, [[3, 5\ , 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, \ 11], [1, 2], [5, 9]]);\n\ngap> Pla\ narEmbedding(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11, 7 ], [ 7 ]\ , [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(IsMutableDigraph, [\ [2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1, 2, 3, \ 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9],\ [2, 3, 5, 6, 7, 8], [3, 5]]);\n\n\ gap> PlanarEmbedding(D);\nfail\n]]>\n \n\ \n\n \n \ \n A list or fail.\n \n If digr\ aph is an outer planar digraph, then\n OuterPlanarEmbedding retu\ rns the immutable list of lists\n of out-neighbours of a subdigraph of d\ igraph such that each\n vertex's neighbours are given in clockwise orde\ r. If digraph is not\n outer planar, then fail is returned. <\ P/>\n \n The directions and multiplicities of any edges in digraph are\n ignored by OuterPlanarEmbedding.\n \n\n See also <\ Ref Prop=\"IsOuterPlanarDigraph\"/>.\n \n\n This method uses the ref\ erence implementation in\n &edge-addition-planarity-suite; by John Boyer of\ the algorithms described\n in .\n\n D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], \n> [1, 7, 11], [\ 4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> D := Digraph([[\ 2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10], \n> [2, 5, 8, 9], [1, 2, 3, 4, \ 6, 7, 9, 10], [3, 4, 5, 7, 9, 10], \n> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [\ 2, 3, 5, 6, 7, 8], [3, 5]]);\n\n\ gap> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbedding(CompleteBiparti\ teDigraph(2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\ngap> D := Digraph(IsMut\ ableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [4, 7], [\ 6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> D := Digraph(IsMutableDig\ raph, [[2, 4, 7, 9, 10],\n> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],\n> [1,\ 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],\n> [3, 4, 5, 6, 9, 10], [3, 4, 5,\ 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);\n\ngap> OuterPlanarEmbedding(D);\nfail\ngap> OuterPlanarEmbedding(Complete\ BipartiteDigraph(2, 2));\n[ [ 3, 4 ], [ 4, 3 ], [ ], [ ] ]\n]]>\n \ \n\n\n \n \n \n \n A list or fail.\n \n The\ se attributes return the immutable list of lists of\n out-neighbours of a s\ ubdigraph of the digraph digraph which is\n homeomorphic to one of t\ he following:\n the complete bipartite graph with vertex sets of sizes 2\ and\n 3; the complete bipartite graph with vertex sets of sizes\ 3\n and 3; or the complete graph with 4 vertices. If\ \n digraph has no such subdigraphs, then fail is returned. \n\ \n\n See also and for more details. \n\n This method uses the r\ eference implementation in\n &edge-addition-planarity-suite; by John Boyer \ of the algorithms described\n in .\n\n D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], [1, 7, 11], \n> \ [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\n[ [ 3, 5, 10 ], [ 9\ , 8, 10 ], [ 4 ], [ 6 ], [ 7, 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ]\ ]\ngap> SubdigraphHomeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ],\ [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digrap\ h([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], [1, 11], \n> [4, 7], [6, 8], [2, 7\ ], [2, 11], [1, 2], [5, 9]]);\n\ \ngap> SubdigraphHomeomorphicToK4(D);\nfail\ngap> SubdigraphHomeomorphicToK23(\ D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ \ ], [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\nfail\ngap> Subd\ igraphHomeomorphicToK23(NullDigraph(0));\nfail\ngap> SubdigraphHomeomorphicToK\ 33(CompleteDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK33(CompleteBipart\ iteDigraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ ], [ ], [ ] \ ]\ngap> SubdigraphHomeomorphicToK4(CompleteDigraph(3));\nfail\ngap> D := Digra\ ph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 7, 11], [\ 4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\n[ [ 3, 5, 10 ], [ 9, 8\ , 10 ], [ 4 ], [ 6 ], [ 7, 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ \ngap> SubdigraphHomeomorphicToK23(D);\n[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4 ], [\ 6 ], [ 11 ], [ 7 ], [ 8 ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> D := Digraph(\ IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],\n> [3, 6], [1, 11], [4, 7],\ [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);\n\ngap> SubdigraphHomeomorphicToK4(D);\nfail\ngap> SubdigraphHomeomo\ rphicToK23(D);\n[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4 ], [ 6 ], [ 11 ], [ 7 ], [ 8\ ], \n [ ], [ 11 ], [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK33(D);\nfail\ \ngap> SubdigraphHomeomorphicToK23(NullDigraph(0));\nfail\ngap> SubdigraphHome\ omorphicToK33(CompleteDigraph(5));\nfail\ngap> SubdigraphHomeomorphicToK33(Com\ pleteBipartiteDigraph(3, 3));\n[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ ], \ [ ], [ ] ]\ngap> SubdigraphHomeomorphicToK4(CompleteDigraph(3));\nfail\n]]><\ /Example>\n \n\n\n \n\n\n \n Properties of digraphs\n\n Edge properties\n \n \ \n true or false.\n \n Retu\ rns true if the digraph digraph has loops, and\n false\ if it does not. A loop is an edge with equal source and range.\n \n \ \n If the argument digraph is mutable, then the return value of\n\ this property is recomputed every time it is called. \n\n D := Digraph([[1, 2], [2]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 1 ], [ 1, 2 ], [ 2, 2 ] ]\ngap> DigraphHas\ Loops(D);\ntrue\ngap> D := Digraph([[2, 3], [1], [2]]);\n\ngap> DigraphEdges(D);\n[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ]\ , [ 3, 2 ] ]\ngap> DigraphHasLoops(D);\nfalse\ngap> D := CompleteDigraph(IsMut\ ableDigraph, 4);\n\ngap> DigraphHas\ Loops(D);\nfalse]]>\n \n\n\n \n \n \n true or fal\ se.\n \n This property is true if the di\ graph digraph\n is antisymmetric, and false if it is not.\n \ \n\n A digraph is antisymmetric if whenever there is an edge w\ ith source\n u and range v, and an edge with source v \ and range\n u, then the vertices u and v are equal.\n \ \n \n If the argument digraph is mutable, then the retur\ n value of\n this property is recomputed every time it is called. \n\n <\ Example> gr1 := Digraph([[2], [1, 3], [2, 3]]);\n\ngap> IsAntisymmetricDigraph(gr1);\nfalse\ngap>\ DigraphEdges(gr1){[1, 2]};\n[ [ 1, 2 ], [ 2, 1 ] ]\ngap> gr2 := Digraph([[1, \ 2], [3, 3], [1]]);\n\ngap> Is\ AntisymmetricDigraph(gr2);\ntrue\ngap> DigraphEdges(gr2);\n[ [ 1, 1 ], [ 1, 2 \ ], [ 2, 3 ], [ 2, 3 ], [ 3, 1 ] ]\n]]>\n \n\n\n \n \ \n true or false.\n \n This p\ roperty is true if the digraph digraph is bipartite, and\n f\ alse if it is not. A digraph is bipartite if and only if the\n vertices \ of digraph can be partitioned into two non-empty sets such\n that the \ source and range of any edge of digraph lie in distinct sets.\n Equiva\ lently, a digraph is bipartite if and only if it is 2-colorable; see\n . \n\n See also <\ Ref Attr=\"DigraphBicomponents\"/>.\n \n \n If the argument \ digraph is mutable, then the return value of\n this property is recompu\ ted every time it is called. \n D := ChainDigraph(\ 4);\n\ngap> IsBipartiteDigraph(D);\nt\ rue\ngap> D := CycleDigraph(3);\n\nga\ p> IsBipartiteDigraph(D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutable\ Digraph, 5, 4);\n\ngap> IsBipartite\ Digraph(D);\ntrue]]>\n \n\n\n \n \n true or false.\n \n Returns tru\ e if the digraph digraph is a complete bipartite\n digraph, and \ false if it is not. \n\n A digraph is a complete bipartite di\ graph if it is bipartite, see\n , and\ there exists a unique edge with\n source i and range j if an\ d only if i and j lie\n in different bicomponents of digra\ ph, see . \n\n Equivalent\ ly, a bipartite digraph with bicomponents of size m and\n n i\ s complete precisely when it has 2mn edges, none of which\n are mult\ iple edges. \n\n See also .\n \ \n \n If the argument digraph is mutable, then the return\ value of\n this property is recomputed every time it is called. \n\n D := CycleDigraph(2);\n\ngap> IsCompleteBipartiteDigraph(D);\ntrue\ngap> D := CycleDigraph(4\ );\n\ngap> IsBipartiteDigraph(D);\ntr\ ue\ngap> IsCompleteBipartiteDigraph(D);\nfalse\ngap> D := CompleteBipartiteDig\ raph(IsMutableDigraph, 5, 4);\n\nga\ p> IsCompleteBipartiteDigraph(D);\ntrue]]>\n \n\n\n \n \n true or false.\n \n \ Returns true if the digraph digraph is complete, and\n fal\ se if it is not. \n\n A digraph is complete if it has no loo\ ps, and for all\n distinct vertices i and j,\n ther\ e is exactly one edge with source i and range j.\n\n Equivale\ ntly, a digraph with n vertices is complete precisely when\n it has \ n(n - 1) edges, no loops, and no multiple edges.\n \n \n \ If the argument digraph is mutable, then the return value of\n this\ property is recomputed every time it is called. \n\n D := Digraph([[2, 3], [1, 3], [1, 2]]);\n\ngap> IsCompleteDigraph(D);\ntrue\ngap> D := Digraph([[2, 2], [1]]\ );\n\ngap> IsCompleteDigraph(\ D);\nfalse\ngap> D := CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteDigraph(D);\nfalse]]>\n \n\n\n \n \n true or <\ K>false.\n \n This property returns true\ if digraph is a complete\n multipartite digraph, and false i\ f not. \n\n A digraph is a complete multipartite digraph if and \ only if\n its vertices can be partitioned into at least two maximal indepen\ dent sets,\n where every possible edge between these independent sets occur\ s in the\n digraph exactly once.\n \n \n If the argument \ digraph is mutable, then the return value of\n this property is recompu\ ted every time it is called. \n\n D := CompleteMul\ tipartiteDigraph([2, 4, 6]);\n\ngap> IsCompleteMultipartiteDigraph(D);\ntrue\ngap> D := \ CompleteBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsCompleteMultipartiteDigraph(D);\ntrue]]>\n \ \n\n\n \n \n \n <\ Returns>true or false.\n \n Returns <\ K>true if the digraph digraph is empty, and\n false if it\ is not. A digraph is empty if it has no\n edges. \n\n IsNu\ llDigraph is a synonym for IsEmptyDigraph.\n \n \n I\ f the argument digraph is mutable, then the return value of\n this p\ roperty is recomputed every time it is called. \n\n D := Digraph([[], []]);\n\ngap> IsE\ mptyDigraph(D);\ntrue\ngap> IsNullDigraph(D);\ntrue\ngap> D := Digraph([[], [1\ ]]);\n\ngap> IsEmptyDigraph(D);\nfa\ lse\ngap> IsNullDigraph(D);\nfalse]]>\n \n\n\n \n \n true or false.\n \n A d\ igraph is an equivalence digraph if and only if the digraph satisifies\n al\ l of ,\n and .\n A partial order digraph corresponds to an equivalence relation.\n \n \n If the argume\ nt digraph is mutable, then the return value of\n this property is r\ ecomputed every time it is called. \n\n D := Digraph([\ [1, 3], [2], [1, 3]]);\n\ngap> IsE\ quivalenceDigraph(D);\ntrue\n]]>\n \n\n\ \n \n \n \ true or false.\n \n This pro\ perty is true if the digraph digraph is\n functional. \n\ \n A digraph is functional if every vertex is the source of a\n u\ nique edge.\n \n \n If the argument digraph is mutable, \ then the return value of\n this property is recomputed every time it is cal\ led. \n gr1 := Digraph([[3], [2], [2], [1], [6], [\ 5]]);\n\ngap> IsFunctionalDigraph(\ gr1);\ntrue\ngap> gr2 := Digraph([[1, 2], [1]]);\n\ngap> IsFunctionalDigraph(gr2);\nfalse\ngap> gr3 := Digraph(3\ , [1, 2, 3], [2, 3, 1]);\n\ngap> I\ sFunctionalDigraph(gr3);\ntrue\n]]>\n \n\ \n\n \n \n true or false.\n \n A multid\ igraph is one that has at least two\n edges with equal source and range\ . \n\n If the argument digraph is mutable, then the return value \ of\n this property is recomputed every time it is called. \n\n \ D := Digraph([\"a\", \"b\", \"c\"], [\"a\", \"b\", \"b\"], [\"\ b\", \"c\", \"a\"]);\n\ngap> IsMul\ tiDigraph(D);\nfalse\ngap> D := DigraphFromDigraph6String(\"&Bug\");\n\ngap> IsDuplicateFree(DigraphEdges(D));\n\ true\ngap> IsMultiDigraph(D);\nfalse\ngap> D := Digraph([[1, 2, 3, 2], [2, 1],\ [3]]);\n\ngap> IsDuplicateFr\ ee(DigraphEdges(D));\nfalse\ngap> IsMultiDigraph(D);\ntrue\ngap> D := DigraphM\ utableCopy(D); \n\ngap> IsMulti\ Digraph(D);\ntrue]]>\n \n\n\n \n \n tr\ ue or false.\n \n This property is tr\ ue if the digraph digraph is\n reflexive, and false if it\ is not.\n A digraph is reflexive if it has a loop at every vertex. \ \n \n \n If the argument digraph is mutable, then th\ e return value of\n this property is recomputed every time it is called. \n\ D := Digraph([[1, 2], [2]]);\n\ngap> IsReflexiveDigraph(D);\ntrue\ngap> D := Digrap\ h([[3, 1], [4, 2], [3], [2, 1]]);\n\ngap> IsReflexiveDigraph(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n \ This property is true if the digraph digraph\n is symmetric, \ and false if it is not. \n\n A symmetric digraph is one wh\ ere for each non-loop edge, having\n source u and range v, th\ ere is a corresponding edge with\n source v and range u. If \ there are n edges with\n source u and range v, then th\ ere must be precisely n\n edges with source v and range u<\ /C>. In other words, a symmetric\n digraph has a symmetric adjacency matri\ x .\n \n \n If the argument d\ igraph is mutable, then the return value of\n this property is recomput\ ed every time it is called. \n gr1 := Digraph([[2]\ , [1, 3], [2, 3]]);\n\ngap> IsSymm\ etricDigraph(gr1);\ntrue\ngap> adj1 := AdjacencyMatrix(gr1);;\ngap> Display(ad\ j1);\n[ [ 0, 1, 0 ],\n [ 1, 0, 1 ],\n [ 0, 1, 1 ] ]\ngap> adj1 = Tr\ ansposedMat(adj1);\ntrue\ngap> gr1 = DigraphReverse(gr1);\ntrue\ngap> gr2 := D\ igraph([[2, 3], [1, 3], [2, 3]]);\n\ngap> IsSymmetricDigraph(gr2);\nfalse\ngap> adj2 := AdjacencyMatrix(gr2);;\n\ gap> Display(adj2);\n[ [ 0, 1, 1 ],\n [ 1, 0, 1 ],\n [ 0, 1, 1 ] ]\ \ngap> adj2 = TransposedMat(adj2);\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n Th\ is property is true if the digraph digraph is a tournament,\n \ and false if it is not. \n\n A tournament is an orientation of \ a complete (undirected) graph.\n Specifically, a tournament is a digraph wh\ ich has a unique directed edge\n (of some orientation) between any pair of \ distinct vertices, and no loops.\n \n \n If the argument dig\ raph is mutable, then the return value of\n this property is recomputed\ every time it is called. \n D := Digraph([[2, 3, \ 4], [3, 4], [4], []]);\n\ngap> IsT\ ournament(D);\ntrue\ngap> D := Digraph([[2], [1], [3]]);\n\ngap> IsTournament(D);\nfalse\ngap> D := CycleDigraph\ (IsMutableDigraph, 3);\n\ngap> IsTou\ rnament(D);\ntrue\ngap> DigraphRemoveEdge(D, 1, 2);\n\ngap> IsTournament(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n This property is true if the digraph digraph\n is \ transitive, and false if it is not.\n\n A digraph is transitive if whenever [ i, j ] and\n [ j, k ] are edges of the digra\ ph, then [ i, k ] is also an\n edge of the digraph. \n\n Let \ n be the number of vertices of an arbitrary digraph, and let\n <\ M>m be the number of edges.\n For general digraphs, the methods used fo\ r this property use a version\n of the Floyd-Warshall algorithm, and have c\ omplexity O(n^3).\n\n However for digraphs which are topologically s\ ortable\n [], then methods with\n \ complexity O(m + n + m \\cdot n) will be used when appropriate.\n \n \n If the argument digraph is mutable, then the return val\ ue of\n this property is recomputed every time it is called. \n \n \ D := Digraph([[1, 2], [3], [3]]);\n\ngap> IsTransitiveDigraph(D);\nfalse\ngap> gr2 :=\ Digraph([[1, 2, 3], [3], [3]]);\n\ \ngap> IsTransitiveDigraph(gr2);\ntrue\ngap> gr2 = DigraphTransitiveClosure(D)\ ;\ntrue\ngap> gr3 := Digraph([[1, 2, 2, 3], [3, 3], [3]]);\n\ngap> IsTransitiveDigraph(gr3);\ntrue\n]]>\n \n\n\n \n \n \n true or false.\n \n A digraph is a preorder digraph if and only if the digraph satisifies\ both\n and .\n A preorder digraph (or quasiorder digraph) digraph corres\ ponds to\n the preorder relation \\leq defined by x \\leq y i\ f and only\n if [x, y] is an edge of digraph.\n \n \ \n If the argument digraph is mutable, then the return value of\n \ this property is recomputed every time it is called. \n\n D := Digraph([[1], [2, 3], [2, 3]]);\n\ngap> IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[1 .. 4], [1\ .. 4], [1 .. 4], [1 .. 4]]);\n\n\ gap> IsPreorderDigraph(D);\ntrue\ngap> D := Digraph([[2], [3], [4], [5], [1]])\ ;\n\ngap> IsPreorderDigraph(D);\nf\ alse\ngap> D := Digraph([[1], [1, 2], [2, 3]]);\n\ngap> IsQuasiorderDigraph(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n <\ Description>\n A digraph is a partial order digraph if and only if the digr\ aph satisifies\n all of ,\n and .\n A p\ artial order digraph corresponds\n to the partial order relation \ \\leq defined by x \\leq y if and\n only if [x, y] is an \ edge of digraph.\n \n \n If the argument digraph \ is mutable, then the return value of\n this property is recomputed every ti\ me it is called. \n\n D := Digraph([[1, 3], [2, 3], [3\ ]]);\n\ngap> IsPartialOrderDigraph\ (D);\ntrue\ngap> D := CycleDigraph(5);\n\ngap> IsPartialOrderDigraph(D);\nfalse\ngap> D := Digraph([[1, 1], [1, 1, \ 2], [3], [3, 3, 4, 4]]);\n\n\ gap> IsPartialOrderDigraph(D);\ntrue\n]]>\n \n\n\n \n \n \n \n true or fa\ lse.\n \n IsMeetSemilatticeDigraph retur\ ns true if the digraph\n digraph is a meet semilattice; Is\ JoinSemilatticeDigraph\n returns true if the digraph digraph<\ /A> is a join semilattice;\n and IsLatticeDigraph returns true if the digraph\n digraph is both a meet and a join semilattice.\n\ \n\n For a partial order digraph the\n corresponding partial order is the relation \\leq, define\ d by\n x \\leq y if and only if [x, y] is an edge.\n A dig\ raph is a meet semilattice if it is a partial order and every\n pair\ of vertices has a greatest lower bound (meet) with respect to the\n aforem\ entioned relation. A join semilattice is a partial order where\n eve\ ry pair of vertices has a least upper bound (join) with respect to\n the re\ lation.\n \n \n If the argument digraph is mutable, then\ the return value of\n this property is recomputed every time it is called.\ \n\n D := Digraph([[1, 3], [2, 3], [3]]);\n\ngap> IsMeetSemilatticeDigraph(D);\nfal\ se\ngap> IsJoinSemilatticeDigraph(D);\ntrue\ngap> IsLatticeDigraph(D);\nfalse\ \ngap> D := Digraph([[1], [2], [1 .. 3]]);\n\ngap> IsJoinSemilatticeDigraph(D);\nfalse\ngap> IsMeetSemilatticeDi\ graph(D);\ntrue\ngap> IsLatticeDigraph(D);\nfalse\ngap> D := Digraph([[1 .. 4]\ , [2, 4], [3, 4], [4]]);\n\ngap> I\ sMeetSemilatticeDigraph(D);\ntrue\ngap> IsJoinSemilatticeDigraph(D);\ntrue\nga\ p> IsLatticeDigraph(D);\ntrue\ngap> D := Digraph([[1, 1, 1], [1, 1, 2, 2],\n> \ [1, 3, 3], [1, 2, 3, 3, 4]]);\n\ngap> IsMeetSemilatticeDigraph(D);\ntrue\ngap> IsJoinSem\ ilatticeDigraph(D);\ntrue\ngap> IsLatticeDigraph(D);\ntrue\n]]>\n <\ /Description>\n\n\n \n\n Regularity\ \n \n \n true or false.\n \n \ This property is true if there is an integer n such that\n f\ or every vertex v of digraph digraph there are exactly\n n\ edges terminating in v.\n\n See also and\n .\n \n \n If\ the argument digraph is mutable, then the return value of\n this pr\ operty is recomputed every time it is called. \n I\ sInRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsInRegularDigraph(ChainDig\ raph(4));\nfalse\n]]>\n \n\n\n \n \n \ true or false.\n \n This property is \ true if there is an integer n such that\n for every vertex v<\ /C> of digraph digraph there are exactly\n n edges starting a\ t v.\n \n\n See also and\n\ .\n \n \n If the argument <\ A>digraph is mutable, then the return value of\n this property is recom\ puted every time it is called. \n IsOutRegularDigr\ aph(CompleteDigraph(4));\ntrue\ngap> IsOutRegularDigraph(ChainDigraph(4));\nfa\ lse\n]]>\n \n\n\n \n \n true or f\ alse.\n \n This property is true if ther\ e is an integer n such that\n for every vertex v of digraph <\ A>digraph there are exactly\n n edges starting and terminating a\ t v. In other words,\n the property is true if digraph\ is both in-regular and\n and out-regular.\n\n See also and\n .\n \n \ \n If the argument digraph is mutable, then the return value of\ \n this property is recomputed every time it is called. \n IsRegularDigraph(CompleteDigraph(4));\ntrue\ngap> IsRegularDigraph\ (ChainDigraph(4));\nfalse\n]]>\n \n\n\n \ \n \n\ true or false.\n \n If \ digraph is a connected symmetric graph, this property returns\n true\ if for any two vertices u and v of digraph\n and \ any two integers i and j between 0 and the\n diameter \ of digraph, the number of vertices at distance i\n from u<\ /C> and distance j from v depends only on\n i, j, and the distance between vertices u and\n v. \n\n Al\ ternatively, a distance regular graph is a graph for which there exist\n in\ tegers b_i, c_i, and i such that for any two\n vertice\ s u, v in digraph which are distance i\n apart,\ there are exactly b_i neighbors of v which are at\n distance\ i - 1 away from u, and c_i neighbors of\n v wh\ ich are at distance i + 1 away from u. This\n definition is u\ sed to check whether digraph is distance regular. \n\n In the cas\ e where digraph is not symmetric or not connected, the\n property is\ false.\n \n \n If the argument digraph is mutabl\ e, then the return value of\n this property is recomputed every time it is \ called. \n\n D := DigraphSymmetricClosure(ChainDig\ raph(5));;\ngap> IsDistanceRegularDigraph(D);\nfalse\ngap> D := Digraph([[2, 3\ , 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]]);\n\ngap> IsDistanceRegularDigraph(D);\ntrue\n]]>\n \n\n\n \n\n Connectivity and c\ ycles\n \n \n true or false.\n \n\ This property is true if the digraph digraph is acyclic, and\ \n false if it is not. A digraph is acyclic if every directed\ \n cycle on the digraph is trivial. See Section for the definition of a directed cycle, and of a\ trivial\n directed cycle.\n\n The method used in this operation has\ complexity O(m+n) where\n m is the number of edges (counting\ multiple edges as one) and\n n is the number of vertices in the dig\ raph. \n\n If the argument digraph is mutable, then the return v\ alue of\n this property is recomputed every time it is called. \n\n Petersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,\n>\ function(x, y)\n> return IsEmpty(Intersection(x, y));\n> end);;\ngap> D := \ Digraph(Petersen);\n\ngap> IsAcy\ clicDigraph(D);\nfalse\ngap> D := DigraphFromDiSparse6String(\n> \".b_OGCIDBaP\ GkULEbQHCeRIdrHcuZMfRyDAbPhTi|zF\");\n\ngap> IsAcyclicDigraph(D);\ntrue\ngap> IsAcyclicDigraph(ChainDigraph(10\ ));\ntrue\ngap> D := CompleteDigraph(IsMutableDigraph, 4);\n\ngap> IsAcyclicDigraph(D);\nfalse\ngap> IsAcyclicDig\ raph(CycleDigraph(10));\nfalse]]>\n \n\n\ \n \n \n true or false.\n \n IsChainDig\ raph returns true if the digraph digraph is\n isomorphic \ to the chain digraph with the same number of vertices as\n digraph, \ and false if it is not; see . \n\ \n A digraph is a chain if and only if it is a directed tree, in whi\ ch\n every vertex has out degree at most one; see \n and .\n \n \n If the argum\ ent digraph is mutable, then the return value of\n this property is \ recomputed every time it is called. \n\n D := Digraph(\ [[1, 3], [2, 3], [3]]);\n\ngap> Is\ ChainDigraph(D);\nfalse\ngap> D := ChainDigraph(5);\n\ngap> IsChainDigraph(D);\ntrue\ngap> D := DigraphReverse(D);\ \n\ngap> IsChainDigraph(D);\ntrue\ \ngap> D := ChainDigraph(IsMutableDigraph, 5);\n\ngap> IsChainDigraph(D);\ntrue\ngap> DigraphReverse(D);\n\ngap> IsChainDigraph(D);\ntrue]]>\ \n \n\n\n \n \n true or false.\n \n This property is true if the digraph digra\ ph\n is weakly connected and false if it is not. A digraph\n \ digraph is weakly connected if it is possible to travel\n fr\ om any vertex to any other vertex by traversing edges in either\n direction\ (possibly against the orientation of some of them). \n\n The method us\ ed in this function has complexity O(m) if the\n digraph's attribute is set, otherwise it has\n complexity O(m\ +n) (where\n m is the number of edges and\n n is the n\ umber of vertices of the digraph).\n \n \n If the argument d\ igraph is mutable, then the return value of\n this property is recomput\ ed every time it is called. \n\n D := Digraph([[2]\ , [3], []]);;\ngap> IsConnectedDigraph(D);\ntrue\ngap> D := Digraph([[1, 3], [\ 4], [3], []]);;\ngap> IsConnectedDigraph(D);\nfalse\ngap> D := Digraph(IsMutab\ leDigraph, [[2], [3], []]);;\ngap> IsConnectedDigraph(D);\ntrue\ngap> D := Dig\ raph(IsMutableDigraph, [[1, 3], [4], [3], []]);;\ngap> IsConnectedDigraph(D);\ \nfalse]]>\n \n\n\n \n <\ Prop Name=\"IsBiconnectedDigraph\" Arg=\"digraph\"/>\n true o\ r false.\n \n A connected digraph is bico\ nnected if it is still connected (in the\n sense of ) when any vertex is removed. \n If D has at least \ 3 vertices, then IsBiconnectedDigraph\n implies ;\n see or for a more\n detailed explanation.\n \n\n IsBiconn\ ectedDigraph returns true if the digraph\n digraph is bic\ onnected, and false if it is not. In\n particular, IsBiconnectedD\ igraph returns false if\n digraph is not connected. \ \n \n If the argument digraph is mutable, then the return value\ of\n this property is recomputed every time it is called. \n\n Mult\ iple edges are ignored by this method. \n\n The method used in this ope\ ration has complexity O(m+n) where\n m is the number of edges\ and n is the number of vertices in\n the digraph.\n \n\n \ See also , , and \n \ .\n \n \n IsBiconnectedDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBico\ nnectedDigraph(CycleDigraph(5));\ntrue\ngap> D := Digraph([[1, 1], [1, 1, 2], \ [3], [3, 3, 4, 4]]);;\ngap> IsBiconnectedDigraph(D);\nfalse\ngap> D := Complet\ eBipartiteDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBiconnectedDigraph(D);\ntrue]]>\n \ \n\n\n \n \n true or false.\n \n \ A connected digraph is bridgeless if it is still connected (in the\n\ sense of ) when any edge is removed. \n \ If D has at least 3 vertices, then implies IsBridgelessDigraph;\n see or for a more\n detailed explanation.\n \ \n\n IsBridgelessDigraph returns true if the digraph\n\ D is bridgeless, and false if it is not. In\n particular,\ IsBridgelessDigraph returns false if\n D is not conne\ cted. \n \n If the argument D is mutable, then the return v\ alue of\n this property is recomputed every time it is called. \n\n \ Multiple edges are ignored by this method. \n\n The method used in this\ operation has complexity O(m+n) where\n m is the number of e\ dges and n is the number of vertices in\n the digraph.\n \n\n\ See also , , and \ \n .\n \n \ IsBridgelessDigraph(Digraph([[1, 3], [2, 3], [3]]));\nfalse\ngap> IsBridgeless\ Digraph(CycleDigraph(5));\ntrue\ngap> D := Digraph([[1, 1], [1, 1, 2], [3], [3\ , 3, 4, 4]]);;\ngap> IsBridgelessDigraph(D);\nfalse\ngap> D := CompleteBiparti\ teDigraph(IsMutableDigraph, 5, 4);\n\ngap> IsBridgelessDigraph(D);\ntrue\ngap> D := Digraph([[2, 5], [1, 3, 4, 5]\ , [2, 4], [2, 3], [1, 2]]);\n\nga\ p> IsBridgelessDigraph(D);\ntrue\ngap> IsBiconnectedDigraph(D);\nfalse\ngap> D\ := Digraph([[2], [3], [4], [2]]);\n\ngap> IsBridgelessDigraph(D);\nfalse\ngap> IsBiconnectedDigraph(D);\nfalse\ \ngap> IsBridgelessDigraph(ChainDigraph(2));\nfalse\ngap> IsBiconnectedDigraph\ (ChainDigraph(2));\ntrue\n]]>\n \n\n\n \ \n \ \n true or false.\n \n This\ property is true if the digraph digraph is strongly\n connec\ ted and false if it is not. \n\n A digraph digraph is \ strongly connected if there is a directed\n path from every vertex to e\ very other vertex. See Section for the definition of a directed\n path. \n\n The metho\ d used in this operation is based on Gabow's Algorithm and has complexity O(m+n), where m is\n the number of e\ dges (counting multiple edges as one) and n is the\n number of verti\ ces in the digraph.\n \n \n If the argument digraph is m\ utable, then the return value of\n this property is recomputed every time i\ t is called. \n\n D := CycleDigraph(250000);\n\ngap> IsStronglyConnectedDigraph(D)\ ;\ntrue\ngap> D := DigraphRemoveEdges(D, [[250000, 1]]);\n\ngap> IsStronglyConnectedDigraph(D);\nfalse\ \ngap> D := CycleDigraph(IsMutableDigraph, 250000);\n\ngap> IsStronglyConnectedDigraph(D);\ntrue\ngap> D\ igraphRemoveEdge(D, [250000, 1]);\n\ngap> IsStronglyConnectedDigraph(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n This property is true if the digraph digraph\n \ is aperiodic, i.e. if its is equal to 1.\n \ Otherwise, the property is false.\n \n \n If the argum\ ent digraph is mutable, then the return value of\n this property is \ recomputed every time it is called. \n D := Digrap\ h([[6], [1], [2], [3], [4, 4], [5]]);\n\ngap> IsAperiodicDigraph(D);\nfalse\ngap> D := Digraph([[2], [3, 5]\ , [4], [5], [1, 2]]);\n\ngap> IsAp\ eriodicDigraph(D);\ntrue\ngap> D := Digraph(IsMutableDigraph, [[2], [3, 5], [4\ ], [5], [1, 2]]);\n\ngap> IsAperiodi\ cDigraph(D);\ntrue]]>\n \n\n\n \n \n true<\ /K> or false.\n \n Returns true if th\ e digraph digraph is a directed tree, and\n false if it is no\ t. \n\n A directed tree is an acyclic digraph with precisely 1 s\ ource,\n such that no two vertices share an out-neighbour.\n Note the em\ pty digraph is not considered a directed\n tree as it has no source. \n\ \n See also .\n \n \n If the a\ rgument digraph is mutable, then the return value of\n this property\ is recomputed every time it is called. \n\n D := \ Digraph([[], [2]]);\n\ngap> IsDirec\ tedTree(D);\nfalse\ngap> D := Digraph([[3], [3], []]);\n\ngap> IsDirectedTree(D);\nfalse\ngap> D := Digraph([[2]\ , [3], []]);\n\ngap> IsDirectedTre\ e(D);\ntrue\ngap> D := Digraph([[2, 3], [6], [4, 5], [], [], []]);\n\ngap> IsDirectedTree(D);\ntrue\n]]>\n \n\n\n \n \n \n true or false.\n \n \ The property IsUndirectedTree returns true if the digraph\n \ digraph is an undirected tree, and the property\n IsUndirectedF\ orest returns true if digraph is an\n undirected forest; \ otherwise, these properties return false. \n\n An undirected \ tree is a symmetric digraph without loops, in which for\n any pair of d\ istinct vertices u and v, there is exactly one\n directed pat\ h from u to v. See and\ , and see\n Section for the definition of\n directed path. This defin\ ition implies that an undirected tree has\n no multiple edges. \n\n \ An undirected forest is a digraph, each of whose connected components\n\ is an undirected tree. In other words, an undirected forest is isomorphic \ to\n a disjoint union of undirected trees. See and . In particular, every\n undirected tree is an un\ directed forest. \n\n Please note that the digraph with zero vertices i\ s considered to be neither\n an undirected tree nor an undirected forest.\n\ \n \n If the argument digraph is mutable, then the retu\ rn value of\n this property is recomputed every time it is called. \n\n \ D := Digraph([[3], [3], [1, 2]]);\n\ngap> IsUndirectedTree(D);\ntrue\ngap> IsSymmetricD\ igraph(D) and not DigraphHasLoops(D);\ntrue\ngap> D := Digraph([[3], [5], [1, \ 4], [3], [2]]);\n\ngap> IsConnecte\ dDigraph(D);\nfalse\ngap> IsUndirectedTree(D);\nfalse\ngap> IsUndirectedForest\ (D);\ntrue\ngap> D := Digraph([[1, 2], [1], [2]]);\n\ngap> IsUndirectedTree(D) or IsUndirectedForest(D);\nfalse\ \ngap> IsSymmetricDigraph(D) or not DigraphHasLoops(D);\nfalse]]>\n \ \n\n\n \n \n true or false.\ \n \n This property returns true if the digraph digraph is Eulerian.\n \n\n A connected digraph is called Eulerian \ if there exists a directed\n circuit on the digraph which includes every ed\ ge exactly once. See\n Section for the definition of\n a directed circuit. Note that the empty digra\ ph with at most one vertex is\n considered to be Eulerian.\n \n \ \n If the argument digraph is mutable, then the return value of\n \ this property is recomputed every time it is called. \n\n D := Digraph([[]]);\n\ngap> I\ sEulerianDigraph(D);\ntrue\ngap> D := Digraph([[2], []]);\n\ngap> IsEulerianDigraph(D);\nfalse\ngap> D := Digraph\ ([[3], [], [2]]);\n\ngap> IsEuleri\ anDigraph(D);\nfalse\ngap> D := Digraph([[2], [3], [1]]);\n\ngap> IsEulerianDigraph(D);\ntrue\n]]>\n \ \n\n\n \n \n true or false.\n \n If digraph is Hamiltonian, then this property r\ eturns\n true, and false if it is not. \n\n A digraph \ with n vertices is Hamiltonian if it has a\n directed cycle o\ f length n. See Section for the definition of a directed cycle.\n Note the empty digraphs o\ n 0 and 1 vertices are considered to be\n Hamiltonian. \n\n The metho\ d used in this operation has the worst case complexity as\n .\n \n \n If the argument digraph is\ mutable, then the return value of\n this property is recomputed every time\ it is called. \n\n g := Digraph([[]]);\n\ngap> IsHamiltonianDigraph(g);\ntrue\ngap> g :=\ Digraph([[2], [1]]);\n\ngap> IsHa\ miltonianDigraph(g);\ntrue\ngap> g := Digraph([[3], [], [2]]);\n\ngap> IsHamiltonianDigraph(g);\nfalse\ngap> g :\ = Digraph([[2], [3], [1]]);\n\ngap\ > IsHamiltonianDigraph(g);\ntrue\n]]>\n \n\n\n \n \n <\ Returns>true or false.\n \n IsCycl\ eDigraph returns true if the digraph digraph is\n isomorp\ hic to the cycle digraph with the same number of vertices as\n digraph, and false if it is not; see . \n\n A digraph is a cycle if and only if it is strongly connected a\ nd has\n the same number of edges as vertices. \n \n \n If t\ he argument digraph is mutable, then the return value of\n this prop\ erty is recomputed every time it is called. \n\n D := \ Digraph([[1, 3], [2, 3], [3]]);\n\ \ngap> IsCycleDigraph(D);\nfalse\ngap> D := CycleDigraph(5);\n\ngap> IsCycleDigraph(D);\ntrue\ngap> D := OnDigraphs\ (D, (1, 2, 3));\n\ngap> D = CycleD\ igraph(5);\nfalse\ngap> IsCycleDigraph(D);\ntrue]]>\n \n\n\n \n \n Planarity\ \n \n \n true or false.\n \n A planar<\ /E> digraph is a digraph that can be embedded in the plane in\n such a way \ that its edges do not intersect. A digraph is planar if and only\n if it do\ es not have a subdigraph that is homeomorphic to either the\n complete grap\ h on 5 vertices or the complete bipartite graph with\n vertex sets o\ f sizes 3 and 3.\n \n \n IsPlanarDigraph ret\ urns true if the digraph digraph is\n planar and false\ if it is not. The directions and multiplicities of\n any edges in digra\ ph are ignored by IsPlanarDigraph.\n \n\n See also .\n \n\n This method uses the referen\ ce implementation in\n &edge-addition-planarity-suite; by John Boyer of the\ algorithms described\n in .\n\n IsPlanarDigraph(CompleteDigraph(4));\ntrue\ngap> IsPlanarDigraph(Comple\ teDigraph(5));\nfalse\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(2, 3));\n\ true\ngap> IsPlanarDigraph(CompleteBipartiteDigraph(3, 3));\nfalse\ngap> IsPla\ narDigraph(CompleteDigraph(IsMutableDigraph, 4));\ntrue\ngap> IsPlanarDigraph(\ CompleteDigraph(IsMutableDigraph, 5));\nfalse\ngap> IsPlanarDigraph(CompleteBi\ partiteDigraph(IsMutableDigraph, 2, 3));\ntrue\ngap> IsPlanarDigraph(CompleteB\ ipartiteDigraph(IsMutableDigraph, 3, 3));\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n An outer planar digraph is a digraph that can be embedded \ in the\n plane in such a way that its edges do not intersect, and all verti\ ces\n belong to the unbounded face of the embedding. A digraph is outer pl\ anar\n if and only if it does not have a subdigraph that is homeomorphic to\ either\n the complete graph on 4 vertices or the complete bipartite\ graph\n with vertex sets of sizes 2 and 3.\n \n \n\ IsOuterPlanarDigraph returns true if the digraph\n dig\ raph is outer planar and false if it is not. The\n directions an\ d multiplicities of any edges in digraph are ignored by\n IsPlana\ rDigraph. \n\n See also .\n\n Th\ is method uses the reference implementation in\n &edge-addition-planarity-s\ uite; by John Boyer of the algorithms described\n in .\ \n\n IsOuterPlanarDigraph(CompleteDigraph(4));\nfalse\ \ngap> IsOuterPlanarDigraph(CompleteDigraph(5));\nfalse\ngap> IsOuterPlanarDig\ raph(CompleteBipartiteDigraph(2, 3));\nfalse\ngap> IsOuterPlanarDigraph(Comple\ teBipartiteDigraph(3, 3));\nfalse\ngap> IsOuterPlanarDigraph(CycleDigraph(10))\ ;\ntrue\ngap> IsOuterPlanarDigraph(CompleteDigraph(IsMutableDigraph, 4));\nfal\ se\ngap> IsOuterPlanarDigraph(CompleteDigraph(IsMutableDigraph, 5));\nfalse\ng\ ap> IsOuterPlanarDigraph(CompleteBipartiteDigraph(IsMutableDigraph,\n> \ 2, 3));\nfalse\ngap> IsOuterPlanarDi\ graph(CompleteBipartiteDigraph(IsMutableDigraph,\n> \ 3, 3));\nfalse\ngap> IsOuterPlanarDigraph(CycleDigraph(\ IsMutableDigraph, 10));\ntrue\n\n]]>\n \n\n\n \n\n Homomorphisms and transformations\n \n \n \ true or false.\n \n This pro\ perty returns true if digraph is a core,\n and false i\ f it is not.\n\n A digraph D is a core if and only if it \ has no proper\n subdigraphs A such that there exists a homomorphism \ from D\n to A. In other words, a digraph D is a core i\ f and only if\n every endomorphism on D is an automorphism on D.\n \n \n If the argument digraph is mutable, then the\ return value of\n this property is recomputed every time it is called. \n\ \n D := CompleteDigraph(6);\n\ngap> IsDigraphCore(D);\ntrue\ngap> D := DigraphSymmet\ ricClosure(CycleDigraph(6));\n\ngap> DigraphHomomorphism(D, CompleteDigraph(2));\nTransformation( [ 1\ , 2, 1, 2, 1, 2 ] )\ngap> IsDigraphCore(D);\nfalse\n]]>\n \n\n\n \n \n true or false.\n \n If digraph is a digraph without multiple edges, then\n IsEdgeTransitive returns true if digraph\n is edge trans\ itive, and false otherwise. A digraph is\n edge transitive if\ its automorphism group acts\n transitively on its edges (via the action\n \ ).\n \n \n If the ar\ gument digraph is mutable, then the return value of\n this property \ is recomputed every time it is called. \n\n IsEdge\ Transitive(CompleteDigraph(2));\ntrue\ngap> IsEdgeTransitive(ChainDigraph(3));\ \nfalse\ngap> IsEdgeTransitive(Digraph([[2], [3, 3, 3], []]));\nError, the arg\ ument must be a digraph with no multiple edges,\n]]>\n \n\n\n \n \n true or false.\n \n If digraph is a digraph, then IsVertexTransitive r\ eturns\n true if digraph is vertex transitive, and false\n otherwise. A digraph is vertex transitive if its automorphism g\ roup\n acts transitively on its vertices.\n \n \n If the arg\ ument digraph is mutable, then the return value of\n this property i\ s recomputed every time it is called. \n\n IsVerte\ xTransitive(CompleteDigraph(2));\ntrue\ngap> IsVertexTransitive(ChainDigraph(3\ ));\nfalse\n]]>\n \n\n\n \n\n\ \n\n \n Homomorphisms\n \n\n Acting on digraphs<\ /Heading>\n \n \n \n A digraph.\n \n If digraph is a digraph, \ and the second argument perm is a\n permutation of the vertic\ es of digraph, then this operation\n returns a digraph constructed b\ y relabelling the vertices of\n digraph according to perm. N\ ote that for an automorphism\n f of a digraph, we have OnDigraphs\ (digraph, f) =\n digraph.\n \n\n If the second \ argument is a transformation trans of the\n vertices of di\ graph, then this operation returns a digraph\n constructed by transform\ ing the source and range of each edge according to\n trans. Thus a v\ ertex which does not appear in the image of\n trans will be isolated\ in the returned digraph, and the returned\n digraph may contain multiple e\ dges, even if digraph does not.\n If trans is mathematically \ a permutation, then the result coincides\n with OnDigraphs(digraph, AsPermutation(trans)).\n \n\n The of digraph will not be retained\n in the returned di\ graph. \n\n If digraph belongs to , then\n relabelling of the vertices is performed directly on digraph<\ /A>.\n If digraph belongs to , an \ immutable\n copy of digraph with the vertices relabelled is returned\ . \n\n D := Digraph([[3], [1, 3, 5], [1], [1, 2\ , 4], [2, 3, 5]]);\n\ngap> new :=\ OnDigraphs(D, (1, 2));\n\ngap> O\ utNeighbours(new);\n[ [ 2, 3, 5 ], [ 3 ], [ 2 ], [ 2, 1, 4 ], [ 1, 3, 5 ] ]\ng\ ap> D := Digraph([[2], [], [2]]);\n\ngap> t := Transformation([1, 2, 1]);;\ngap> new := OnDigraphs(D, t);\n\ngap> OutNeighbours(new);\n[ [ 2,\ 2 ], [ ], [ ] ]\ngap> ForAll(DigraphEdges(D),\n> e -> IsDigraphEdge(new, [\ e[1] ^ t, e[2] ^ t]));\ntrue\n]]>\n \n\n\ \n \n \n\ \n A digraph.\n \n If digraph is a digraph, and pair is a pair consisting \ of a\n permutation of the vertices and a permutation of the edges of\n <\ A>digraph, then this operation returns a digraph\n constructed by relab\ elling the vertices and edges of digraph\n according to perm[1] and perm[2], respectively. \n\n In its second form, OnMult\ iDigraphs returns a digraph with vertices\n and edges permuted by pe\ rm1 and perm2, respectively. \n\n Note that OnDigraphs(digraph,\n perm)=OnMultiDigraphs(digraph, [perm, ()]) whe\ re perm is\n a permutation of the vertices of digraph. If you\ are only interested\n in the action of a permutation on the vertices of a \ digraph, then you can\n use OnDigraphs instead of OnMultiDigraphs\ .\n\n If digraph belongs to , th\ en\n relabelling of the vertices is performed directly on digraph.\n\ If digraph belongs to , an immuta\ ble\n copy of digraph with the vertices relabelled is returned. \ \n\n D1 := Digraph([\n> [3, 6, 3], [], [3], [9, 10\ ], [9], [], [], [10, 4, 10], [], []]);\n\ngap> p := BlissCanonicalLabelling(D1);\n[ (1,7)(3,6)(4,10)(5,9\ ), () ]\ngap> D2 := OnMultiDigraphs(D1, p);\n\ngap> OutNeighbours(D2);\n[ [ ], [ ], [ ], [ ], [ ], [\ 6 ], [ 6, 3, 6 ], [ 4, 10, 4 ], \n [ 5 ], [ 5, 4 ] ]]]>\n \n\n\n \n\n \n Isomorphisms and canonical labellings\n From version 0.11.0 of &Digraphs; it is possible to use either &b\ liss; or\n &nauty; (via &NautyTracesInterface;) to calculate canonical labe\ llings and\n automorphism groups of digraphs; see and \ for more details about &bliss; and &nauty;,\n r\ espectively. \n\n \n \n \n Nothing.\ \n \n These functions can be used to specify whether &nauty; \ or &bliss; should be\n used by default by &Digraphs;. If &NautyTracesInterf\ ace; is not available,\n then these functions do nothing. Otherwise, by cal\ ling\n DigraphsUseNauty subsequent computations will default to usin\ g\n &nauty; rather than &bliss;, where possible.\n \n\n You can c\ all these functions at any point in a &GAP; session, as many\n times as you\ like, it is guaranteed that existing digraphs remain\n valid, and that com\ parison of existing digraphs and newly created digraphs\n via\n ,\n ,\n , and\n are \ also valid. \n\n It is also possible to compute the automorphism group o\ f a specific digraph\n using both &nauty; and &bliss; using \n and , respectively.\n \n\n\n \n \n A permutation group.\n \n If digraph is a digraph, then this attribute contains the group of\n auto\ morphisms of digraph. An automorphism of digraph\n is\ an isomorphism from digraph to itself. See for more information\n about isomor\ phisms of digraphs. \n\n If digraph is not a multidigraph then t\ he automorphism group is\n returned as a group of permutations on the set o\ f vertices of\n digraph. \n\n If digraph is a multidig\ raph then the automorphism group is returned\n as the direct product of a g\ roup of permutations on the set of vertices of\n digraph with a grou\ p of permutations on the set of edges of\n digraph. These groups can\ be accessed using on the\n returned group. \n\n B\ y default, the automorphism group is found using &bliss; by Tommi Junttila\n \ and Petteri Kaski. If &NautyTracesInterface; is available, then &nauty; by\n\ Brendan Mckay and Adolfo Piperno can be used instead; see\n , \n , ,\n and .\n \n\n If the argument digraph is mutabl\ e, then the return value of\n this attribute is recomputed every time it is\ called. \n\n johnson := DigraphFromGraph6String(\ \"E}lw\");\n\ngap> G := Automorph\ ismGroup(johnson);\nGroup([ (3,4), (2,3)(4,5), (1,2)(5,6) ])\ngap> cycle := Cy\ cleDigraph(9);\n\ngap> G := Automorph\ ismGroup(cycle);\nGroup([ (1,2,3,4,5,6,7,8,9) ])\ngap> IsCyclic(G) and Size(G)\ = 9;\ntrue]]>\n \n \n\n \n \n \n \n A permutation group\ .\n \n If digraph is a digraph, then this at\ tribute contains the group of\n automorphisms of digraph as calculat\ ed using &bliss; by Tommi Junttila\n and Petteri Kaski. \n\n The att\ ribute and\n oper\ ation returns\n the value of either BlissAutomorphismGroup or . These groups are, of cours\ e, equal but\n their generating sets may differ. \n The attribute returns the value of\n BlissAutomorphismGroup as it is\ not implemented for &nauty; The \n requirements for the optional arguments\ vertex_colours and \n edge_colours are documented in .\n\n See also , and .\n \n\n If the argument digraph is mutable\ , then the return value of\n this attribute is recomputed every time it is \ called. \n\n G := BlissAutomorphismGroup(JohnsonDi\ graph(5, 2));;\ngap> IsSymmetricGroup(G);\ntrue\ngap> Size(G);\n120]]>\n \n \n\n \n \n A permuta\ tion group.\n \n If digraph is a digraph, th\ en this attribute contains the group of\n automorphisms of digraph a\ s calculated using &nauty; by\n Brendan Mckay and Adolfo Piperno via &Nauty\ TracesInterface;.\n\n \n\n The attribute and\n operation returns\n the value of either Nau\ tyAutomorphismGroup or .\n These groups are, of course, equal but their generat\ ing sets may differ. \n\n See also , and\ .\n \n\n If the argument digraph\ is mutable, then the return value of\n this attribute is recomputed ev\ ery time it is called. \n\n NautyAutomorphismGroup(Joh\ nsonDigraph(5, 2));\nGroup([ (3,4)(6,7)(8,9), (2,3)(5,6)(9,10), (2,5)(3,6)(4,7\ ),\n (1,2)(6,8)(7,9) ])]]>\n \n \n\n <\ ManSection>\n \n A permutatio\ n group.\n \n\n This operation computes the automor\ phism group of a vertex-coloured digraph.\n A vertex-coloured digraph can b\ e specified by its underlying digraph\n digraph and its colouring vert_colours. Let n be the\n number of vertices of digraph. The colouring vert_colours may\n have one of the following two f\ orms:\n\n \n \n a list of n integers, where \ vert_colours[i] is the\n colour of vertex i, using\ the colours [1 .. m] for some\n m <= n; or\n \n \n a list of non-empty disjoint lists whose union is\ \n DigraphVertices(digraph), such that\n vert_c\ olours[i] is the list of all vertices with colour\n i\ .\n \n \n\n The automorphism group of a colour\ ed digraph digraph with\n colouring vert_colours is the group\ consisting of its automorphisms; an\n automorphism of digraph is an isomorphism of coloured\n digraphs from digraph to itself. \ This group is equal to the\n subgroup of AutomorphismGroup(digraph) consisting of those\n automorphisms that preserve the colouring spec\ ified by vert_colours. See\n , and see \n for more information about isom\ orphisms of coloured digraphs. \n\n If digraph is not a multidig\ raph then the automorphism group is\n returned as a group of permutations o\ n the set of vertices of\n digraph. \n\n If digraph is\ a multidigraph then the automorphism group is returned\n as the direct pro\ duct of a group of permutations on the set of vertices of\n digraph \ with a group of permutations on the set of edges of\n digraph. These\ groups can be accessed using on the\n returned group.\n\n By default, the automorphism group is found using &bliss; by Tommi J\ unttila\n and Petteri Kaski. If &NautyTracesInterface; is available, then &\ nauty; by\n Brendan Mckay and Adolfo Piperno can be used instead; see\n \ ,\n , ,\n and .\n\n cycle := CycleDigraph(9);\n\ \ngap> G := AutomorphismGroup(cycle);;\ngap> IsCyclic(G) and Size(G) = 9;\ntru\ e\ngap> colours := [[1, 4, 7], [2, 5, 8], [3, 6, 9]];;\ngap> H := Automorphism\ Group(cycle, colours);;\ngap> Size(H);\n3\ngap> H = AutomorphismGroup(cycle, [\ 1, 2, 3, 1, 2, 3, 1, 2, 3]);\ntrue\ngap> H = SubgroupByProperty(G, p -> OnTupl\ esSets(colours, p) = colours);\ntrue\ngap> IsTrivial(AutomorphismGroup(cycle, \ [1, 1, 2, 2, 2, 2, 2, 2, 2]));\ntrue]]>\n \n\n\n \n \n A permutation group.\n \n\n \ This operation computes the automorphism group of a vertex- and/or\n edge\ -coloured digraph.\n A coloured digraph can be specified by its underlying \ digraph\n digraph and colourings vert_colours, edge_colour\ s.\n Let n be the number of vertices of digraph. The colo\ urings\n must have the following forms:\n\n \n \n \ vert_colours must be fail or a list of n integers,\n \ where vert_colours[i] is the colour of vertex i,\n \ using the colours [1 .. m] for some m <= n;\n \n \n edge_colours must be fail or a list o\ f n lists of\n integers of the same shape as OutNeighbours(di\ graph), where\n edge_colours[i][j] is the colour of t\ he edge\n OutNeighbours(digraph)[i][j], using the colours [1 \ .. k]\n for some k <= n;\n \n \n\n \ Giving vert_colours [edge_colours] as fail is\n equ\ ivalent to setting all vertices [edges] to be the same colour. \n\n Unl\ ike , it is\n poss\ ible to obtain the automorphism group of an edge-coloured multidigraph\n (s\ ee ) when no two edges share the same\n sour\ ce, range, and colour.\n\n The automorphism group of a vertex/edge-c\ oloured digraph\n digraph with colouring c is the group consi\ sting of its\n vertex/edge-colour preserving automorphisms; an\n auto\ morphism of digraph is an isomorphism of vertex/edge-coloured\n \ digraphs from digraph to itself. This group is equal to the\n subgr\ oup of AutomorphismGroup(digraph) consisting of those\n autom\ orphisms that preserve the colouring specified by colours. See\n , and see \n \ for more information about isomorphisms of coloured digraphs. \n\n I\ f digraph is not a multidigraph then the automorphism group is\n ret\ urned as a group of permutations on the set of vertices of\n digraph\ . \n\n If digraph is a multidigraph then the automorphism group \ is returned\n as the direct product of a group of permutations on the set o\ f vertices of\n digraph with a group of permutations on the set of e\ dges of\n digraph. These groups can be accessed using on the\n returned group. \n\n By default, the automorphism group\ is found using &bliss; by Tommi Junttila\n and Petteri Kaski. If &NautyTra\ cesInterface; is available, then &nauty; by\n Brendan Mckay and Adolfo Pipe\ rno can be used instead; see\n ,\n , , and\n .\n\n cycle := CycleDigraph(12\ );\n\ngap> vert_colours := List([1 .\ . 12], x -> x mod 3 + 1);;\ngap> edge_colours := List([1 .. 12], x -> [x mod 2\ + 1]);;\ngap> Size(AutomorphismGroup(cycle));\n12\ngap> Size(AutomorphismGrou\ p(cycle, vert_colours));\n4\ngap> Size(AutomorphismGroup(cycle, fail, edge_col\ ours));\n6\ngap> Size(AutomorphismGroup(cycle, vert_colours, edge_colours));\n\ 2\ngap> IsTrivial(AutomorphismGroup(cycle,\n> vert_colours, List([1 .. 12], x \ -> [x mod 4 + 1])));\ntrue\n]]>\n \n\n\n\ \n \n \n A permutation, or a list of two permu\ tations.\n \n A function \\rho that maps a d\ igraph to a digraph is a canonical\n representative map if the fol\ lowing two conditions hold for all\n digraphs G and H: \n\ \n \n \n \\rho(G) and G are isomorphi\ c as digraphs; and\n \n \n \\rho(G)=\\rho(H)<\ /M> if and only if G and H are\n isomorphic as digraphs.\ \n \n \n\n A canonical labelling of a digraph \ G (under \\rho) is an\n isomorphism of G onto its c\ anonical representative,\n \\rho(G). See \n for more information about isomorph\ isms of digraphs. \n\n BlissCanonicalLabelling returns a canonic\ al labelling of the digraph\n digraph found using &bliss; by Tommi J\ unttila and Petteri Kaski.\n NautyCanonicalLabelling returns a canon\ ical labelling of the digraph\n digraph found using &nauty; by Brend\ an McKay and Adolfo Piperno.\n Note that the canonical labellings returned \ by &bliss; and &nauty; are not\n usually the same (and may depend of the ve\ rsion used). \n\n BlissCanonicalLabelling can only be computed if\ digraph has\n no multiple edges; see . \n\n digraph1 := DigraphFromDiSparse6String\ (\".ImNS_AiB?qRN\");\n\ngap> Blis\ sCanonicalLabelling(digraph1);\n(1,9,5,7)(3,6,4,10)\ngap> p := (1, 2, 7, 5)(3,\ 9)(6, 10, 8);;\ngap> digraph2 := OnDigraphs(digraph1, p);\n\ngap> digraph1 = digraph2;\nfalse\ngap> OnDigraphs\ (digraph1, BlissCanonicalLabelling(digraph1)) =\n> OnDigraphs(digraph2, Bli\ ssCanonicalLabelling(digraph2));\ntrue]]>\n \n\n\n \n \n \n A permutation.\n \n A functi\ on \\rho that maps a coloured digraph to a coloured digraph is\n a <\ E>canonical representative map if the following two conditions hold\n f\ or all coloured digraphs G and H:\n\n \n \n \ \\rho(G) and G are isomorphic as coloured digraphs; and\n\ \n \n \\rho(G)=\\rho(H) if and only if <\ M>G and H are\n isomorphic as coloured digraphs.\n \n \n\n A canonical labelling of a coloured digraph \ G (under\n \\rho) is an isomorphism of G onto its cano\ nical\n representative, \\rho(G). See for more inf\ ormation about\n isomorphisms of coloured digraphs. \n\n A coloured \ digraph can be specified by its underlying digraph digraph\n and its\ colouring colours. Let n be the number of vertices of\n \ digraph. The colouring colours may have one of the following\n t\ wo forms: \n\n \n \n a list of n integer\ s, where colours[i] is the\n colour of vertex i, u\ sing the colours [1 .. m] for some\n m <= n; or\n \ \n \n a list of non-empty disjoint lists whose union\ is\n DigraphVertices(digraph), such that\n col\ ours[i] is the list of all vertices with colour\n i.\ \n \n \n\n If digraph and colours toget\ her form a coloured digraph,\n BlissCanonicalLabelling returns a can\ onical labelling of the digraph\n digraph found using &bliss; by Tom\ mi Junttila and Petteri Kaski.\n Similarly, NautyCanonicalLabelling \ returns a canonical labelling of\n the digraph digraph found using &\ nauty; by Brendan McKay and Adolfo\n Piperno. Note that the canonical labe\ llings returned by &bliss; and\n &nauty; are not usually the same (and may \ depend of the version used). \n\n BlissCanonicalLabelling can onl\ y be computed if digraph has\n no multiple edges; see . The canonical\n labelling of digraph is given as \ a permutation of its vertices.\n The canonical representative of digraph\ can be created from\n digraph and its canonical labelling p<\ /C> by using the\n operation :\n gap> OnDigraphs(digraph, p);\n\n The colo\ uring of the canonical representative can easily be\n constructed. A vertex\ v (in digraph) has colour i if\n and only if the vert\ ex v ^ p (in the canonical representative) has\n colour i, wh\ ere p is the permutation of the canonical\n labelling that acts on t\ he vertices of digraph. In particular, if\n colours has the f\ irst form that is described above, then the\n colouring of the canonical re\ presentative is given by:\n\n gap> List(DigraphVertices(digraph), i ->\ colours[i / p]);\n\n On the other hand, if colours has the se\ cond form above, then the\n canonical representative has colouring:\n\n \ gap> OnTuplesSets(colours, p);\n\n \n\n digraph := DigraphFromDiSparse6String(\".ImNS_AiB?qRN\");\n\ngap> colours := [[1, 2, 8, 9, 10], [3, 4, \ 5, 6, 7]];;\ngap> p := BlissCanonicalLabelling(digraph, colours);\n(1,5,8,4,10\ ,3,9)(6,7)\ngap> OnDigraphs(digraph, p);\n\ngap> OnTuplesSets(colours, p);\n[ [ 1, 2, 3, 4, 5 ], [ 6, 7, 8, 9, \ 10 ] ]\ngap> colours := [1, 1, 1, 1, 2, 3, 1, 3, 2, 1];;\ngap> p := BlissCanon\ icalLabelling(digraph, colours);\n(1,6,9,7)(3,4,5,8,10)\ngap> OnDigraphs(digra\ ph, p);\n\ngap> List(DigraphVerti\ ces(digraph), i -> colours[i / p]);\n[ 1, 1, 1, 1, 1, 1, 2, 2, 3, 3 ]]]>\n \n\n\n \n \n \n A digraph.\n \n The a\ ttribute BlissCanonicalLabelling returns the canonical\n representat\ ive found by applying . The digraph returned is canonical in the sense\n that\n\n \ \n \n BlissCanonicalDigraph(digraph) a\ nd digraph are\n isomorphic as digraphs; and\n \n \ \n If gr is any digraph then BlissCanonicalDigraph(g\ r)\n and BlissCanonicalDigraph(digraph) are equal if \ and only\n if gr and digraph are isomorphic as digraphs.\ \n \n \n\n Analogously, the attribute NautyCanonic\ alLabelling returns the\n canonical representative found by applying .\n \n\n If the argument digraph is mutable, then the return value of\n\ this attribute is recomputed every time it is called. \n\n digraph := Digraph([[1], [2, 3], [3], [1, 2, 3]]);\n\ngap> canon := BlissCanonicalDigraph(digraph);\ \n\ngap> OutNeighbours(canon);\n[ \ [ 1 ], [ 2 ], [ 3, 2 ], [ 1, 3, 2 ] ]\n]]>\n \n\n\n \n \n\ A permutation group.\n \n If digraph<\ /A> is immutable and was created knowing a subgroup of\n its automorphism g\ roup, then this group is stored in the attribute\n DigraphGroup. If \ digraph is mutable, or was not created\n knowing a subgroup of its a\ utomorphism group, then DigraphGroup\n returns the entire automorphi\ sm group of digraph. \n Note that if digraph is mutable, t\ hen the automorphism group is\n recomputed every time this function is call\ ed.\n \n\n Note that certain other constructor operations such as <\ Ref\n Oper=\"CayleyDigraph\"/>, , a\ nd , may not require a group as one of the\ arguments,\n but use the standard constructor method using a group, and he\ nce set the\n DigraphGroup attribute for the resulting digraph.\n\n \ n := 4;;\ngap> adj := function(x, y)\n> retur\ n (((x - y) mod n) = 1) or (((x - y) mod n) = n - 1);\n> end;;\ngap> group \ := CyclicGroup(IsPermGroup, n);\nGroup([ (1,2,3,4) ])\ngap> D := Digraph(IsMut\ ableDigraph, group, [1 .. n], \\^, adj);\n\ngap> HasDigraphGroup(D);\nfalse\ngap> DigraphGroup(D);\nGroup([ (2,4),\ (1,2)(3,4) ])\ngap> AutomorphismGroup(D);\nGroup([ (2,4), (1,2)(3,4) ])\ngap>\ D := Digraph(group, [1 .. n], \\^, adj);\n\ngap> HasDigraphGroup(D);\ntrue\ngap> DigraphGroup(D);\nGroup([ (1,2\ ,3,4) ])\ngap> D := DoubleDigraph(D);\n\ngap> HasDigraphGroup(D);\ntrue\ngap> DigraphGroup(D);\nGroup([ (1,2,3,\ 4)(5,6,7,8), (1,5)(2,6)(3,7)(4,8) ])\ngap> AutomorphismGroup(D) =\n> Group([(6\ , 8), (5, 7), (4, 6), (3, 5), (2, 4),\n> (1, 2)(3, 4)(5, 6)(7, 8)]);\nt\ rue\ngap> D := Digraph([[2, 3], [], []]);\n\ngap> HasDigraphGroup(D);\nfalse\ngap> HasAutomorphismGroup(D);\nfal\ se\ngap> DigraphGroup(D);\nGroup([ (2,3) ])\ngap> HasAutomorphismGroup(D);\ntr\ ue\ngap> group := DihedralGroup(8);\n\ng\ ap> D := CayleyDigraph(group);\n\ \ngap> HasDigraphGroup(D);\ntrue\ngap> GeneratorsOfGroup(DigraphGroup(D));\n[ \ (1,2)(3,5)(4,6)(7,8), (1,7,4,3)(2,5,6,8), (1,4)(2,6)(3,7)(5,8) ]]]>\ \n \n\n\n \n \n \n An immutable list of lists of inte\ gers.\n \n \n DigraphOrbits returns the orb\ its of the action of the\n on the set of verti\ ces of digraph.\n\n G := Group([(2, 3)(7, 8\ , 9), (1, 2, 3)(4, 5, 6)(8, 9)]);;\ngap> D := EdgeOrbitsDigraph(G, [1, 2]);\n<\ immutable digraph with 9 vertices, 6 edges>\ngap> DigraphOrbits(D);\n[ [ 1, 2,\ 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]\ngap> D := DigraphMutableCopy(D);\n\ngap> DigraphOrbits(D);\n[ [ 1, 2, 3 ], [ 4, \ 5, 6, 7, 8, 9 ] ]\n]]>\n \n\n\n \n \n \n \ An immutable list of integers.\n \n \n Digraph\ OrbitReps returns a list of orbit representatives\n of the action of th\ e on the set of vertices of\n digraph.\ \n\n D := CayleyDigraph(AlternatingGroup(4));\n\ngap> DigraphOrbitReps(D);\n[ 1 ]\ \ngap> D := DigraphMutableCopy(D);\n\ngap> DigraphOrbitReps(D);\n[ 1 ]\ngap> D := DigraphFromDigraph6String(\"&I\ GO??S?`?_@?a?CK?O\");\n\ngap> Di\ graphOrbitReps(D);\n[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]\ngap> DigraphOrbitReps(D\ igraphMutableCopy(D));\n[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]\n]]>\n \n\n\n \n \n \n An immutable list of integers.\n\ \n \n DigraphSchreierVector returns the so\ -called Schreier vector\n of the action of the on the set of vertices of\n digraph. The Schreier vector is \ a list sch of integers with\n length DigraphNrVertices(digraph\ ) where:\n \n sch[i] < 0:\n \ \n implies that i is an orbit representative and\n \ DigraphOrbitReps(digraph)[-sch[i]] = i.\n \n\n \ sch[i] > 0:\n \n implies that i / \ gens[sch[i]]\n is one step closer to the root (or representative) o\ f the tree, where\n gens is the generators of DigraphGroup(digraph).\n \n \n\n \ n := 4;;\ngap> adj := function(x, y)\n> return (((x - y) mod n) = 1) or \ (((x - y) mod n) = n - 1);\n> end;;\ngap> group := CyclicGroup(IsPermGroup,\ n);\nGroup([ (1,2,3,4) ])\ngap> D := Digraph(IsMutableDigraph, group, [1 .. n\ ], \\^, adj);\n\ngap> sch := Digraph\ SchreierVector(D);\n[ -1, 2, 2, 1 ]\ngap> D := CayleyDigraph(AlternatingGroup(\ 4));\n\ngap> sch := DigraphSchre\ ierVector(D);\n[ -1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1 ]\ngap> DigraphOrbitReps(\ D);\n[ 1 ]\ngap> gens := GeneratorsOfGroup(DigraphGroup(D));\n[ (1,7,5)(2,10,9\ )(3,4,11)(6,8,12), (1,3,2)(4,5,6)(7,9,8)(10,11,12) ]\ngap> 10 / gens[sch[10]];\ \n2\ngap> 7 / gens[sch[7]];\n1\ngap> 5 / gens[sch[5]];\n7]]>\n \n\n\n \n \n \n A permutation group.\n \ \n \n DigraphStabilizer returns the stabilizer of the \ vertex v\n under of the action of the o\ n the set of\n vertices of digraph.\n\n \ D := DigraphFromDigraph6String(\"&GYHPQgWTIIPW\");\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\ngap> DigraphSta\ bilizer(D, 2);\nGroup(())\ngap> D := DigraphMutableCopy(D);\n\ngap> DigraphStabilizer(D, 8);\nGroup(())\ngap> Dig\ raphStabilizer(D, 2);\nGroup(())\n]]>\n \n\n\n \n \n true or false.\n \n This operation returns true if there exis\ ts an isomorphism from the\n digraph digraph1 to the digraph digr\ aph2. See for more information\n about isomorphisms of digraphs. \n\n By d\ efault, an isomorphism is found using the canonical labellings of the\n dig\ raphs obtained from &bliss; by Tommi Junttila\n and Petteri Kaski. If &Naut\ yTracesInterface; is available, then &nauty; by\n Brendan Mckay and Adolfo \ Piperno can be used instead; see\n , and .\n\n digraph1 := Cy\ cleDigraph(4);\n\ngap> digraph2 := Cy\ cleDigraph(5);\n\ngap> IsIsomorphicDi\ graph(digraph1, digraph2);\nfalse\ngap> digraph2 := DigraphReverse(digraph1);\ \n\ngap> IsIsomorphicDigraph(digra\ ph1, digraph2);\ntrue\ngap> digraph1 := Digraph([[3], [], []]);\n\ngap> digraph2 := Digraph([[], [], [2]]);\n\ngap> IsIsomorphicDigraph(digraph1, di\ graph2);\ntrue]]>\n \n\n\n \n \n t\ rue or false.\n \n This operation tests \ for isomorphism of coloured digraphs. A coloured\n digraph can be specifie\ d by its underlying digraph digraph1 and its\n colouring colours1\ . Let n be the number of vertices of\n digraph1. The col\ ouring colours1 may have one of the following\n two forms:\n\n \n \n a list of n integers, where colours<\ C>[i] is the\n colour of vertex i, using the colours [1 .\ . m] for some\n m <= n; or\n \n \n\ a list of non-empty disjoint lists whose union is\n DigraphV\ ertices(digraph), such that\n colours[i] is th\ e list of all vertices with colour\n i.\n \n \n\n If digraph1 and digraph2 are digraphs without multipl\ e edges,\n and colours1 and colours2 are colourings of dig\ raph1\n and digraph2, respectively, then this operation returns \ true\n if there exists an isomorphism between these two coloured dig\ raphs. See\n for more information about isomorphisms of coloure\ d digraphs. \n\n By default, an isomorphism is found using the canonica\ l labellings of the\n digraphs obtained from &bliss; by Tommi Junttila\n \ and Petteri Kaski. If &NautyTracesInterface; is available, then &nauty; by\n \ Brendan Mckay and Adolfo Piperno can be used instead; see\n , and .\n\n digraph1 := ChainDigraph(4);\n\ngap> digraph2 := ChainDigraph(3);\n\ngap> IsIsomorphicDigraph(digraph1, digraph2,\n> [[1, 4], [2, 3]], [[\ 1, 2], [3]]);\nfalse\ngap> digraph2 := DigraphReverse(digraph1);\n\ngap> IsIsomorphicDigraph(digraph1, digraph2,\ \n> [1, 1, 1, 1], [1, 1, 1, 1]);\ntrue\ngap> IsIsomorphicDigraph(digraph1, di\ graph2,\n> [1, 2, 2, 1], [1, 2, 2, 1]);\ntrue\ngap> IsIsomorphicDigraph(digra\ ph1, digraph2,\n> [1, 1, 2, 2], [1, 1, 2, 2]);\nfalse]]>\n \n\n\n \n \n A permutat\ ion, or a pair of permutations, or fail.\n \n \ This operation returns an isomorphism between the digraphs digraph1\n\ and digraph2 if one exists, else this operation returns fail\ .\n \n\n An isomorphism from a digraph digraph1 to a d\ igraph\n digraph2 is a bijection p from the vertices of\n \ digraph1 to the vertices of digraph2 with the following\n pro\ perty: for all vertices i and j of digraph1,\n [i, \ j] is an edge of digraph1 if and only if [i ^ p, j\n ^ p] is an edge of digraph2. \n\n If there exists such an isomorph\ ism, then this operation returns one.\n The form of this isomorphism is a p\ ermutation p of the vertices of\n digraph1 such that \n\n\ OnDigraphs(digraph1, p) = digraph2.\n\n By default, an is\ omorphism is found using the canonical labellings of the\n digraphs obtaine\ d from &bliss; by Tommi Junttila\n and Petteri Kaski. If &NautyTracesInterf\ ace; is available, then &nauty; by\n Brendan Mckay and Adolfo Piperno can b\ e used instead; see\n , and .\n\n digraph1 := CycleDigraph(4)\ ;\n\ngap> digraph2 := CycleDigraph(5)\ ;\n\ngap> IsomorphismDigraphs(digraph\ 1, digraph2);\nfail\ngap> digraph1 := CompleteBipartiteDigraph(10, 5);\n\ngap> digraph\ 2 := CompleteBipartiteDigraph(5, 10);\n\ngap> p := IsomorphismDigraphs(digraph1, digra\ ph2);\n(1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)\ngap> OnDigraphs(digraph1, p)\ = digraph2;\ntrue\n]]>\n \n\n\n \n \n A permutation, or fail.\n \n This operati\ on searches for an isomorphism between coloured digraphs. A\n coloured dig\ raph can be specified by its underlying digraph digraph1\n and its c\ olouring colours1. Let n be the number of vertices\n of d\ igraph1. The colouring colours1 may have one of the\n following \ two forms:\n\n \n \n a list of n integers, w\ here colours[i] is the\n colour of vertex i, using\ the colours [1 .. m] for some\n m <= n; or\n \n \n a list of non-empty disjoint lists whose union is\ \n DigraphVertices(digraph), such that\n colour\ s[i] is the list of all vertices with colour\n i.\n \ \n \n\n An isomorphism between coloured digraph\ s is an isomorphism between\n the underlying digraphs that preserves the co\ lourings. See for more information\n about isomorphisms of digraphs. More precisely, l\ et f be an\n isomorphism of digraphs from the digraph digraph1 (with colouring\n colours1) to the digraph digraph2 (with c\ olouring\n colours2), and let p be the permutation of the ver\ tices of\n digraph1 that corresponds to f. Then f pre\ serves the\n colourings of digraph1 and digraph2 – and henc\ e is an\n isomorphism of coloured digraphs – if colours1[i] =\n\ colours2[i ^ p] for all vertices i in digraph1.\ \n \n\n This operation returns such an isomorphism if one exists, el\ se it returns\n fail. \n\n By default, an isomorphism is foun\ d using the canonical labellings of the\n digraphs obtained from &bliss; by\ Tommi Junttila\n and Petteri Kaski. If &NautyTracesInterface; is available\ , then &nauty; by\n Brendan Mckay and Adolfo Piperno can be used instead; s\ ee\n , and \ .\n\n digraph1 := ChainDigraph(4);\n\ngap> digraph2 := ChainDigraph(3);\n\ngap> IsomorphismDigraphs(digraph1, digraph2,\n> \ [[1, 4], [2, 3]], [[1, 2], [3]]);\nfail\ngap> digraph2 := DigraphReverse(digra\ ph1);\n\ngap> colours1 := [1, 1, 1\ , 1];;\ngap> colours2 := [1, 1, 1, 1];;\ngap> p := IsomorphismDigraphs(digraph\ 1, digraph2, colours1, colours2);\n(1,4)(2,3)\ngap> OnDigraphs(digraph1, p) = \ digraph2;\ntrue\ngap> List(DigraphVertices(digraph1), i -> colours1[i ^ p]) = \ colours2;\ntrue\ngap> colours1 := [1, 1, 2, 2];;\ngap> colours2 := [2, 2, 1, 1\ ];;\ngap> p := IsomorphismDigraphs(digraph1, digraph2, colours1, colours2);\n(\ 1,4)(2,3)\ngap> OnDigraphs(digraph1, p) = digraph2;\ntrue\ngap> List(DigraphVe\ rtices(digraph1), i -> colours1[i ^ p]) = colours2;\ntrue\ngap> IsomorphismDig\ raphs(digraph1, digraph2,\n> [1, 1, 2, 2], [1, 1, 2, 2]);\nfail]]>\ \n \n\n\n \n \n An immutable list of lists\ .\n \n This function returns the list out of\ out-neighbours of each\n representative of the orbits of the action\ of on\n the vertex set of the digraph digr\ aph. \n\n More specifically, if reps is the list of orbit re\ presentatives,\n then a vertex j appears in out[i] each time\ \n there exists an edge with source reps[i] and range j in\n \ digraph.\n \n\n If is trivial\ , then \n is returned.\n\n D := Digraph([\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2,\ 3, 4]]);\n\ngap> DigraphGroup(D)\ ;\nGroup(())\ngap> RepresentativeOutNeighbours(D);\n[ [ 2, 1, 3, 4, 5 ], [ 3, \ 5 ], [ 2 ], [ 1, 2, 3, 5 ], [ 1, 2, 3, 4 ] ]\ngap> D := Digraph(IsMutableDigra\ ph, [\n> [2, 1, 3, 4, 5], [3, 5], [2], [1, 2, 3, 5], [1, 2, 3, 4]]);\n\ngap> DigraphGroup(D);\nGroup(())\ngap> R\ epresentativeOutNeighbours(D);\n[ [ 2, 1, 3, 4, 5 ], [ 3, 5 ], [ 2 ], [ 1, 2, \ 3, 5 ], [ 1, 2, 3, 4 ] ]\ngap> D := DigraphFromDigraph6String(\"&GYHPQgWTIIPW\ \");\n\ngap> G := DigraphGroup(D)\ ;;\ngap> GeneratorsOfGroup(G);\n[ (1,2)(3,4)(5,6)(7,8), (1,3,2,4)(5,7,6,8), (1\ ,5)(2,6)(3,8)(4,7) ]\ngap> Set(RepresentativeOutNeighbours(D), Set);\n[ [ 2, 3\ , 5 ] ]]]>\n \n\n\n \n <\ Oper Name=\"IsDigraphIsomorphism\" Arg=\"src, ran, x\" \n Label=\"for d\ igraphs and transformation or permutation\"/>\n \n \n \n true or false.\n \n \ IsDigraphIsomorphism returns true if the permutation or\n tra\ nsformation x is an isomorphism from the digraph src to\n the\ digraph ran.\n \n\n IsDigraphAutomorphism returns \ true if the permutation or\n transformation x is an automorphism\ of the digraph digraph.\n \n\n A permutation or transformati\ on x is an isomorphism from a\n digraph src to a digra\ ph ran if the following hold:\n \n \n x is a bijection from the vertices of src to those of\n ran<\ /A>;\n \n \n [u ^ x, v ^ x]\ is an edge of\n ran if and only if [u, v] is an\n \ edge of src; and\n \n \n x fixes \ every i which is not a vertex of src.\n \n \n See also . \n \n If col1 and col2, or col, are given, \ then they must\n represent vertex colourings; see \n \n for detail\ s of the permissible values for\n these arguments. The homomorphism must th\ en also have the property:\n \n \n \n col1[i] =\ col2[i ^ x] for all vertices i of src,\n for IsDi\ graphIsomorphism. \n \n col[i] = col[i ^ x] for all vertices i of digraph,\n for IsDigraphAutomo\ rphism. \n \n\n For some digraphs, it can be faster to\ use IsDigraphAutomorphism\n than to test membership in the automorp\ hism group of digraph.\n\n src := Digraph([\ [1], [1, 2], [1, 3]]);\n\ngap> IsD\ igraphAutomorphism(src, (1, 2, 3));\nfalse\ngap> IsDigraphAutomorphism(src, (2\ , 3));\ntrue\ngap> IsDigraphAutomorphism(src, (2, 3), [2, 1, 1]);\ntrue\ngap> \ IsDigraphAutomorphism(src, (2, 3), [2, 2, 1]);\nfalse\ngap> IsDigraphAutomorph\ ism(src, (2, 3)(4, 5));\nfalse\ngap> IsDigraphAutomorphism(src, (1, 4));\nfals\ e\ngap> IsDigraphAutomorphism(src, ());\ntrue\ngap> ran := Digraph([[2, 1], [2\ ], [2, 3]]);\n\ngap> IsDigraphIsom\ orphism(src, ran, (1, 2));\ntrue\ngap> IsDigraphIsomorphism(ran, src, (1, 2));\ \ntrue\ngap> IsDigraphIsomorphism(ran, src, (1, 2));\ntrue\ngap> IsDigraphIsom\ orphism(src, Digraph([[3], [1, 3], [2]]), (1, 2, 3));\nfalse\ngap> IsDigraphIs\ omorphism(src, ran, (1, 2), [1, 2, 3], [2, 1, 3]);\ntrue\ngap> IsDigraphIsomor\ phism(src, ran, (1, 2), [1, 2, 2], [2, 1, 3]);\nfalse\n]]>\n \n\n\n \n \n \n true\ or false. \n \n The operation IsDigr\ aphColouring verifies whether or not \n the list list describe\ s a proper colouring of the digraph \n digraph. \n \n\n \ A list list describes a proper colouring of a digraph\n \ digraph if list consists of positive integers, the length of\n \ list equals the number of vertices in digraph, and for any\ \n vertices u, v of digraph if u and v are ad\ jacent,\n then list[u] >< list[v].\n \ \n\n A transformation t describes a proper colouring of a digraph\ \n digraph, if ImageListOfTransformation(t,\n Digra\ phNrVertices(digraph)) is a proper colouring of \n digraph<\ /A>. \n\n See also .\n \n \ D := JohnsonDigraph(5, 3);\n\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2\ , 1, 4, 5, 6, 7]);\ntrue\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, \ 6, 7]);\nfalse\ngap> IsDigraphColouring(D, [1, 2, 3, 3, 2, 1, 2, 5, 6, -1]);\n\ false\ngap> IsDigraphColouring(D, [1, 2, 3]);\nfalse\ngap> IsDigraphColouring(\ D, IdentityTransformation);\ntrue\n]]>\n \n \n\n \n\n Homomorphisms of digraphs\n\n The following methods exist to find homomorphisms between digraphs\ .\n If an argument to one of these methods is a digraph with multiple edges\ ,\n then the multiplicity of edges will be ignored in order to perform the\ \n calculation; the digraph will be treated as if it has no multiple edges.\ \n\n \n \n The argument user_param.\n \n This function finds homomorphisms from the digrap\ h D1 to the digraph\n D2 subject to the conditions imposed by\ the other arguments as\n described below.\n\n If f and g<\ /C> are homomorphisms found by\n HomomorphismDigraphsFinder, then f cannot be obtained from\n g by right multiplying by an automo\ rphism of D2 in\n aut_grp.\n\n \n hook\ \n \n This argument should be a function or fa\ il. \n\n If hook is a function, then it must have two arg\ uments\n user_param (see below) and a transformation t. T\ he\n function hook(user_param, t) is called every \ time\n a new homomorphism t is found by\n Homomorphism\ DigraphsFinder. If the function returns true,\n then Homo\ morphismDigraphsFinder stops and does not find any\n further homomo\ rphisms. This feature might be useful if you are\n searching for a homo\ morphism that satisfies some condition that you\n cannot specify via th\ e other arguments to\n HomomorphismDigraphsFinder.\n \ \n\n If hook is fail, then a default function is used whi\ ch\n simply adds every new homomorphism found by\n Homomorphi\ smDigraphsFinder to user_param, which must be a\n mutable li\ st in this case.\n \n\n user_param\n \ \n If hook is a function, then user_param can be a\ ny &GAP;\n object. The object user_param is used as the first ar\ gument of\n the function hook. For example, user_param mi\ ght be a\n transformation semigroup, and hook(user_param, t)\n might set user_param to be the closure of user_p\ aram and\n t. \n\n If the value of hook is\ fail, then the value of\n user_param must be a mutable l\ ist.\n \n\n max_results\n \n \ This argument should be a positive integer or infinity.\n \ HomomorphismDigraphsFinder will return after it has found\n m\ ax_results homomorphisms or the search is complete, whichever\n hap\ pens first.\n \n\n hint\n \n \ This argument should be a positive integer or fail. \n\n \ If hint is a positive integer, then only homorphisms of rank\n \ hint are found. \n\n If hint is fail, then no\ restriction is put on the rank\n of homomorphisms found.\n \n\n injective\n \n This argument \ should be 0, 1, or 2. If it is\n 2, then o\ nly embeddings are found, if it is 1, then only\n injective homo\ morphisms are found, and if it is 0 there are no\n restrictions \ imposed by this argument.\n \n\n For backwards compatibility\ , injective can also be false\n or true which corr\ espond to the values 0 and 1\n described in the previous \ paragraph, respectively. \n \n\n image\n \ \n This argument should be a subset of the vertices of the g\ raph D2.\n HomomorphismDigraphsFinder only finds homomorp\ hisms from\n D1 to the subgraph of D2 induced by the vert\ ices\n image.\n \n\n partial_map\n \n This argument should be a partial map from D1<\ /A> to D2,\n that is, a (not necessarily dense) list of vertices\ of the digraph\n D2 of length no greater than the number vertic\ es in the digraph\n D1. HomomorphismDigraphsFinder only f\ inds homomorphisms\n extending partial_map (if any).\n \n \n colors1\n \n This sho\ uld be a list representing possible colours of vertices in the\n digrap\ h D1; see \n \n for details of the permis\ sible values for this argument.\n \n \n colors2\ \n \n This should be a list representing possible\ colours of vertices in the\n digraph D2; see \n \n for details of the permissible values for this argument.\n \ \n order\n \n The optiona\ l argument order specifies the order the\n vertices in D1\ appear in the search for homomorphisms. \n The value of this parameter\ can have a large impact\n on the runtime of the function. It seems in \ many cases to be a good\n idea for this to be the , i.e.\n vertices ordered from highest to lowest degr\ ee.\n \n \n The optional argument aut_grp\ should be a subgroup of the\n automorphism group of D2. This fu\ nction returns unique\n representatives of the homomorphisms found up t\ o right multiplication\n by aut_grp. If this argument is not spe\ cific, it defaults to the\n full automorphism group of D2, which\ may be costly to calculate.\n \n \n\n D := ChainDigraph(10);\n\ \ngap> D := DigraphSymmetricClosure(D);\n\ngap> HomomorphismDigraphsFinder(D, D, fail, [], infinity,\ 2, 0,\n> [3, 4], [], fail, fail);\n[ Transformation( [ 3, 4, 3, 4, 3, 4, 3, 4\ , 3, 4 ] ), \n Transformation( [ 4, 3, 4, 3, 4, 3, 4, 3, 4, 3 ] ) ]\ngap> D2 \ := CompleteDigraph(6);;\ngap> HomomorphismDigraphsFinder(D, D2, fail, [], 1, f\ ail, 0,\n> [1 .. 6], [1, 2, 1], fail, fail);\n[ Transformation( [ 1, 2, 1, 3, \ 4, 5, 6, 1, 2, 1 ] ) ]\ngap> func := function(user_param, t)\n> Add(user_param\ , t * user_param[1]);\n> end;;\ngap> HomomorphismDigraphsFinder(D, D2, func, [\ Transformation([2, 2])],\n> 3, fail, 0, [1 .. 6], [1, 2, 1], fail, fail);\n[ T\ ransformation( [ 2, 2 ] ), \n Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 2 \ ] ), \n Transformation( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 3 ] ), \n Transformatio\ n( [ 2, 2, 2, 3, 4, 5, 6, 2, 2, 4 ] ) ]\ngap> HomomorphismDigraphsFinder(NullD\ igraph(2), NullDigraph(3), fail,\n> [], infinity, fail, 1, [1, 2, 3], fail, fa\ il, fail, fail,\n> Group(()));\n[ IdentityTransformation, Transformation( [ 1,\ 3, 3 ] ), \n Transformation( [ 2, 1 ] ), Transformation( [ 2, 3, 3 ] ), \n \ Transformation( [ 3, 1, 3 ] ), Transformation( [ 3, 2, 3 ] ) ]\ngap> Homomorph\ ismDigraphsFinder(NullDigraph(2), NullDigraph(3), fail,\n> [], infinity, fail,\ 1, [1, 2, 3], fail, fail, fail, fail,\n> Group((1, 2)));\n[ IdentityTransform\ ation, Transformation( [ 1, 3, 3 ] ), \n Transformation( [ 3, 1, 3 ] ) ]]]>\n \n\n\n \n \n A transformati\ on, or fail.\n \n A homomorphism from dig\ raph1 to digraph2 is a mapping from\n the vertex set of digra\ ph1 to a subset of the vertices of\n digraph2, such that every p\ air of vertices [i,j] which has\n an edge i->j is mapped to a\ pair of vertices [a,b] which\n has an edge a->b. Note that \ non-adjacent vertices can still be\n mapped to adjacent vertices. \n\n \ DigraphHomomorphism returns a single homomorphism between\n di\ graph1 and digraph2 if it exists, otherwise it returns\n fail\ .\n gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph(\ [[3, 5], [2], [3, 1], [], [4]]);\n\ \ngap> DigraphHomomorphism(gr1, gr1);\nIdentityTransformation\ngap> map := Dig\ raphHomomorphism(gr1, gr2);\nTransformation( [ 3, 1, 5, 4, 5 ] )\ngap> IsDigra\ phHomomorphism(gr1, gr2, map);\ntrue\n]]>\n \n\n\n \n \n \n A list of transformations.\n \ \n HomomorphismsDigraphsRepresentatives finds every\n \ between digraph1 and\n digr\ aph2, up to right multiplication by an element of the\n of digraph2.\n In other w\ ords, every homomorphism f between digraph1 and\n digraph2\ can be written as the composition f = g * x,\n where g i\ s one of the HomomorphismsDigraphsRepresentatives\n and x is \ an automorphism of digraph2.\n \n\n HomomorphismsDigraphs<\ /C> returns all homomorphisms between\n digraph1 and digraph2\ .\n gr1 := ChainDigraph(3);;\ngap> gr2 := Digraph(\ [[3, 5], [2], [3, 1], [], [4]]);\n\ \ngap> HomomorphismsDigraphs(gr1, gr2);\n[ Transformation( [ 1, 3, 1 ] ), Tran\ sformation( [ 1, 3, 3 ] ), \n Transformation( [ 1, 5, 4, 4, 5 ] ), Transforma\ tion( [ 2, 2, 2 ] ), \n Transformation( [ 3, 1, 3 ] ), Transformation( [ 3, 1\ , 5, 4, 5 ] ), \n Transformation( [ 3, 3, 1 ] ), Transformation( [ 3, 3, 3 ] \ ) ]\ngap> HomomorphismsDigraphsRepresentatives(gr1, CompleteDigraph(3));\n[ Tr\ ansformation( [ 2, 1 ] ), Transformation( [ 2, 1, 2 ] ) ]\n]]>\n \n\n\n \n \n A transformation, or fail<\ /K>.\n \n DigraphMonomorphism returns a sing\ le injective between digr\ aph1 and digraph2\n if one exists, otherwise it returns fail<\ /K>.\n gr1 := ChainDigraph(3);;\ngap> gr2 := Digra\ ph([[3, 5], [2], [3, 1], [], [4]]);\n\ngap> DigraphMonomorphism(gr1, gr1);\nIdentityTransformation\ngap> Digraph\ Monomorphism(gr1, gr2);\nTransformation( [ 3, 1, 5, 4, 5 ] )\n]]>\n \ \n\n\n \n \n \n A list of transfor\ mations.\n \n These operations behave the same as <\ Ref Oper=\"HomomorphismsDigraphs\"/> and\n , except they only return\n injective homomorphi\ sms.\n gr1 := ChainDigraph(3);;\ngap> gr2 := Digra\ ph([[3, 5], [2], [3, 1], [], [4]]);\n\ngap> MonomorphismsDigraphs(gr1, gr2);\n[ Transformation( [ 1, 5, 4, 4, 5 \ ] ), \n Transformation( [ 3, 1, 5, 4, 5 ] ) ]\ngap> MonomorphismsDigraphsRepr\ esentatives(gr1, CompleteDigraph(3));\n[ Transformation( [ 2, 1 ] ) ]\n]]>\n \n\n\n \n \n A transformation, \ or fail.\n \n DigraphEpimorphism retu\ rns a single surjective betw\ een digraph1 and digraph2\n if one exists, otherwise it retur\ ns fail.\n gr1 := DigraphReverse(ChainDigra\ ph(4));\n\ngap> gr2 := DigraphRemo\ veEdge(CompleteDigraph(3), [1, 2]);\n\ngap> DigraphEpimorphism(gr2, gr1);\nfail\ngap> DigraphEpimorphism(gr1, gr\ 2);\nTransformation( [ 3, 1, 2, 3 ] )\n]]>\n \n\n\n \n \n \n A list of transformations.\n \n These operations behave the same as and\n , exc\ ept they only return\n surjective homomorphisms.\n gr1 := DigraphReverse(ChainDigraph(4));\n\ngap> gr2 := DigraphSymmetricClosure(CycleDigraph(3));\n\ngap> EpimorphismsDigraphs\ Representatives(gr1, gr2);\n[ Transformation( [ 3, 1, 2, 1 ] ), Transformation\ ( [ 3, 1, 2, 3 ] ), \n Transformation( [ 2, 1, 2, 3 ] ) ]\ngap> EpimorphismsD\ igraphs(gr1, gr2);\n[ Transformation( [ 1, 2, 1, 3 ] ), Transformation( [ 1, 2\ , 3, 1 ] ), \n Transformation( [ 1, 2, 3, 2 ] ), Transformation( [ 1, 3, 1, 2\ ] ), \n Transformation( [ 1, 3, 2, 1 ] ), Transformation( [ 1, 3, 2, 3 ] ), \ \n Transformation( [ 2, 1, 2, 3 ] ), Transformation( [ 2, 1, 3, 1 ] ), \n Tr\ ansformation( [ 2, 1, 3, 2 ] ), Transformation( [ 2, 3, 1, 2 ] ), \n Transfor\ mation( [ 2, 3, 1, 3 ] ), Transformation( [ 2, 3, 2, 1 ] ), \n Transformation\ ( [ 3, 1, 2, 1 ] ), Transformation( [ 3, 1, 2, 3 ] ), \n Transformation( [ 3,\ 1, 3, 2 ] ), Transformation( [ 3, 2, 1, 2 ] ), \n Transformation( [ 3, 2, 1,\ 3 ] ), Transformation( [ 3, 2, 3, 1 ] ) ]\n]]>\n \n<\ /ManSection>\n\n \n \n A transformation, or fail.\n \ \n An embedding of a digraph digraph1 into another dig\ raph\n digraph2 is a from dig\ raph1\n to digraph2 which has the additional property that a pai\ r of\n vertices [i, j] which have no edge i -> j in digrap\ h1 are\n\n mapped to a pair of vertices [a, b] which have no edg\ e a->b in\n digraph2. \n\n In other words, an embedding\ t is an isomorphism from\n digraph1 to the of digraph2\n on the image of t. \n\n \ DigraphEmbedding returns a single embedding if one exists, otherwise\n\ it returns fail.\n gr := ChainDigraph(3\ );\n\ngap> DigraphEmbedding(gr, Compl\ eteDigraph(4));\nfail\ngap> DigraphEmbedding(gr, Digraph([[3], [1, 4], [1], [3\ ]]));\nTransformation( [ 2, 4, 3, 4 ] )\n]]>\n \n\n\n \n \n \n \ A list of transformations.\n \n These oper\ ations behave the same as and\n , except they only return\n \ embeddings of D1 into D2. \n\n See also .\n D1 := NullDigraph(2);\n\ngap> D2 := CycleDigraph(5);\n\ngap> EmbeddingsDigraphsRepresentatives(D1, D2);\ \n[ Transformation( [ 1, 3, 3 ] ), Transformation( [ 1, 4, 3, 4 ] ) ]\ngap> Em\ beddingsDigraphs(D1, D2);\n[ Transformation( [ 1, 3, 3 ] ), Transformation( [ \ 1, 4, 3, 4 ] ), \n Transformation( [ 2, 4, 4, 5, 1 ] ), \n Transformation( [\ 2, 5, 4, 5, 1 ] ), \n Transformation( [ 3, 1, 5, 1, 2 ] ), \n Transformatio\ n( [ 3, 5, 5, 1, 2 ] ), \n Transformation( [ 4, 1, 1, 2, 3 ] ), \n Transform\ ation( [ 4, 2, 1, 2, 3 ] ), \n Transformation( [ 5, 2, 2, 3, 4 ] ), \n Trans\ formation( [ 5, 3, 2, 3, 4 ] ) ]\n]]>\n \n\n\n \n \n \ \n \n \n \n \n \n \n true or <\ K>false.\n \n IsDigraphHomomorphism retu\ rns true if the permutation\n or transformation x is a homomo\ rphism from the digraph\n src to the digraph ran.\n \n\ \n IsDigraphEpimorphism returns true if the permutation\n \ or transformation x is a surjective homomorphism from the digraph\n \ src to the digraph ran.\n \n\n IsDigraphMonomorphis\ m returns true if the permutation\n or transformation x i\ s an injective homomorphism from the digraph\n src to the digraph ran.\n \n\n IsDigraphEndomorphism returns true if\ the permutation\n or transformation x is an endomorphism of the dig\ raph\n digraph.\n \n\n A permutation or transformation \ x is a homomorphism from a\n digraph src to a digraph \ ran if the following hold:\n \n \n [u ^ x\ , v ^ x] is an edge of\n ran whenever [u, v] is an\n edge of src; and \n \n x<\ /A> fixes every i which is not a vertex of src.\n \n\ \n See also . \ \n\n\n If col1 and col2, or col, are given, then they \ must\n represent vertex colourings; see \n \n for details of the p\ ermissible values for these argument. The\n homomorphism must then also hav\ e the property:\n \n \n \n col[i] = col[i ^ x]<\ /C> for all vertices i of digraph,\n in the case of Is\ DigraphEndomorphism.\n \n col1[i] = col2[i ^ x\ ] for all vertices i of src,\n in the cases of the ot\ her operations.\n \n\n See also .\n \n src := Digraph([[1], [1, \ 2], [1, 3]]);\n\ngap> ran := Digra\ ph([[1], [1, 2]]);\n\ngap> IsDigra\ phHomomorphism(src, ran, Transformation([1, 2, 2]));\ntrue\ngap> IsDigraphHomo\ morphism(src, ran, Transformation([2, 1, 2]));\nfalse\ngap> IsDigraphHomomorph\ ism(src, ran, Transformation([3, 3, 3]));\nfalse\ngap> IsDigraphHomomorphism(s\ rc, src, Transformation([3, 3, 3]));\ntrue\ngap> IsDigraphHomomorphism(src, ra\ n, Transformation([1, 2, 2]),\n> [1, 2, 2], [1, 2]);\ \ntrue\ngap> IsDigraphHomomorphism(src, ran, Transformation([1, 2, 2]),\n> \ [2, 1, 1], [1, 2]);\nfalse\ngap> IsDigraphEndomorphism(s\ rc, Transformation([3, 3, 3]));\ntrue\ngap> IsDigraphEndomorphism(src, Transfo\ rmation([3, 3, 3]), [1, 1, 1]);\ntrue\ngap> IsDigraphEndomorphism(src, Transfo\ rmation([3, 3, 3]), [1, 1, 2]);\nfalse\ngap> IsDigraphEpimorphism(src, ran, Tr\ ansformation([3, 3, 3]));\nfalse\ngap> IsDigraphMonomorphism(src, ran, Transfo\ rmation([1, 2, 2]));\nfalse\ngap> IsDigraphEpimorphism(src, ran, Transformatio\ n([1, 2, 2]));\ntrue\ngap> IsDigraphMonomorphism(ran, src, ());\ntrue]]>\n \n\n\n \n \n \n true or false.\n \n IsDigraphEmbedding returns true if the permutat\ ion\n or transformation x is a embedding of the digraph\n src<\ /A> into the digraph ran, while respecting the colourings\n col1<\ /A> and col2 if given.\n \n\n A permutation or transformation\ x is a embedding of a digraph\n src into a digraph ran if x is a monomorphism from\n src to ran and \ the inverse of x is a monomorphism from\n the subdigraph of ran induced by the image of x to\n src.\n\n See also . \n\n src := Di\ graph([[1], [1, 2]]);\n\ngap> ran \ := Digraph([[1], [1, 2], [1, 3]]);\n\ngap> IsDigraphMonomorphism(src, ran, ());\ntrue\ngap> IsDigraphEmbedding(s\ rc, ran, ());\ntrue\ngap> IsDigraphEmbedding(src, ran, (), [2, 1], [2, 1, 1]);\ \ntrue\ngap> IsDigraphEmbedding(src, ran, (), [2, 1], [1, 2, 1]);\nfalse\ngap>\ ran := Digraph([[1, 2], [1, 2], [1, 3]]);\n\ngap> IsDigraphMonomorphism(src, ran, IdentityTransformation);\ntru\ e\ngap> IsDigraphEmbedding(src, ran, IdentityTransformation);\nfalse]]>\n \n\n\n \n \n true or false. \n \n The operation \ DigraphsRespectsColouring verifies whether or not \n the permutati\ on or transformation x respects the vertex colourings\n col1 and col2 of the digraphs src and range.\n That is,\ DigraphsRespectsColouring returns true if and only if for\n \ all vertices i of src, col1[i] = col2[i ^ x].\n \n\n src := Digraph([[1], [1, 2]]);\n\ngap> ran := Digraph([[1], [1, 2], [1, 3]]\ );\n\ngap> DigraphsRespectsColouri\ ng(src, ran, (1, 2), [2, 1], [1, 2, 1]);\ntrue\ngap> DigraphsRespectsColouring\ (src, ran, (1, 2), [2, 1], [2, 1, 1]);\nfalse\n]]>\n \n \n\n \n \n \n A list of transformation\ s.\n \n An endomorphism of digraph is a homo\ morphism\n from digraph back to \ itself.\n\n GeneratorsOfEndomorphismMonoid, called with a single arg\ ument,\n returns a generating set for the monoid of all endomorphisms of\n \ digraph. If digraph belongs to , then the value of\n GeneratorsOfEndomorphismMonoid wil\ l not be recomputed on future\n calls. \n\n If the colors argu\ ment is specified, then\n GeneratorsOfEndomorphismMonoid will return\ a generating\n set for the monoid of endomorphisms which respect the given\ colouring.\n The colouring colors can be in one of two forms: \ \n\n \n \n A list of positive integers of size the \ number of vertices of\n digraph, where colors[i] i\ s the colour of vertex\n i.\n \n \n \ A list of lists, such that colors[i] is a list of all\n \ vertices with colour i.\n \n \n\n If the l\ imit argument is specified, then it will return only the\n first lim\ it homomorphisms, where limit must be a positive\n integer or infinity. \n\n gr := Digraph(List([1 .. 3\ ], x -> [1 .. 3]));;\ngap> GeneratorsOfEndomorphismMonoid(gr);\n[ Transformati\ on( [ 1, 3, 2 ] ), Transformation( [ 2, 1 ] ), \n IdentityTransformation, Tra\ nsformation( [ 1, 2, 1 ] ), \n Transformation( [ 1, 2, 2 ] ), Transformation(\ [ 1, 1, 2 ] ), \n Transformation( [ 1, 1, 1 ] ) ]\ngap> GeneratorsOfEndomorp\ hismMonoid(gr, 3);\n[ Transformation( [ 1, 3, 2 ] ), Transformation( [ 2, 1 ] \ ), \n IdentityTransformation ]\ngap> gr := CompleteDigraph(3);;\ngap> Generat\ orsOfEndomorphismMonoid(gr);\n[ Transformation( [ 2, 3, 1 ] ), Transformation(\ [ 2, 1 ] ), \n IdentityTransformation ]\ngap> GeneratorsOfEndomorphismMonoid\ (gr, [1, 2, 2]);\n[ Transformation( [ 1, 3, 2 ] ), IdentityTransformation ]\ng\ ap> GeneratorsOfEndomorphismMonoid(gr, [[1], [2, 3]]);\n[ Transformation( [ 1,\ 3, 2 ] ), IdentityTransformation ]\n]]>\n \n\n\n \n \n A trans\ formation, or fail.\n \n A proper colouri\ ng of a digraph is a labelling of its vertices in\n such a way that adj\ acent vertices have different labels. A proper\n n-colouring is a proper colouring that uses exactly n\n colours. Equivalently\ , a proper (n-)colouring of a digraph can be\n defined to be a from a digraph onto the\n complete digraph (w\ ith n vertices); see . Note that\ a digraph with loops () does not have a\ proper n-colouring for\n any value n.\n \n\n If digraph is a digraph and n is a non-negative integer, then\n DigraphColouring(digraph, n) returns an epimorphism\n fr\ om digraph onto the complete digraph with n vertices if one\n \ exists, else it returns fail. \n\n See also and \n \n\n Note that a \ digraph with at least two vertices has a 2-colouring if and only\n if it is\ bipartite, see .\n\n DigraphColouring(CompleteDigraph(5), 4);\nfail\ngap> DigraphColouring(C\ hainDigraph(10), 1);\nfail\ngap> D := ChainDigraph(10);;\ngap> t := DigraphCol\ ouring(D, 2);\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\ngap> IsDigr\ aphColouring(D, t); \ntrue\ngap> DigraphGreedyColouring(D);\nTransformation( [\ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1 ] )\n]]>\n \n\n\n \n \n \n \n\ A transformation, or fail.\n \n \ A proper colouring of a digraph is a labelling of its vertices in\n \ such a way that adjacent vertices have different labels. Note that a digraph \ \n with loops () does not have any proper \n\ colouring.\n \n \n If digraph is a digraph and orde\ r is a dense list consisting \n of all of the vertices of digraph (in any order), then \n DigraphGreedyColouring\n uses a greedy\ algorithm with the specified order to obtain some proper\n colouring of digraph, which may not use the minimal number of\n colours. \n \ \n If digraph is a digraph and func is a function whose argum\ ent \n is a digraph, and that returns a dense list order, then \n \ DigraphGreedyColouring(digraph, func) returns\n Di\ graphGreedyColouring(digraph, func(digraph)).\n \n\n If the optional second argument (either a list or a function), is no\ t \n specified, then is used by def\ ault. \n \n \n See also\n .\n \n\n DigraphGreedyColouring(ChainDigraph(10));\nTransformation\ ( [ 2, 1, 2, 1, 2, 1, 2, 1, 2, 1 ] )\ngap> DigraphGreedyColouring(ChainDigraph\ (10), [1 .. 10]);\nTransformation( [ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2 ] )\n]]>\n \n\n\n \n \n A list of the vertices.\ \n \n DigraphWelshPowellOrder returns a list\ of all of the vertices of \n the digraph digraph ordered according \ to the sum of the number of \n out- and in-neighbours, from highest to lowe\ st. \n \n\n DigraphWelshPowellOrder(Digraph\ ([[4], [9], [9], [], \n> [4, 6, 9], [1], [\ ], [], \n> [4, 5], [4, 5]]));\n[ 5, 9, 4, \ 1, 6, 10, 2, 3, 7, 8 ]\n]]>\n \n\n\n \ \n \n \ A non-negative integer.\n \n A proper colouring<\ /E> of a digraph is a labelling of its\n vertices in such a way that adjace\ nt vertices have different labels.\n Equivalently, a proper digraph colouri\ ng can be defined to be a from a dig\ raph onto a complete digraph. \n\n If digraph is a digraph witho\ ut loops (see , then ChromaticNumber<\ /C> returns the least\n non-negative integer n such that there is a \ proper colouring of\n digraph with n colours. In other words\ , for a digraph with at\n least one vertex, ChromaticNumber returns \ the least number n\n such that DigraphColouring(digraph, n\ ) does not return\n fail. See .\n\n ChromaticNumber(NullDigraph(10));\n1\ngap> ChromaticNumber(CompleteDi\ graph(10));\n10\ngap> ChromaticNumber(CompleteBipartiteDigraph(5, 5));\n2\ngap\ > ChromaticNumber(Digraph([[], [3], [5], [2, 3], [4]]));\n3\ngap> ChromaticNum\ ber(NullDigraph(0));\n0\ngap> D := PetersenGraph(IsMutableDigraph);\n\ngap> ChromaticNumber(D);\n3\n]]>\n \n\n\n \n \n A list of positive integers.\n \n If D is a digraph, then DigraphCore \ returns a list of vertices\n corresponding to the core of D\ . In particular, the subdigraph\n of D induced by this list is iso\ morphic to the core of D. \n\n The core of a digraph D<\ /C> is the minimal subdigraph C of\n D which is a homomorphic\ image of D. The core of a digraph \n is unique up to isomorphism.\n\ D := DigraphSymmetricClosure(CycleDigraph(8));\ \n\ngap> DigraphCore(D)\ ;\n[ 1, 2 ]\ngap> D := PetersenGraph();\n\ngap> DigraphCore(D);\n[ 1 .. 10 ]\ngap> D := Digraph(IsMutableDigra\ ph, [[3], [3], [4], [5], [2]]);\n\ng\ ap> DigraphCore(D);\n[ 2, 3, 4, 5 ]\n]]>\n \n \n\n \n\n\n \n \n Finding cliques and\ independent sets\n\n In &Digraphs;, a clique of a digraph\ is a set of mutually adjacent\n vertices of the digraph, and an indepen\ dent set is a set of mutually\n non-adjacent vertices of the digraph. A\ maximal clique is a clique\n which is not properly contained in ano\ ther clique, and a maximal\n independent set is an independent set\ which is not properly contained\n in another independent set. Using this \ definition in &Digraphs;, cliques\n and independent sets are both permitted\ , but not required, to contain vertices\n at which there is a loop. Anothe\ r name for a clique is a complete\n subgraph.\n \n\n &Di\ graphs; provides extensive functionality for computing cliques and\n indepe\ ndent sets of a digraph, whether maximal or not. The fundamental\n algorit\ hm used in most of the methods in &Digraphs; to calculate cliques\n and ind\ ependent sets is a version of the Bron-Kerbosch algorithm.\n Calculating th\ e cliques and independent sets of a digraph is a well-known\n and hard prob\ lem, and searching for cliques or independent sets in a\n digraph can be ve\ ry lengthy, even for a digraph with a small number of\n vertices. &Digraph\ s; uses several strategies to increase the performance\n of these calculati\ ons.\n \n\n From the definition of cliques and independent sets, it \ follows that the\n presence of loops and multiple edges in a digraph is irr\ elevant to the\n existence of cliques and independent sets in the digraph. \ See and \ for more\n information about these properties. Therefore given a digraph\ \n digraph, the cliques and independent sets of digraph are\n\ equal to the cliques and independent sets of the digraph:\n\n \n \ \n DigraphRemoveLoops(DigraphRemoveAllMultipleEdges(<\ A>digraph)).\n \n \n\n See and f\ or more information about these\n attributes. Furthermore, the cliques of \ this digraph are equal to the\n cliques of the digraph formed by removing a\ ny edge [u,v] for which\n the corresponding reverse edge [v,u] is not present. Therefore,\n overall, the cliques of digraph are \ equal to the cliques of the\n symmetric digraph:\n\n \n \n MaximalSymmetricSubdigraphWithoutLoops(digraph).\n \n \n\n See for more\n information about this. The of this symmetric digraph con\ tains the\n automorphism group of digraph as a subgroup. By perform\ ing the\n search for maximal cliques with the help of this larger automorph\ ism group\n to reduce the search space, the computation time may be reduced\ . The\n functions and attributes which return representatives of cliques o\ f\n digraph will return orbit representatives of cliques under the\n\ action of the automorphism group of the maximal symmetric subdigraph\n \ without loops on sets of vertices. \n\n The independent sets of\ a digraph are equal to the independent sets of the\n . Therefore, overall, the independent\n sets of digra\ ph are equal to the independent sets of the symmetric\n digraph:\n\n \ \n \n DigraphSymmetricClosure(DigraphRemoveLoops(D\ igraphRemoveAllMultipleEdges(\n digraph))).\n \n \n\n Again, the automorphism group of this symmetric digr\ aph contains the\n automorphism group of digraph. Since a search for\ independent sets\n is equal to a search for cliques in the , the\n methods used in &Digraphs; usually transform a search f\ or independent sets\n into a search for cliques, as described above. The f\ unctions and\n attributes which return representatives of independent sets \ of\n digraph will return orbit representatives of independent sets u\ nder\n the action of the automorphism group of the symmetric closure\ of the\n digraph formed by removing loops and multiple edges. \n\n P\ lease note that in &Digraphs;, cliques and independent sets are not\n requi\ red to be maximal. Some authors use the word clique to mean\n maximal clique, and some authors use the phrase independent set to\n mean maxi\ mal independent set, but please note that &Digraphs; does\n not use thi\ s definition.\n \n\n Finding cliques\n \ \n\n\ntrue or false.\n\n If digraph is a digraph and l is a duplic\ ate-free list of\n vertices of digraph, then\n IsClique(dig\ raph,l) returns true\n if l is a \ clique of digraph and false if it is\n not. Similarly,\n\ IsMaximalClique(digraph,l) returns\n \ true if l is a maximal clique of digraph and\n \ false if it is not. \n\n A clique of a digraph is a set\ of mutually adjacent vertices of the\n digraph. A maximal clique is\ a clique that is not properly\n contained in another clique. A clique is p\ ermitted, but not required, to\n contain vertices at which there is a loop.\ \n D := CompleteDigraph(4);;\ngap> IsClique(D, [1,\ 3, 2]);\ntrue\ngap> IsMaximalClique(D, [1, 3, 2]);\nfalse\ngap> IsMaximalCliq\ ue(D, DigraphVertices(D));\ntrue\ngap> D := Digraph([[1, 2, 4, 4], [1, 3, 4], \ [2, 1], [1, 2]]);\n\ngap> Is\ Clique(D, [2, 3, 4]);\nfalse\ngap> IsMaximalClique(D, [1, 2, 4]);\ntrue\ngap> \ D := CompleteDigraph(IsMutableDigraph, 4);;\ngap> IsClique(D, [1, 3, 2]);\ntru\ e]]>\n \n\n\n \n \n The argument user_param.\n \n This function finds cliques of the digraph digraph\ subject to the\n conditions imposed by the other arguments as describe\ d below. Note\n that a clique is represented by the immutable list of the v\ ertices that\n it contains.\n \n\n Let G denote the automo\ rphism group of the maximal symmetric\n subdigraph of digraph withou\ t loops (see \ and ).\n\n \n hook\n \n This argument shoul\ d be a function or fail. \n\n If hook is a function, t\ hen it should have two arguments\n user_param (see below) and a \ clique c. The function\n hook(user_param, c) is called every time a new\n clique c is found by CliquesFi\ nder. \n\n If hook is fail, then a default functio\ n is used that\n simply adds every new clique found by CliquesFinder\ to\n user_param, which must be a list in this case.\n \ \n\n user_param\n \n If h\ ook is a function, then user_param can be any &GAP;\n object\ . The object user_param is used as the first argument for\n the \ function hook. For example, user_param might be a\n list,\ and hook(user_param, c) might add the size\n of t\ he clique c to the list user_param. \n\n If the value\ of hook is fail, then the value of\n user_param m\ ust be a list.\n \n\n limit\n \ \n This argument should be a positive integer or infinity.\n \ CliquesFinder will return after it has found\n limit \ cliques or the search is complete.\n \n\n include and exclude\n \n These arguments should each\ be a (possibly empty) duplicate-free list\n of vertices of digraph<\ /A> (i.e. positive integers less than the\n number of vertices of di\ graph). \n\n CliquesFinder will only look for cliques co\ ntaining all of the\n vertices in include and containing none of\ the vertices in\n exclude. \n\n Note that the search\ may be much more efficient if each of these lists\n is invariant under\ the action of G on sets of vertices.\n \n\n \ max\n \n This argument should be true or <\ K>false. If max is\n true then CliquesFinder will on\ ly search for maximal\n cliques. If max is false t\ hen non-maximal cliques may be\n found.\n \n\n <\ A>size\n \n This argument should be fail o\ r a positive integer.\n If size is a positive integer then Cl\ iquesFinder will\n only search for cliques that contain precisely <\ A>size vertices.\n If size is fail then cliques of an\ y size may be found.\n \n\n reps\n <\ Item>\n This argument should be true or false. \n\n \ If reps is true then the arguments include and\n \ exclude are each required to be invariant under the action of\n \ G on sets of vertices. In this case, CliquesFinder will\n \ find representatives of the orbits of the desired cliques under the\n \ action of G, although representatives may be returned that\n \ are in the same orbit.\n If reps is false then Cli\ quesFinder will not take this into\n consideration. \n\n \ For a digraph such that G is non-trivial, the search for\n cliqu\ e representatives can be much more efficient than the search for\n all \ cliques.\n \n \n\n This function uses a version of th\ e Bron-Kerbosch algorithm.\n D := CompleteDigraph(\ 5);\n\ngap> user_param := [];;\nga\ p> f := function(a, b) # Calculate size of clique\n> AddSet(user_param, Siz\ e(b));\n> end;;\ngap> CliquesFinder(D, f, user_param, infinity, [], [], false,\ fail,\n> true);\n[ 1, 2, 3, 4, 5 ]\ngap> CliquesFinder(D, fa\ il, [], 5, [2, 4], [3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ], [ 2, 4,\ 5 ], [ 1, 2, 4, 5 ] ]\ngap> CliquesFinder(D, fail, [], 2, [2, 4], [3], false,\ fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ] ]\ngap> CliquesFinder(D, fail, [], inf\ inity, [], [], true, 5, false);\n[ [ 1, 2, 3, 4, 5 ] ]\ngap> CliquesFinder(D, \ fail, [], infinity, [1, 3], [], false, 3, false);\n[ [ 1, 2, 3 ], [ 1, 3, 4 ],\ [ 1, 3, 5 ] ]\ngap> CliquesFinder(D, fail, [], infinity, [1, 3], [], true, 3,\ false);\n[ ]\ngap> D := CompleteDigraph(IsMutableDigraph, 5);\n\ngap> user_param := [];;\ngap> f := function(a,\ b) # Calculate size of clique\n> AddSet(user_param, Size(b));\n> end;;\nga\ p> CliquesFinder(D, f, user_param, infinity, [], [], false, fail,\n> \ true);\n[ 1, 2, 3, 4, 5 ]\ngap> CliquesFinder(D, fail, [], 5, [2, 4], \ [3], false, fail, false);\n[ [ 2, 4 ], [ 1, 2, 4 ], [ 2, 4, 5 ], [ 1, 2, 4, 5 \ ] ]\ngap> CliquesFinder(D, fail, [], 2, [2, 4], [3], false, fail, false);\n[ [\ 2, 4 ], [ 1, 2, 4 ] ]\ngap> CliquesFinder(D, fail, [], infinity, [], [], true\ , 5, false);\n[ [ 1, 2, 3, 4, 5 ] ]\ngap> CliquesFinder(D, fail, [], infinity,\ [1, 3], [], false, 3, false);\n[ [ 1, 2, 3 ], [ 1, 3, 4 ], [ 1, 3, 5 ] ]\ngap\ > CliquesFinder(D, fail, [], infinity, [1, 3], [], true, 3, false);\n[ ]]]>\n \n\n\n \n \n \ \n An immutable list of positive integers, or fail.\ \n \n If digraph is a digraph, then these functions re\ turns a clique\n of digraph if one exists that satisfies the argumen\ ts, else it\n returns fail. A clique is defined by the set of verti\ ces that it\n contains; see and . \n\n The optional arguments include and exclude\ must each be a\n (possibly empty) duplicate-free list of vertices of <\ A>digraph, and\n the optional argument size must be a positive i\ nteger. By default,\n include and exclude are empty. These f\ unctions will search for\n a clique of digraph that includes the ver\ tices of include\n but does not include any vertices in exclude; if the argument\n size is supplied, then additionally the clique\ will be required to\n contain precisely size vertices. \n\n I\ f include is not a clique, then these functions return fail.\n \ Otherwise, the functions behave in the following way, depending on the\n \ number of arguments:\n\n \n One or two arguments\n \ \n If one or two arguments are supplied, then DigraphCliq\ ue and\n DigraphMaximalClique greedily enlarge the clique include\n until it can not continue. The result is guaranteed to \ be a maximal\n clique. This may or may not return an answer more quickl\ y than using\n . with a limit of \ 1.\n \n Three arguments\n \n I\ f three arguments are supplied, then DigraphClique greedily\n en\ larges the clique include until it can not continue, although\n \ this clique may not be maximal. \n\n Given three arguments, Digra\ phMaximalClique returns the maximal\n clique returned by Digraph\ MaximalCliques(digraph,\n include, e\ xclude, 1) if one exists,\n else fail.\n \ \n Four arguments\n \n If four arguments a\ re supplied, then DigraphClique returns the\n clique returned by\ DigraphCliques(digraph,\n include, exclude, 1,\n size) if one exists, else <\ K>fail. This clique may\n not be maximal. Given four arguments,\ DigraphMaximalClique\n returns the maximal clique returned by\n\ DigraphMaximalCliques(digraph, include,\ \n exclude, 1, size) if one exists, els\ e\n fail.\n \n \n\n D := Digraph([[2, 3, 4], [1, 3], [1, 2], [1, 5], []]);\n\ngap> IsSymmetricDigraph(D);\nfalse\ngap> Digrap\ hClique(D);\n[ 5 ]\ngap> DigraphMaximalClique(D);\n[ 5 ]\ngap> DigraphClique(D\ , [1, 2]);\n[ 1, 2, 3 ]\ngap> DigraphMaximalClique(D, [1, 3]);\n[ 1, 3, 2 ]\ng\ ap> DigraphClique(D, [1], [2]);\n[ 1, 4 ]\ngap> DigraphMaximalClique(D, [1], [\ 3, 4]);\nfail\ngap> DigraphClique(D, [1, 5]);\nfail\ngap> DigraphClique(D, [],\ [], 2);\n[ 1, 2 ]\ngap> D := Digraph(IsMutableDigraph,\n> [[2\ , 3, 4], [1, 3], [1, 2], [1, 5], []]);\n\ngap> IsSymmetricDigraph(D);\nfalse\ngap> DigraphClique(D);\n[ 5 ]]]>\n \n\n\n \n \ \n \n \n \n \n \n An immutable list of lists \ of positive integers.\n \n If digraph is dig\ raph, then these functions and attributes use to return cliques of digraph. A\n clique is defined by the set \ of vertices that it contains; see and . \n\n The optional arguments include and \ exclude must each be a\n (possibly empty) list of vertices of dig\ raph, the optional argument\n limit must be either a positive in\ teger or infinity, and the\n optional argument size must be a\ positive integer. If not\n specified, then include and exclude<\ /A> are chosen to be empty\n lists, and limit is set to infinity<\ /C>. \n\n The functions will return as many suitable cliques as possibl\ e, up to the\n number limit. These functions will find cliques that\ contain all\n of the vertices of include but do not contain any of \ the\n vertices of exclude. The argument size restricts the s\ earch\n to those cliques that contain precisely size vertices.\n \ If the function or attribute has Maximal in its name, then only\n ma\ ximal cliques will be returned; otherwise non-maximal cliques may be\n retu\ rned. \n\n Let G denote the automorphism group of maximal symmet\ ric subdigraph\n of digraph without loops (see and ).\n\n \n Distinct cliques<\ /Mark>\n \n DigraphMaximalCliques and DigraphCliqu\ es each return a\n duplicate-free list of at most limit cliq\ ues of digraph\n that satisfy the arguments. \n\n The \ computation may be significantly faster if include and\n excl\ ude are invariant under the action of G\n on sets of vertice\ s.\n \n\n Orbit representatives of cliques\n \ \n To use DigraphMaximalCliquesReps or DigraphCliques\ Reps,\n the arguments include and exclude must each b\ e invariant\n under the action of G on sets of vertices. \n\n\ If this is the case, then DigraphMaximalCliquesReps and\n \ DigraphCliquesReps each return a duplicate-free list of at most\n \ limit orbits representatives (under the action of G on\n \ sets vertices) of cliques of digraph that satisfy the\n argu\ ments. \n\n The representatives are not guaranteed to be in distinct\ orbits.\n However, if fewer than lim results are returned, then\ there\n will be at least one representative from each orbit of maximal\ cliques.\n \n \n\n D := Dig\ raph([\n> [2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]]);\n\ngap> IsSymmetricDigraph(D);\ntrue\ngap> G \ := AutomorphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4) ])\ngap> Digra\ phMaximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]\ngap> DigraphMaxi\ malCliquesReps(D);\n[ [ 1, 2, 3 ], [ 3, 4 ] ]\ngap> Orbit(G, [1, 2, 3], OnSets\ );\n[ [ 1, 2, 3 ], [ 4, 5, 6 ] ]\ngap> Orbit(G, [3, 4], OnSets);\n[ [ 3, 4 ] ]\ \ngap> DigraphMaximalCliquesReps(D, [3, 4], [], 1);\n[ [ 3, 4 ] ]\ngap> Digrap\ hMaximalCliques(D, [1, 2], [5, 6], 1, 2);\n[ ]\ngap> DigraphCliques(D, [1], [\ 5, 6], infinity, 2);\n[ [ 1, 2 ], [ 1, 3 ] ]\ngap> D := Digraph(IsMutableDigra\ ph, [\n> [2, 3], [1, 3], [1, 2, 4], [3, 5, 6], [4, 6], [4, 5]]);\n\ngap> IsSymmetricDigraph(D);\ntrue\ngap> G := \ AutomorphismGroup(D);\nGroup([ (5,6), (1,2), (1,5)(2,6)(3,4) ])\ngap> DigraphM\ aximalCliques(D);\n[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 3, 4 ] ]]]>\n \n\n\n \n \n A non-negative integer.\n \n If digraph is a digraph, then CliqueNumber(digraph)\n returns the largest integer n such that digraph contains \ a\n clique with n vertices as an induced subdigraph.\n \n\n \ A clique of a digraph is a set of mutually adjacent vertices of the\n\ digraph. Loops and multiple edges are ignored for the purpose of\n dete\ rmining the clique number of a digraph. \n D := Co\ mpleteDigraph(4);;\ngap> CliqueNumber(D);\n4\ngap> D := Digraph([[1, 2, 4, 4],\ [1, 3, 4], [2, 1], [1, 2]]);\n\ngap> CliqueNumber(D);\n3\ngap> D := CompleteDigraph(IsMutableDigraph, 4);\ ;\ngap> CliqueNumber(D);\n4]]>\n\n\n\n <\ /Section>\n\n Finding independent sets\n \n \n \n true or false.\n \n If digraph is a digraph and\ l is a duplicate-free list of\n vertices of digraph, then\n \ IsIndependentSet(digraph,l) returns\n \ true if l is an independent set of digraph and\n \ false if it is not. Similarly,\n IsMaximalIndependentSet(\ digraph,l)\n returns true if l is\ a maximal independent set of\n digraph and false if i\ t is not. \n\n An independent set of a digraph is a set of mutu\ ally non-adjacent\n vertices of the digraph. A maximal independent set is an independent\n set that is not properly contained in another indepe\ ndent set. An\n independent set is permitted, but not required, to contain \ vertices at\n which there is a loop.\n D := Cyc\ leDigraph(4);;\ngap> IsIndependentSet(D, [1]);\ntrue\ngap> IsMaximalIndependen\ tSet(D, [1]);\nfalse\ngap> IsIndependentSet(D, [1, 4, 3]);\nfalse\ngap> IsInde\ pendentSet(D, [2, 4]);\ntrue\ngap> IsMaximalIndependentSet(D, [2, 4]);\ntrue\n\ gap> D := CycleDigraph(IsMutableDigraph, 4);;\ngap> IsIndependentSet(D, [1]);\ \ntrue]]>\n \n\n\n \n \n \n An immutable list of positive\ integers, or fail.\n \n If digraph i\ s a digraph, then these functions returns an independent\n set of digrap\ h if one exists that satisfies the arguments, else it\n returns fail\ . An independent set is defined by the set of vertices\n that it conta\ ins; see and . \n\n The optional arguments include and exclud\ e must each be a\n (possibly empty) duplicate-free list of vertices of \ digraph, and the\n optional argument size must be a positive \ integer. By default,\n include and exclude are empty. These \ functions will search\n for an independent set of digraph that inclu\ des the vertices of\n include but does not include any vertices in <\ A>exclude;\n if the argument size is supplied, then additionally\ the independent\n set will be required to contain precisely size ve\ rtices. \n\n If include is not an independent set, then these fun\ ctions return\n fail. Otherwise, the functions behave in the follow\ ing way,\n depending on the number of arguments:\n\n \n One or two arguments\n \n If one or two arguments ar\ e supplied, then DigraphIndependentSet\n and DigraphMaximalIn\ dependentSet greedily enlarge the\n independent set include \ until it can not continue. The result\n is guaranteed to be a maximal \ independent set. This may or may not\n return an answer more quickly th\ an using . with a lim\ it of 1.\n \n Three arguments\n \n \ If three arguments are supplied, then DigraphIndependentSet\n \ greedily enlarges the independent set include until it can not\n \ continue, although this independent set may not be maximal. \n\n \ Given three arguments, DigraphMaximalIndependentSet returns the\n \ maximal independent set returned by\n DigraphMaximalIndependentSe\ ts(digraph,\n include, exclude, 1) if one exists,\n else fail.\n \n Four arguments\n \n If four arguments are supplied,\ then DigraphIndependentSet\n returns the independent set return\ ed by\n DigraphIndependentSets(digraph, include\ ,\n exclude, 1, size) if one exi\ sts, else\n fail. This independent set may not be maximal. G\ iven four\n arguments, DigraphMaximalIndependentSet returns the \ maximal\n independent set returned by\n DigraphMaximalIndepen\ dentSets(digraph,\n include, exclu\ de, 1,\n size) if one exists, else fail\ .\n \n \n D := ChainDigraph(\ 6);\n\ngap> DigraphIndependentSet(D);\ \n[ 6, 4, 2 ]\ngap> DigraphMaximalIndependentSet(D);\n[ 6, 4, 2 ]\ngap> Digrap\ hIndependentSet(D, [2, 4]);\n[ 2, 4, 6 ]\ngap> DigraphMaximalIndependentSet(D,\ [1, 3]);\n[ 1, 3, 6 ]\ngap> DigraphIndependentSet(D, [2, 4], [6]);\n[ 2, 4 ]\ \ngap> DigraphMaximalIndependentSet(D, [2, 4], [6]);\nfail\ngap> DigraphIndepe\ ndentSet(D, [1], [], 2);\n[ 1, 3 ]\ngap> DigraphMaximalIndependentSet(D, [1], \ [], 2);\nfail\ngap> DigraphMaximalIndependentSet(D, [1], [], 3);\n[ 1, 3, 5 ]\ \ngap> D := ChainDigraph(IsMutableDigraph, 6);\n\ngap> DigraphIndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphMaximal\ IndependentSet(D);\n[ 6, 4, 2 ]\ngap> DigraphIndependentSet(D, [2, 4]);\n[ 2, \ 4, 6 ]\ngap> DigraphMaximalIndependentSet(D, [1, 3]);\n[ 1, 3, 6 ]\ngap> Digra\ phIndependentSet(D, [2, 4], [6]);\n[ 2, 4 ]\ngap> DigraphMaximalIndependentSet\ (D, [2, 4], [6]);\nfail\ngap> DigraphIndependentSet(D, [1], [], 2);\n[ 1, 3 ]\ \ngap> DigraphMaximalIndependentSet(D, [1], [], 2);\nfail\ngap> DigraphMaximal\ IndependentSet(D, [1], [], 3);\n[ 1, 3, 5 ]]]>\n \n\n\n \n \n \n \n \n\ \n \n A\ n immutable list of lists of positive integers.\n \n \ If digraph is digraph, then these functions and attributes use to return independent sets of digraph. A\ n\n independent set is defined by the set of vertices that it contains; see\ \n and . \n\n The optional arguments include and exclude must each be a\n (possibly empty) list of vertices of digraph, th\ e optional argument\n limit must be either a positive integer or \ infinity, and the\n optional argument size must be a positive in\ teger. If not\n specified, then include and exclude are chos\ en to be empty\n lists, and limit is set to infinity. \n\ \n The functions will return as many suitable independent sets as possible,\ up\n to the number limit. These functions will find independent se\ ts\n that contain all of the vertices of include but do not\n con\ tain any of the vertices of exclude The argument size\n restr\ icts the search to those cliques that contain precisely size\n verti\ ces. If the function or attribute has Maximal in its name,\n then o\ nly maximal independent sets will be returned; otherwise non-maximal\n inde\ pendent sets may be returned.\n \n\n Let G denote the \n of the of the digraph formed from\n digraph by rem\ oving loops and ignoring the multiplicity of edges.\n\n \n Distinct independent sets\n \n DigraphMaximalInde\ pendentSets and DigraphIndependentSets\n each return a dupli\ cate-free list of at most limit independent\n sets of digraph\ that satisfy the arguments. \n\n The computation may be signifi\ cantly faster if include and\n exclude are invariant unde\ r the action of G on sets of\n vertices.\n \n\n \ Representatives of distinct orbits of independent sets\n \n To use DigraphMaximalIndependentSetsReps or\n D\ igraphIndependentSetsReps, the arguments include and\n ex\ clude must each be invariant under the action of G on\n sets\ of vertices. \n\n If this is the case, then DigraphMaximalIndepe\ ndentSetsReps and\n DigraphIndependentSetsReps each return a\ list of\n at most limit orbits representatives (under the actio\ n of\n G on sets of vertices) of independent sets of digraph<\ /A>\n that satisfy the arguments. \n\n The representatives a\ re not guaranteed to be in distinct orbits.\n However, if lim is\ not specified, or fewer than lim\n results are returned, then t\ here will be at least one representative\n from each orbit of maximal i\ ndependent sets.\n \n \n\n D\ := CycleDigraph(5);\n\ngap> DigraphM\ aximalIndependentSetsReps(D);\n[ [ 1, 3 ] ]\ngap> DigraphIndependentSetsReps(D\ );\n[ [ 1 ], [ 1, 3 ] ]\ngap> Set(DigraphMaximalIndependentSets(D));\n[ [ 1, 3\ ], [ 1, 4 ], [ 2, 4 ], [ 2, 5 ], [ 3, 5 ] ]\ngap> DigraphMaximalIndependentSe\ ts(D, [1]);\n[ [ 1, 3 ], [ 1, 4 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5]\ );\n[ [ 1 ], [ 2 ], [ 3 ], [ 1, 3 ] ]\ngap> DigraphIndependentSets(D, [], [4, \ 5], 1, 2);\n[ [ 1, 3 ] ]\ngap> D := CycleDigraph(IsMutableDigraph, 5);\n\ngap> DigraphMaximalIndependentSetsReps(D\ );\n[ [ 1, 3 ] ]\ngap> DigraphIndependentSetsReps(D);\n[ [ 1 ], [ 1, 3 ] ]\nga\ p> Set(DigraphMaximalIndependentSets(D));\n[ [ 1, 3 ], [ 1, 4 ], [ 2, 4 ], [ 2\ , 5 ], [ 3, 5 ] ]\ngap> DigraphMaximalIndependentSets(D, [1]);\n[ [ 1, 3 ], [ \ 1, 4 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5]);\n[ [ 1 ], [ 2 ], [ 3 ], \ [ 1, 3 ] ]\ngap> DigraphIndependentSets(D, [], [4, 5], 1, 2);\n[ [ 1, 3 ] ]]]>\ \n \n\n\n \n\n \n Visualising and\ IO\n\n Visualising a digraph\n \n \n Noth\ ing.\n \n This function attempts to convert the\ string str into a pdf\n document and open this document, i.e. to \ splash it all over your monitor.\n\n The string str must corre\ spond to a valid dot or\n LaTeX text file and you must have\ have GraphViz and\n pdflatex installed on your computer. \ For details about these file\n formats, see http://www.latex-project\ .org and\n http://www.graphviz.org. \n\n This fu\ nction is provided to allow convenient, immediate viewing of the\n pictur\ es produced by the function . \n\n The optio\ nal second argument opts should be a record with\n components corr\ esponding to various options, given below.\n\n \n path\ \n \n this should be a string representing the pa\ th to the directory where\n you want Splash to do its work. Th\ e default value of this\n option is \"~/\".\n \n\ \n directory\n \n this should be a s\ tring representing the name of the directory in\n path where y\ ou want Splash to do its work. This function\n will create thi\ s directory if does not already exist. \n\n The default value of \ this option is \"tmp.viz\" if the option\n path is pres\ ent, and the result of\n is used otherwise.\n \n\n filename\n\ \n this should be a string representing the name of the\ file where\n str will be written. The default value of this \ option is\n \"vizpicture\".\n \n\n \ viewer\n \n this should be a string representing \ the name of the program which\n should open the files produced by \ GraphViz or pdflatex.\n \n\n type\ \n \n this option can be used to specify that the string\ str contains\n a &LaTeX; or dot document. You can spec\ ify this option in\n str directly by making the first line \ \"%latex\" or\n \"//dot\". There is no default value for \ this option, this\n option must be specified in str or in o\ pt.type.\n \n\n engine\n \n\ this option can be used to specify the GraphViz engine to use\n \ to render a dot document. The valid choices are \"dot\",\n \ \"neato\", \"circo\", \"twopi\", \"fdp\",\ \n \"sfdp\", and \"patchwork\". Please refer to the\n \ GraphViz documentation for details on these engines.\n The de\ fault value for this option is \"dot\", and it\n must be speci\ fied in opt.engine.\n \n\n filetype\n\ \n this should be a string representing the type of fil\ e which\n Splash should produce. For &LaTeX; files, this optio\ n is\n ignored and the default value \"pdf\" is used.\n \ \n \n\n This function was written by Attila Egri-Nagy\ and Manuel Delgado with some\n minor changes by J. D. Mitchell. \n \ Splash(DotDigraph(RandomDigraph(4)));\n]]>\n \ \n \n\n \n \n \n \ \n \n \n A strin\ g.\n \n DotDigraph produces a graphical \ representation of the digraph\n digraph. Vertices are displayed as\ circles, numbered consistently\n with digraph. Edges are displaye\ d as arrowed lines between\n vertices, with the arrowhead of each line po\ inting towards the range\n of the edge. \n\n DotColoredDigraph\ differs from DotDigraph only in\n that the values in given in\ the two lists are used to color the vertices and\n edges of the graph wh\ en displayed. The list for vertex colours should be\n a list of length eq\ ual to the number of vertices, containing strings that\n are accepted by \ the graphviz software, which is the one used for graph\n representation. \ The list for edge colours should be a list of lists\n with the same shape\ of the outneighbours of the digraph that contains strings \n that corres\ pond to colours accepted by the graphviz software. If the lists \n are no\ t the appropriate size, or have holes then the function will return\n an \ error. \n\n DotVertexColoredDigraph differs from DotDigraph<\ /C> only in\n that the values in given in the list are used to color the \ vertices\n of the graph when displayed. The list for vertex colours shoul\ d be\n a list of length equal to the number of vertices, containing strin\ gs that\n are accepted by the graphviz software, which is the one used fo\ r graph\n representation. If the list is not the appropriate size, or has\ holes \n then the function will return an error. \n\n DotEdge\ ColoredDigraph differs from DotDigraph only in\n that the valu\ es in given in the list are used to color the vertices and\n edges of the\ graph when displayed. The list for edge colours should be a list\n of li\ sts with the same shape of the outneighbours of the digraph that contains\n \ strings that correspond to colours accepted by the graphviz software. If th\ e \n list is not the appropriate size, or has holes then the function wil\ l return\n an error. \n\n DotVertexLabelledDigraph differs\ from DotDigraph only in\n that the values in are used to label\n the vertices in the produced picture \ rather than the numbers 1 to\n the number of vertices of the digra\ ph. \n\n The output is in dot format (also known as GraphVi\ z)\n format. For details about this file format, and information abou\ t how to\n display or edit this format see http://www.graphviz.org. \n\n The string returned by DotDigraph or\n DotV\ ertexLabelledDigraph can be written to a file using\n the command . \n a\ dj := List([1 .. 4], x -> [1, 1, 1, 1]);\n[ [ 1, 1, 1, 1 ], [ 1, 1, 1, 1 ], [ \ 1, 1, 1, 1 ], [ 1, 1, 1, 1 ] ]\ngap> adj[1][3] := 0;\n0\ngap> gr := DigraphByA\ djacencyMatrix(adj);\n\ngap> D :=\ CompleteDigraph(4);\n\ngap> vertc\ olors := [];;\ngap> vertcolors[1] := \"blue\";; vertcolors[2] := \"red\";; \ng\ ap> vertcolors[3] := \"green\";; vertcolors[4] := \"yellow\";;\ngap> edgecolor\ s := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];;\ngap> edgecolors[3\ ] := [];; edgecolors[4] := [];; \ngap> edgecolors[1][2] := \"lightblue\";;\nga\ p> edgecolors[1][3] := \"pink\";;\ngap> edgecolors[1][4] := \"purple\";;\ngap>\ edgecolors[2][1] := \"lightblue\";;\ngap> edgecolors[2][3] := \"pink\";; \nga\ p> edgecolors[2][4] := \"purple\";; \ngap> edgecolors[3][1] := \"lightblue\";;\ \ngap> edgecolors[3][2] := \"pink\";; \ngap> edgecolors[3][4] := \"purple\";;\ \ngap> edgecolors[4][1] := \"lightblue\";; \ngap> edgecolors[4][2] := \"pink\"\ ;;\ngap> edgecolors[4][3] := \"purple\";;\ngap> Print(DotColoredDigraph(D, ver\ tcolors, edgecolors));\n//dot\ndigraph hgn{\nnode [shape=circle]\n1[color=blue\ , style=filled]\n2[color=red, style=filled]\n3[color=green, style=filled]\n4[c\ olor=yellow, style=filled]\n1 -> 2[color=lightblue]\n1 -> 3[color=pink]\n1 -> \ 4[color=purple]\n2 -> 1[color=lightblue]\n2 -> 3[color=pink]\n2 -> 4[color=pur\ ple]\n3 -> 1[color=lightblue]\n3 -> 2[color=pink]\n3 -> 4[color=purple]\n4 -> \ 1[color=lightblue]\n4 -> 2[color=pink]\n4 -> 3[color=purple]\n}\ngap> D := Emp\ tyDigraph(3);\n\ngap> vertcolors := [\ ];;\ngap> vertcolors[1] := \"blue\";; vertcolors[2] := \"red\";;\ngap> vertcol\ ors[3] := \"green\";;\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edg\ ecolors[2] := [];; \ngap> edgecolors[3] := [];;\ngap> Print(DotColoredDigraph(\ D, vertcolors, edgecolors));\n//dot\ndigraph hgn{\nnode [shape=circle]\n1[colo\ r=blue, style=filled]\n2[color=red, style=filled]\n3[color=green, style=filled\ ]\n}\ngap> D := Digraph([[2], [1, 3], [2]]);\n\ngap> vertcolors := [];;\ngap> vertcolors[1] := \"blue\";;\ngap> \ vertcolors[2] := \"pink\";;\ngap> vertcolors[3] := \"purple\";;\ngap> edgecolo\ rs := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];;\ngap> edgecolors[\ 3] := [];;\ngap> edgecolors[1][2] := \"green\";; edgecolors[2][1] := \"green\"\ ;;\ngap> edgecolors[2][3] := \"red\";; edgecolors[3][2] := \"red\";;\ngap> Pri\ nt(DotSymmetricColoredDigraph(D, vertcolors, edgecolors));\n//dot\ngraph hgn{\ \nnode [shape=circle]\n\n1[color=blue, style=filled]\n2[color=pink, style=fill\ ed]\n3[color=purple, style=filled]\n1 -- 2[color=green]\n2 -- 3[color=red]\n}\ \ngap> D := Digraph([[2, 3], [1, 3], [1]]);\n\ngap> vertcolors := [];;\ngap> vertcolors[1] := \"blue\";; vertcol\ ors[2] := \"red\";;\ngap> vertcolors[3] := \"green\";;\ngap> edgecolors := [];\ ;\ngap> edgecolors[1] := [];; edgecolors[2] := [];;\ngap> edgecolors[3] := [];\ ;\ngap> edgecolors[1][2] := \"orange\";; edgecolors[1][3] := \"yellow\";;\ngap\ > edgecolors[2][1] := \"orange\";; edgecolors[2][3] := \"pink\";;\ngap> edgeco\ lors[3][1] := \"yellow\";;\ngap> Print(DotColoredDigraph(D, vertcolors, edgeco\ lors));;\n//dot\ndigraph hgn{\nnode [shape=circle]\n1[color=blue, style=filled\ ]\n2[color=red, style=filled]\n3[color=green, style=filled]\n1 -> 2[color=oran\ ge]\n1 -> 3[color=yellow]\n2 -> 1[color=orange]\n2 -> 3[color=pink]\n3 -> 1[co\ lor=yellow]\n}\ngap> D := Digraph(IsMutableDigraph, [[2, 3], [1, 3], [1]]);\n<\ mutable digraph with 3 vertices, 5 edges>\ngap> vertcolors := [];;\ngap> vertc\ olors[1] := \"blue\";; vertcolors[2] := \"red\";;\ngap> vertcolors[3] := \"gre\ en\";;\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [\ ];;\ngap> edgecolors[3] := [];;\ngap> edgecolors[1][2] := \"orange\";; edgecol\ ors[1][3] := \"yellow\";;\ngap> edgecolors[2][1] := \"orange\";; edgecolors[2]\ [3] := \"pink\";;\ngap> edgecolors[3][1] := \"yellow\";;\ngap> Print(DotColore\ dDigraph(D, vertcolors, edgecolors));;\n//dot\ndigraph hgn{\nnode [shape=circl\ e]\n1[color=blue, style=filled]\n2[color=red, style=filled]\n3[color=green, st\ yle=filled]\n1 -> 2[color=orange]\n1 -> 3[color=yellow]\n2 -> 1[color=orange]\ \n2 -> 3[color=pink]\n3 -> 1[color=yellow]\n}\ngap> D;\n\ngap> DotSymmetricDigraph(gr2){[12 .. 70]};\n\" hgn{\\nno\ de [shape=circle]\\n\\n1\\n2\\n3\\n4\\n1 -- 2\\n2 -- 3\\n3 -- 3\\n3 -\"\ngap> \ DotSymmetricDigraph(gr1);\nError, the argument must be a symmetric digraph\ ,\ngap> D := CompleteDigraph(4);\n\ \ngap> vertcolors := [];;\ngap> vertcolors[1] := \"blue\";; vertcolors[2] := \ \"red\";; \ngap> vertcolors[3] := \"green\";; vertcolors[4] := \"yellow\";;\ng\ ap> Print(DotVertexColoredDigraph(D, vertcolors));\n//dot\ndigraph hgn{\nnode \ [shape=circle]\n1[color=blue, style=filled]\n2[color=red, style=filled]\n3[col\ or=green, style=filled]\n4[color=yellow, style=filled]\n1 -> 2\n1 -> 3\n1 -> 4\ \n2 -> 1\n2 -> 3\n2 -> 4\n3 -> 1\n3 -> 2\n3 -> 4\n4 -> 1\n4 -> 2\n4 -> 3\n}\ng\ ap> D := CompleteDigraph(4);\n\nga\ p> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];;\ngap> \ edgecolors[3] := [];; edgecolors[4] := [];; \ngap> edgecolors[1][2] := \"light\ blue\";;\ngap> edgecolors[1][3] := \"pink\";;\ngap> edgecolors[1][4] := \"purp\ le\";;\ngap> edgecolors[2][1] := \"lightblue\";;\ngap> edgecolors[2][3] := \"p\ ink\";; \ngap> edgecolors[2][4] := \"purple\";; \ngap> edgecolors[3][1] := \"l\ ightblue\";; \ngap> edgecolors[3][2] := \"pink\";; \ngap> edgecolors[3][4] := \ \"purple\";;\ngap> edgecolors[4][1] := \"lightblue\";; \ngap> edgecolors[4][2]\ := \"pink\";;\ngap> edgecolors[4][3] := \"purple\";;\ngap> Print(DotEdgeColor\ edDigraph(D, edgecolors));\n//dot\ndigraph hgn{\nnode [shape=circle]\n1\n2\n3\ \n4\n1 -> 2[color=lightblue]\n1 -> 3[color=pink]\n1 -> 4[color=purple]\n2 -> 1\ [color=lightblue]\n2 -> 3[color=pink]\n2 -> 4[color=purple]\n3 -> 1[color=ligh\ tblue]\n3 -> 2[color=pink]\n3 -> 4[color=purple]\n4 -> 1[color=lightblue]\n4 -\ > 2[color=pink]\n4 -> 3[color=purple]\n}\ngap> FileString(\"dot/k4.dot\", DotD\ igraph(gr));\n154]]>\n \n \n\n \n \n \n \n \n A s\ tring.\n \n This function produces a graphical \ representation of the symmetric\n digraph digraph. DotSymmetric\ Digraph will return an\n error if digraph is not a symmetric d\ igraph. See\n . \n\n The functi\ on DotSymmetricColoredDigraph differs from DotDigraph\n onl\ y in that the values given in the two lists are used to color the vertices\n \ and edges of the graph when displayed. The list for vertex colours should \ be\n a list of length equal to the number of vertices, containing strings\ that\n are accepted by the graphviz software, which is the one used for \ graph\n representation. The list for edge colours should be a list of li\ sts\n with the same shape of the outneighbours of the digraph that contai\ ns strings \n that correspond to colours accepted by the graphviz softwar\ e.\n If the list is not the appropriate size, or has holes then the funct\ ion\n will return an error. \n\n The function DotSymmetricVe\ rtexColoredDigraph differs from DotDigraph\n only in that the \ values in given in the list is used to color the vertices\n of the graph \ when displayed. The list for vertex colours should be\n a list of length \ equal to the number of vertices, containing strings that\n are accepted b\ y the graphviz software, which is the one used for graph\n representation\ . If the list is not the appropriate size, or has holes \n then the funct\ ion will return an error. \n\n The function DotSymmetricEdgeColor\ edDigraph differs from DotDigraph\n only in that the values gi\ ven in the list are used to color the edges\n of the graph when displayed\ . The list for edge colours should be\n a list of lists with the same sha\ pe of the outneighbours, containing strings that\n are accepted by the gr\ aphviz software, which is the one used for graph\n representation. If the\ list is not the appropriate size, or has holes \n then the function will\ return an error. \n\n Vertices are displayed as circles, numbered c\ onsistently with\n digraph. Since digraph is symmetric, for\ every non-loop\n edge there is a complementary edge with opposite source\ and range.\n DotSymmetricDigraph displays each pair of complement\ ary edges\n as a single line between the relevant vertices, with no arrow\ head. \n\n The output is in dot format (also known as GraphV\ iz)\n format. For details about this file format, and information abo\ ut how\n to display or edit this format see\n http://www.graphv\ iz.org. \n\n The string returned by DotSymmetricDigraph \ can be written to a\n file using the command . \n\n star := Digraph([[2, 2, \ 3, 4], [1, 1], [1], [1, 4]]);\n\ngap> IsSymmetricDigraph(star);\ntrue\ngap> FileString(\"dot/star.dot\", Do\ tSymmetricDigraph(gr));\ngap> D := Digraph([[2], [1, 3], [2]]);\n\ngap> vertcolors := [];;\ngap> vertcolors[1] :\ = \"blue\";;\ngap> vertcolors[2] := \"pink\";;\ngap> vertcolors[3] := \"purple\ \";;\ngap> edgecolors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];\ ;\ngap> edgecolors[3] := [];;\ngap> edgecolors[1][2] := \"green\";; edgecolors\ [2][1] := \"green\";;\ngap> edgecolors[2][3] := \"red\";; edgecolors[3][2] := \ \"red\";;\ngap> Print(DotSymmetricColoredDigraph(D, vertcolors, edgecolors));\ \n//dot\ngraph hgn{\nnode [shape=circle]\n\n1[color=blue, style=filled]\n2[col\ or=pink, style=filled]\n3[color=purple, style=filled]\n1 -- 2[color=green]\n2 \ -- 3[color=red]\n}\ngap> D := Digraph([[2], [1, 3], [2]]);\n\ngap> vertcolors := [];;\ngap> vertcolors[1] := \"b\ lue\";;\ngap> vertcolors[2] := \"pink\";;\ngap> vertcolors[3] := \"purple\";;\ \ngap> Print(DotSymmetricVertexColoredDigraph(D, vertcolors));\n//dot\ngraph h\ gn{\nnode [shape=circle]\n\n1[color=blue, style=filled]\n2[color=pink, style=f\ illed]\n3[color=purple, style=filled]\n1 -- 2\n2 -- 3\n}\ngap> D := Digraph([[\ 2], [1, 3], [2]]);\n\ngap> edgecol\ ors := [];;\ngap> edgecolors[1] := [];; edgecolors[2] := [];;\ngap> edgecolors\ [3] := [];;\ngap> edgecolors[1][2] := \"green\";; edgecolors[2][1] := \"green\ \";;\ngap> edgecolors[2][3] := \"red\";; edgecolors[3][2] := \"red\";;\ngap> P\ rint(DotSymmetricEdgeColoredDigraph(D, edgecolors));\n//dot\ngraph hgn{\nnode \ [shape=circle]\n\n1\n2\n3\n1 -- 2[color=green]\n2 -- 3[color=red]\n}\n83]]>\n \n \n\n \n \n A string.\n \n This function produces a graphical representation \ of a partial order\n digraph digraph. DotPartialOrderDigraph will return an error\n if digraph is not a partial order digraph\ . See . \n\n Since di\ graph is a partial order, it is both reflexive and\n transitive. The \ output of DotPartialOrderDigraph is the\n of the\n of dig\ raph. \n\n The output is in dot format (also known as Gr\ aphViz)\n format. For details about this file format, and information\ about how\n to display or edit this format see\n http://www.gr\ aphviz.org. \n\n The string returned by DotPartialOrderDigra\ ph can be written to a\n file using the command . \n\n poset := Digraph(\ [[1, 4], [2], [2, 3, 4], [4]);\ngap> IsPartialOrderDigraph(gr);\ntrue\ngap> Fi\ leString(\"dot/poset.dot\", DotPartialOrderDigraph(gr));\n83]]>\n \n \n\n \n \n \n A string.\n \n This fu\ nction produces a graphical representation of a preorder\n digraph dig\ raph. DotPreorderDigraph will return an error\n if digraph<\ /A> is not a preorder digraph. See .\ \n\n A preorder digraph is reflexive and transitive but in general it\ is\n not anti-symmetric and may have strongly connected components\n \ containing more than one vertex. The \n \ Q obtained by forming the quotient of digraph by the\n part\ ition of its vertices into the strongly connected components\n satisfies \ . Thus every vertex of\n Q co\ rresponds to a strongly connected component of digraph.\n The outp\ ut of DotPreorderDigraph displays the \n of Q with\n vertices displayed as round\ ed rectangles labelled by all of the vertices\n of digraph in the \ corresponding strongly connected component. \n\n The output is in \ dot format (also known as GraphViz)\n format. For details abou\ t this file format, and information about how\n to display or edit this f\ ormat see\n http://www.graphviz.org. \n\n The string \ returned by DotPreorderDigraph can be written to a\n file using th\ e command . \n\n preset := Digraph([[1, 2, 4, 5], [1, 2, 4, 5], [3, 4], [4], [1, \ 2, 4, 5]);\ngap> IsPreorderDigraph(gr);\ntrue\ngap> FileString(\"dot/preset.do\ t\", DotProrderDigraph(gr));\n83]]>\n \n \ \n\n \n \n A string.\n \n DotHighlightedDigraph produces a graphical represenation\ of the\n digraph digraph, where the vertices in the list verts\ , and\n edges between them, are drawn with colour colour1 and \ all other\n vertices and edges in digraph are drawn with colour colour2.\n If colour1 and colour2 are not given then\n\ DotHighlightedDigraph uses black and grey respectively. \n\n \ Note that DotHighlightedDigraph does not validate the colours\n \ colour1 and colour2 - consult the GraphViz documentation to\n\ see what is available.\n\n See for more \ details on the output. \n\n digraph := Digraph([[\ 2, 3], [2], [1, 3]]);\n\ngap> FileString(\"d\ ot/my_digraph.dot\",\n> DotHighlightedDigraph(digraph, [1, 2], \"red\", \"blac\ k\"));\n264]]>\n \n \n\n \n\n \ Reading and writing digraphs to a file\n This \ section describes different ways to store and read graphs\n from a file in \ the &Digraphs; package.\n \n Graph6 \n \ \n\t Graph6 is a graph data format for storing undirected graphs\n\twi\ th no multiple edges nor loops of size up to 2^{36} - 1 in\n\tprintab\ le chracters. The format consists of two parts. The first part\n\tuses one to \ eight bytes to store the number of vertices. And the second\n\tpart is the upp\ er half of the adjacency matrix converted into ASCII\n\tcharacters. For a more\ detail description see \n\thttp://cs.anu.edu.au/~bdm/data/\ formats.txt.\n \n\n Sparse6 \n \n\t Sparse6 is a graph data format for storing undirected graphs\n\t\ with possibly multiple edges or loops. The maximal number of vertices\n\tallow\ ed is 2^{36} - 1 . The format consists of two parts. The\n\tfirst part\ uses one to eight bytes to store the number of vertices.\n\tAnd the second pa\ rt only stores information about the edges.\n\tTherefore, the Sparse6 \ format return a more compact encoding\n\tthen Graph6 for sparse grap\ h, i.e. graphs where the number\n\tof edges is much less than the number of v\ ertices squared. For a more\n\tdetail description see \n \ http://cs.anu.edu.au/~bdm/data/formats.txt.\n \n\n \ Digraph6 \n \n\tDigraph6 is a new format ba\ sed on Graph6 , but designed\n\tfor digraphs. The entire adjacency mat\ rix is stored, and\n\ttherefore there is support for directed edges and single\ -vertex loops.\n\tHowever, multiple edges are not supported.\n \n\ \n DiSparse6 \n \n\tDiSparse6 is a new fo\ rmat based on Sparse6 , but\n\tdesigned for digraphs. In this format t\ he list of edges is\n\tpartitioned into inceasing and decreasing edges, depend\ ing whether the\n\tedge has its source bigger than the range. Then both sets o\ f edges are\n\twritten separetly in Sparse6 format with a separation \ symbol\n\tin between.\n \n \n \n \n \n A string.\n \n Returns a string s\ tring such that EvalString(string)\n is equal to digraph,\ and has the same mutability.\n See .\n \n\n The methods installed for String make some atte\ mpts to\n ensure that string has as short a length as possible, but\ \n there may exist shorter strings that also evaluate to digraph.\n \ \n\n It is possible that string may contain escaped special\n\ characters. To obtain a representation of digraph that\n can be \ entered as GAP input, please use\n .\ \n Note that Print for a digraph delegates to\n PrintString, which delegates to String.\n D := CycleDigr\ aph(3);\n\ngap> Print(D);\nCycleDigra\ ph(3);\ngap> G := PetersenGraph(IsMutableDigraph);\n\ngap> String(G);\n\"DigraphFromGraph6String(IsMutableDigrap\ h, \\\"IheA@GUAo\\\");\"\ngap> Print(last);\nDigraphFromGraph6String(IsMutable\ Digraph, \"IheA@GUAo\");\ngap> DigraphFromGraph6String(IsMutableDigraph, \"Ihe\ A@GUAo\");\n\n]]>\n \n\n\n \n \n \n \n \n\ A digraph.\n \n If str is a strin\ g encoding a graph in Graph6, Digraph6, Sparse6 or\n DiSparse6 format, then\ the corresponding function returns a digraph. In the\n case of either Grap\ h6 or Sparse6, formats which do not support directed\n edges, this will be \ a digraph such that for every edge, the edge going in\n the opposite direct\ ion is also present. \n\n Each of these functions takes an optional firs\ t argument filt,\n which should be either \n or ,\n and which specifies wh\ ether the output digraph shall\n be mutable or immutable.\n If no first \ argument is provided, then an immutable\n digraph is returned by default.\n\ DigraphFromGraph6String(\"?\");\n\ngap> DigraphFromGraph6String(\"C]\");\n\ngap> DigraphFromGraph6String(\"H?AAEM{\");\n\ \ngap> DigraphFromDigraph6String(\ \"&?\");\n\ngap> DigraphFromDigraph6S\ tring(IsMutableDigraph, \"&DOOOW?\");\n\ngap> DigraphFromDigraph6String(\"&CQFG\");\n\ngap> DigraphFromDigraph6String(\"&IM[SrKLc~lhesbU[F_\");\n\ngap> DigraphFromDiSparse6String(\ \".CaWBGA?b\");\n\n]]>\n \n\n\n \n \n \n \n \n A string.\n \n \ These four functions return a highly compressed string fully describing the\ \n digraph digraph. \n\n Graph6 and Digraph6 are formats best\ used on small, dense graphs, if\n applicable. For larger, sparse graphs us\ e Sparse6 and\n Disparse6 (this latter also preserves multipl\ e edges). \n\n See .\n gr := Digraph([[2, 3], [1], [1]]);\n\ngap> Sparse6String(gr);\n\":Bc\"\ngap> DiSparse6String(gr);\n\".B\ c{f\"\n]]>\n \n\n\n \n <\ Func Name=\"DigraphFile\" Arg=\"filename [, coder][, mode]\"/>\n An \ IO file object.\n \n If filename is a string\ representing the name of a file, then\n DigraphFile returns an &IO;\ package file object for that file.\n \n\n If the optional argument \ coder is specified\n and is a function which either encodes a digrap\ h as a string, or decodes a\n string into a digraph, then this function wil\ l be used when reading or\n writing to the returned file object. If the opt\ ional argument coder\n is not specified, then the encoding of the di\ graphs in the returned file\n object must be specified in the the file exte\ nsion. The file extension must\n be one of: .g6, .s6, .d6<\ /C>, .ds6, .txt,\n .p, or .pickle; more details\ of these file formats is given\n below. \n\n If the optional argume\ nt mode is specified, then it must be one of:\n \"w\" (for wr\ ite), \"a\" (for append), or \"r\" (for read).\n If mode is not specified, then \"r\" is used by default. \n\n If f\ ilename ends in one of: .gz, .bz2, or\n .xz, then \ the digraphs which are read from, or written to, the\n returned file object\ are decompressed, or compressed, appropriately.\n \n\n The file obj\ ect returned by DigraphFile can be given as the first\n argument for\ either of the functions or . The purpose of this is to reduce the overhead of\n recreat\ ing the file object inside the functions \n or \ when, for example, reading or writing many\n \ digraphs in a loop.\n \n\n The currently supported file formats, and\ associated filename extensions,\n are:\n \n graph6 (.g\ 6)\n \n\tA standard and widely-used format for undirected gr\ aphs, with no support\n\tfor loops or multiple edges. Only symmetric graphs a\ re allowed -- each\n\tedge is combined with its converse edge to produce a sin\ gle undirected\n\tedge. This format is best used for \"dense\" graphs -- thos\ e with many\n\tedges per vertex.\n \n sparse6 (.s6)\n \n\tUnlike graph6, sparse6 has support for loops and multiple\ edges.\n\tHowever, its use is still limited to symmetric graphs. This format\ is\n\tbetter-suited to \"sparse\" graphs -- those with few edges per vertex.\ \n \n digraph6 (.d6)\n \n\tThis forma\ t is based on graph6, but stores direction information -\n\ttherefore is not l\ imited to symmetric graphs. Loops are allowed, but\n\tmultiple edges are not.\ Best compression with \"dense\" graphs.\n \n disparse\ 6 (.ds6)\n \n\tAny type of digraph can be encoded in dispars\ e6: directions, loops, and\n\tmultiple edges are all allowed. Similar to spar\ se6, this has the best\n\tcompression rate with \"sparse\" graphs.\n \n plain text (.txt)\n \n\tThis is a human-rea\ dable format which stores graphs in the form\n 0 7 0 8 1 7 2 8 3\ 8 4 8 5 8 6 8 i.e. pairs of vertices\n describing edges in a gr\ aph. More specifically, the vertices making up\n one edge must be separ\ ated by a single space, and pairs of vertices must\n be separated by tw\ o spaces. \n\n See for a more \ flexible way to store\n digraphs in a plain text file. \n \n\n pickled (.p or .pickle)\n \ \n Digraphs are pickled using the &IO; package. This is particularly go\ od\n when the is non-trivial.\n \n \n\n filename := Concatenation(DIGR\ APHS_Dir(), \"/tst/out/man.d6.gz\");;\ngap> file := DigraphFile(filename, \"w\ \");;\ngap> for i in [1 .. 10] do\n> WriteDigraphs(file, Digraph([[1, 3], [2],\ [1, 2]]));\n> od;\ngap> IO_Close(file);;\ngap> file := DigraphFile(filename, \ \"r\");;\ngap> ReadDigraphs(file, 9);\n\ngap> IO_Close(file);;\n]]>\n \n\ \n\n \n \n A digraph, or a list of digraphs.\n \n If filename is a string containing the name of a file containi\ ng\n encoded digraphs or an &IO; file object created using , then ReadDigraphs returns the digraphs\n encoded\ in the file as a list. Note that if filename is a\n compressed fil\ e, which has been compressed appropriately to give a filename\n extension o\ f .gz, .bz2, or .xz, then\n ReadDigraphs can re\ ad filename without it first needing to be\n decompressed.\n \ \n\n If the optional argument decoder is specified\n and is a fun\ ction which decodes a string into a digraph,\n then ReadDigraphs wil\ l use decoder to decode the digraphs\n contained in filename.\ \n\n If the optional argument n is specified, then ReadDigrap\ hs\n returns the nth digraph encoded in the file filename\ .\n \n\n If the optional argument decoder is not specified, t\ hen\n ReadDigraphs will deduce which decoder to use based on the fil\ ename\n extension of filename (after removing the compression-relate\ d\n filename extensions .gz, .bz2, and .xz). For exam\ ple,\n if the filename extension is .g6, then ReadDigraphs wi\ ll use\n the graph6 decoder .\n <\ P/>\n\n The currently supported file formats, and associated filename exten\ sions,\n are:\n \n graph6 (.g6)\n \n\ \tA standard and widely-used format for undirected graphs, with no support\n\t\ for loops or multiple edges. Only symmetric graphs are allowed -- each\n\tedg\ e is combined with its converse edge to produce a single undirected\n\tedge. \ This format is best used for \"dense\" graphs -- those with many\n\tedges per \ vertex.\n \n sparse6 (.s6)\n \n\tUnli\ ke graph6, sparse6 has support for loops and multiple edges.\n\tHowever, its u\ se is still limited to symmetric graphs. This format is\n\tbetter-suited to \ \"sparse\" graphs -- those with few edges per vertex.\n \n digraph6 (.d6)\n \n\tThis format is based on graph6, but\ stores direction information -\n\ttherefore is not limited to symmetric graph\ s. Loops are allowed, but\n\tmultiple edges are not. Best compression with \ \"dense\" graphs.\n \n disparse6 (.ds6)\n <\ Item>\n\tAny type of digraph can be encoded in disparse6: directions, loops, a\ nd\n\tmultiple edges are all allowed. Similar to sparse6, this has the best\n\ \tcompression rate with \"sparse\" graphs.\n \n plain t\ ext (.txt)\n \n\tThis is a human-readable format which store\ s graphs in the form\n 0 7 0 8 1 7 2 8 3 8 4 8 5 8 6 8 i.\ e. pairs of vertices\n describing edges in a graph. More specifically, \ the vertices making up\n one edge must be separated by a single space, \ and pairs of vertices must\n be separated by two spaces. \n\n \ See for a more flexible way to store\n \ digraphs in a plain text file. \n\n \n\n pickled (.p \ or .pickle)\n \n Digraphs are pickled using th\ e &IO; package. This is particularly good\n when the is non-trivial.\n \n \n\n ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"\ ), 10);\n\ngap> ReadDigraphs(\n> C\ oncatenation(DIGRAPHS_Dir(), \"/data/graph5.g6.gz\"), 17);\n\ngap> ReadDigraphs(\n> Concatenation(DIGRAPHS_Dir(\ ), \"/data/tree9.4.txt\"));\n[ , \ \n , \n , \n , \n , \n , \n , \n , \n , \n , \n , \n , \n , \n ]]]>\n \n\n\n\ \n \n \n If digraphs is a list of digr\ aphs or a digraph and filename\n is a string or an &IO; file object \ created using ,\n then WriteDigraphs writ\ es the digraphs to the file represented by\n filename. If the suppl\ ied filename ends in one of the extensions\n .gz, .bz2, or .xz, then the file will be compressed\n appropriately. Excluding thes\ e extensions, if the file ends with an\n extension in the list below, the c\ orresponding graph format will be used to\n encode it. If such an extensio\ n is not included, an appropriate format\n will be chosen intelligently, an\ d an extension appended, to minimise file\n size.\n \n\n \n\n For more verbose information on the progress of\ the function, set the info\n level of InfoDigraphs to 1 or higher, \ using SetInfoLevel. \n\n The currently supported file formats are\ :\n \n graph6 (.g6)\n \n\tA standard and\ widely-used format for undirected graphs, with no support\n\tfor loops or mul\ tiple edges. Only symmetric graphs are allowed -- each\n\tedge is combined wi\ th its converse edge to produce a single undirected\n\tedge. This format is b\ est used for \"dense\" graphs -- those with many\n\tedges per vertex.\n <\ /Item>\n sparse6 (.s6)\n \n\tUnlike graph6, spars\ e6 has support for loops and multiple edges.\n\tHowever, its use is still limi\ ted to symmetric graphs. This format is\n\tbetter-suited to \"sparse\" graphs\ -- those with few edges per vertex.\n \n digraph6 (.d6\ )\n \n\tThis format is based on graph6, but stores direction\ information -\n\ttherefore is not limited to symmetric graphs. Loops are all\ owed, but\n\tmultiple edges are not. Best compression with \"dense\" graphs.\ \n \n disparse6 (.ds6)\n \n\tAny type\ of digraph can be encoded in disparse6: directions, loops, and\n\tmultiple ed\ ges are all allowed. Similar to sparse6, this has the best\n\tcompression rat\ e with \"sparse\" graphs.\n \n\n plain text (.txt)\n \n\tThis is a human-readable format which stores graphs in the\ form\n 0 7 0 8 1 7 2 8 3 8 4 8 5 8 6 8 i.e. pairs of ver\ tices\n describing edges in a graph. More specifically, the vertices ma\ king up\n one edge must be separated by a single space, and pairs of ve\ rtices must\n be separated by two spaces. \n\n See for a more flexible way to store\n digraphs\ in a plain text file. \n\n \n pickled (.p or .pickle)\n\ \n Digraphs are pickled using the &IO; package. This is par\ ticularly good\n when the is non-trivial.\ \n \n \n\n grs := [];;\ngap>\ grs[1] := Digraph([]);\n\ngap> grs[2\ ] := Digraph([[1, 3], [2], [1, 2]]);\n\ngap> grs[3] := Digraph([\n> [6, 7], [6, 9], [1, 3, 4, 5, 8, 9],\n> [1, 2\ , 3, 4, 5, 6, 7, 10], [1, 5, 6, 7, 10], [2, 4, 5, 9, 10],\n> [3, 4, 5, 6, 7, 8\ , 9, 10], [1, 3, 5, 7, 8, 9], [1, 2, 5],\n> [1, 2, 4, 6, 7, 8]]);\n\ngap> filename := Concatenation(DIGRAPHS_D\ ir(), \"/tst/out/man.d6.gz\");;\ngap> WriteDigraphs(filename, grs, \"w\");\nIO\ _OK\ngap> ReadDigraphs(filename);\n[ \ , \n , \n ]]]>\n \n\n\n\ \n \n An iterator.\n \n If fil\ ename is a string representing the name of a file containing\n encoded \ digraphs, then IteratorFromDigraphFile returns an iterator\n for whi\ ch the value of is the\n next\ digraph encoded in the file.\n \n\n If the optional argument dec\ oder is specified and is a function\n which decodes a string into a dig\ raph, then IteratorFromDigraphFile\n will use decoder to deco\ de the digraphs contained in filename.\n \n\n The purpose of \ this function is to easily allow looping over digraphs\n encoded in a file \ when loading all of the encoded digraphs would require\n too much memory. <\ P/>\n\n To see what file types are available, see .\n\n filename := Concatenation(DIGRAPHS_Dir()\ , \"/tst/out/man.d6.gz\");;\ngap> file := DigraphFile(filename, \"w\");;\ngap>\ for i in [1 .. 10] do\n> WriteDigraphs(file, Digraph([[1, 3], [2], [1, 2]])\ );\n> od;\ngap> IO_Close(file);;\ngap> iter := IteratorFromDigraphFile(filenam\ e);\n\ngap> for x in iter do od;\n]]>\n \n<\ /ManSection>\n\n \n \n \n A string.\n \n These two functions return a function which enc\ odes or decodes a digraph in\n a plain text format. \n\n DigraphPl\ ainTextLineEncoder returns a function which takes a single\n digraph as\ an argument. The function returns a string describing the edges\n of that\ digraph; each edge is written as a pair of integers separated by the\n str\ ing delimiter2, and the edges themselves are separated by the\n stri\ ng delimiter1. DigraphPlainTextLineDecoder returns the\n cor\ responding decoder function, which takes a string argument in this format\n \ and returns a digraph. \n\n If only one delimiter is passed as an argum\ ent to\n DigraphPlainTextLineDecoder, it will return a function whic\ h decodes\n a single edge, returning its contents as a list of integers. \n\n The argument offset should be an integer, which will describe \ a\n number to be added to each vertex before it is encoded, or after it is\ \n decoded. This may be used, for example, to label vertices starting at 0\ \n instead of 1. \n\n Note that the number of vertices of a digraph i\ s not stored, and so vertices\n which are not connected to any edge may be \ lost.\n\n gr := Digraph([[2, 3], [1], [1]]);\n\ngap> enc := DigraphPlainTextLineEnco\ der(\" \", \" \", -1);;\ngap> dec := DigraphPlainTextLineDecoder(\" \", \" \ \", 1);;\ngap> enc(gr);\n\"0 1 0 2 1 0 2 0\"\ngap> dec(last);\n\n]]>\n \n\n\n \n \n\ A digraph.\n \n This function takes a st\ ring str, decodes it, and then returns the\n tournament [see ] which it defines, according to\n the following rules\ . \n\n The characters of the string str represent the entries in\ the upper\n triangle of a tournament's adjacency matrix. The number of ver\ tices\n n will be detected from the length of the string and will be\ as\n large as possible. \n\n The first character represents the po\ ssible edge 1 -> 2, the second\n represents 1 -> 3 and so on \ until 1 -> n; then the following\n character represents 2 -> 3, and so on up to the character which\n represents the edge n-1 -> n. \n\n If a character of the string with corresponding edge i -> j<\ /C> is equal\n to 1, then the edge i -> j is present in the t\ ournament.\n Otherwise, the edge i -> j is present instead. In this\ way, all the\n possible edges are encoded one-by-one.\n\n gr := TournamentLineDecoder(\"100001\");\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ ], [ 1, 2, 4 ], [ 1\ , 2 ] ]\n]]>\n \n\n\n \n \ \n A digraph.\n \n This function takes a string str, decodes it, and then returns the\n topologically sorted digraph [\ see ]\n which it defines, according t\ o the following rules. \n\n The characters of the string str rep\ resent the entries in the upper\n triangle of a digraph's adjacency matrix.\ The number of vertices n\n will be detected from the length of the \ string and will be as large as\n possible. \n\n The first character\ represents the possible edge 1 -> 2, the second\n represents 1 -\ > 3 and so on until 1 -> n; then the following\n character repre\ sents 2 -> 3, and so on up to the character which\n represents the e\ dge n-1 -> n. If a character of the string with\n corresponding edg\ e i -> j is equal to 1, then this edge is\n present in the di\ graph. Otherwise, it is not present. In this way, all the\n possible edges\ are encoded one-by-one. \n\n In particular, note that there exists no \ edge [i, j] if j \\leq\n i. In order words, the digraph wi\ ll be topologically sorted.\n\n gr := AdjacencyMat\ rixUpperTriangleLineDecoder(\"100001\");\n\ngap> OutNeighbours(gr);\n[ [ 2 ], [ ], [ 4 ], [ ] ]\ngap> gr := Ad\ jacencyMatrixUpperTriangleLineDecoder(\"111111x111\");\n\ngap> OutNeighbours(gr);\n[ [ 2, 3, 4, 5 ], [ 3, 4 ], [\ 4, 5 ], [ 5 ], [ ] ]\n]]>\n \n\n\n \ \n \n A digrap\ h.\n \n If str is a string consisting of at \ least two non-negative integers\n separated by spaces, then this function w\ ill attempt to return the digraph\n which it defines as a TCode string. \n\n The first integer of the string defines the number of vertices v in\n the digraph, and the second defines the number of edges e. \ The\n following 2e integers should be vertex numbers in the range [0 ..\n v-1]. These integers are read in pairs and define the digrap\ h's\n edges. This function will return an error if str has fewer th\ an\n 2e+2 entries. \n\n Note that the vertex numbers will be \ incremented by 1 in the digraph\n returned. Hence the string fragment 0\ 6 will describe the edge\n [1,7].\n \ gr := TCodeDecoder(\"3 2 0 2 2 1\");\n\ngap> OutNeighbours(gr);\n[ [ 3 ], [ ], [ 2 ] ]\ngap> gr := TCodeDecode\ r(\"12 3 0 10 5 2 8 8\");\n\ngap>\ OutNeighbours(gr);\n[ [ 11 ], [ ], [ ], [ ], [ ], [ 3 ], [ ], [ ], [ 9 \ ], [ ], \n [ ], [ ] ]\n]]>\n \n\n\n \ \n \n \n A string.\ \n \n PlainTextString takes a single digraph, and retu\ rns a string\n describing the edges of that digraph. DigraphFromPlainTe\ xtString\n takes such a string and returns the digraph which it describ\ es. Each edge\n is written as a pair of integers separated by a single spa\ ce. The edges\n themselves are separated by a double space. Vertex number\ s are reduced by\n 1 when they are encoded, so that vertices in the string \ are labelled\n starting at 0. \n\n Note that the number of vertices o\ f a digraph is not stored, and so vertices\n which are not connected to any\ edge may be lost. \n\n \n The operation DigraphFromPlainTextStrin\ g\n takes an optional first argument \n\ or , which specifies whether the output \ digraph shall\n be mutable or immutable. If no first argument is provided, \ then an immutable\n digraph is returned by default.\n\n gr := Digraph([[2, 3], [1], [1]]);\n\ngap> PlainTextString(gr);\n\"0 1 0 2 1 0 2 0\"\ngap> DigraphFr\ omPlainTextString(last);\n\n]]>\n \n\n\n \n \n \n\ \n These functions write and read a single digraph in a huma\ n-readable plain\n text format as follows: each line contains a single edge\ , and each edge is\n written as a pair of integers separated by the string \ delimiter. \n\n filename should be the name of a file whic\ h will be written to or\n read from, and offset should be an integer\ which is added to each\n vertex number as it is written or read. For exam\ ple, if\n WritePlainTextDigraph is called with offset -1, then\n the vertices will be numbered in the file starting from 0 instea\ d of 1 -\n ReadPlainTextDigraph would then need to be called with offset\n 1 to convert back to the original graph. \n\n ignore should be a list of characters which will be ignored when\n rea\ ding the graph.\n gr := Digraph([[1, 2, 3], [1, 1]\ , [2]]);\n\ngap> filename := \ Concatenation(DIGRAPHS_Dir(), \"/tst/out/plain.txt\");;\ngap> WritePlainTextDi\ graph(filename, gr, \",\", -1);\ngap> ReadPlainTextDigraph(filename, \",\", 1,\ ['/', '%']);\n\n]]>\n \n\n\n \n \n \n \n These operations write or read the si\ ngle symmetric digraph digraph\n to or from a file in DIMACS format,\ as appropriate. The operation\n WriteDIMACSDigraph records the vert\ ices and edges of digraph.\n The vertex labels of digraph wil\ l be recorded only if they are\n integers. See and . \n\n The first ar\ gument filename should be the name of the file which will\n be writt\ en to or read from. A file can contain one symmetric digraph in\n DIMACS f\ ormat. If filename ends in one of .gz, .bz2,\n or .\ xz, then the file is compressed, or decompressed, appropriately.\n \ \n\n The DIMACS format is described as follows. Each line in the DIMACS fi\ le has\n one of four types:\n \n \n A line begin\ ning with c and followed by any number of characters\n is a comm\ ent line, and is ignored.\n \n \n A line beginni\ ng with p defines the numbers of vertices and edges\n the digrap\ h. This line has the format p edge <nr_vertices>\n <nr_e\ dges>, where <nr_vertices> and\n <nr_edges>\ ; are replaced by the relevant integers. There\n must be exactly on\ e such line in the file, and it must occur before any\n of the followin\ g kinds of line. \n\n Although it is required to be present, the val\ ue of\n <nr_edges> will be ignored. The correct number of \ edges\n will be deduced from the rest of the information in the file.\n\ \n \n A line of the form e <v> <w&g\ t;, where <v>\n and <w> are integers in t\ he range [1 ..\n <nr_vertices>], specifies that there is\ a (symmetric) edge\n in the digraph between the vertices <v><\ /C> and\n <w>. A symmetric edge only needs to be defined o\ nce; an\n additional line e <v> <w>, or e <w&g\ t;\n <v>, will be interpreted as an additional, multiple, e\ dge.\n Loops are permitted.\n \n \n A lin\ e of the form n <v> <label>, where\n <v> is an integer in the range [1 ..\n <nr_vertices>] and \ <label> is an integer,\n signifies that the vertex <\ v> has the label\n <label> in the digraph. If a lab\ el is not specified for a\n vertex, then ReadDIMACSDigraph will \ assign the label\n 1, according to the DIMACS specification.\n \ \n \n\n A detailed definition of the DIMACS format can\ be found at\n http://mat.gsia.cmu.edu/COLOR/general/ccformat.ps\ , in Section\n 2.1. Note that optional descriptor lines, as described in Se\ ction 2.1, will\n be ignored.\n \n\n\n g\ r := Digraph([[2], [1, 3, 4], [2, 4], [2, 3]]);\n\ngap> filename := Concatenation(DIGRAPHS_Dir(),\n> \ \"/tst/out/dimacs.dimacs\");;\ngap> WriteDIMACSDigraph(filen\ ame, gr);;\ngap> ReadDIMACSDigraph(filename);\n]]>\n \n\n\n \n<\ /Chapter>\n \n\n\n\n \n \ Grape to Digraphs Command Map\n \n Below is a table of &Grape; com\ mands with the &Digraphs; counterparts.\n The sections in this chapter corres\ pond to the chapters in the &Grape;\n manual.\n\n \n \n\ Functions to construct and modify graphs\n \n \n \n The table in this section contains more informat\ ion when viewed in\n html format.\n \n \n\n \n \n \n \n \ &Grape; command\n \n \n &Digraphs; command\ \n \n \n \n \ Comments\n \n \n \n \n\ \n \n \n Graph\n \n \ \n \n \n \n\ \n Used with the arguments of Graph in &Grape;, will return an equivalent \ digraph. \n \n \n \n \n\n \n \n EdgeOrbitsGraph\n <\ /Item>\n \n \n <\ /Item>\n \n \n The operation\ behaves the same as\n EdgeOrbit\ sDigraph in &Grape;.\n \n \n \n \ \n\n \n \n NullGraph\n \ \n \n \n \ \n or\n \ \n \n \n \n \n Note that, unlike in &Grape;, the arguments of and are just\n \ non-negative integers. Since digraphs in &Digraphs; are immutable,\n \ there is no reason (although it is possible) to create a null digrap\ h\n and then add edges, as you might be used to doing in &Grape;.\n\ \n \n \n \n\n \n \n CompleteGraph\n \n <\ Item>\n \n \n \n \n Note that, unlike in &Grape;, \ the argument to is simply a non\ -negative integer\n indicating the number of vertices. By default a\ complete digraph in\n &Digraphs; does not have any loops. If you w\ ant a complete digraph\n with loops at every vertex, then use .\n \n \n \n \n\n \n \n \ JohnsonGraph\n \n \n \n \n \n \n The &Grape; function JohnsonGraph is equivalent to\n \ .\n \n \n\ \n \n\n \n \n Ca\ yleyGraph\n \n \n \n \n \n \n \ The operation behaves the same as\n \ CayleyGraph in &Grape;.\n \n \n \ \n \n\n \n \n AddEd\ geOrbit\n \n \n \n \n \n \ \n Note that, unlike in &Grape;, digraphs in &Digraphs; are immutab\ le,\n and this operation returns a new digraph, i.e. it does not mo\ dify the\n digraph given as an argument.\n \n \ \n \n \n\n \n \n \ RemoveEdgeOrbit\n \n \n \n \n \n <\ Alt Only=\"HTML\">\n Note that, unlike in &Grape;, digraphs in &Dig\ raphs; are immutable,\n and this operation returns a new digraph, i\ .e. it does not modify the\n digraph given as an argument.\n \ \n \n \n \n\n \n \ \n AssignVertexNames\n \n \n \n \n and\n \n \ \n \n \n \ \n Use to set an individual\ \n vertex label and to set a\ ll of\n the vertex labels.\n \n \n \ \n \n \n \n\n \n \n Functions to inspect graphs, vertices and edges\n \ \n \n \n The table in this section contain\ s more information when viewed in\n html format.\n \n \n\n \n \n\n \n <\ Item>\n &Grape; command\n \n \n &\ Digraphs; command\n \n \n \n Comments\n \n \n \n \ \n\n \n \n IsGraph\n \ \n \n \n \n \n \n is a category and not a function like\n IsGraph.\ \n \n \n \n \n\n <\ Row>\n \n OrderGraph\n \n \n \n \n \n \n The operation behaves the same as\n OrderGraph in &Grap\ e;.\n \n \n \n \n\n \ \n \n IsVertex(graph, v)\n \ \n \n v in DigraphVertices(dig\ raph)\n \n \n \ \n The attribute can be used to ret\ urn\n the list of vertices of a digraph, or if v is known to\ be\n a positive integer, then you can simply do\n v <= DigraphNrVertices(digraph)\n \n \ \n \n \n\n \n \n \ VertexName\n \n \n \n \n \n \n Unlike VertexName in &Grape;, the return value o\ f\n is mutable.\n \n \n \n \n\n \n \n VertexNames\n \n \n \ \n \n \n \ \n Unlike VertexNames in &Grape;, the r\ eturn value of\n is mutable.\n \ \n \n \n \n\n \n \n Vertices\n \n \ \n \n \n \n\ \n The vertices of a digraph in &Digra\ phs; is always the range [1\n .. n] where n is the \ number of vertices of the\n digraph. The operation behaves the\n same as Vertices in &Grape;.\ \n \n \n \n \n\n <\ Row>\n \n VertexDegree\n \n \ \n \n \n (see also: )\n \n \n \n \n \ The operation behaves the same as\n \ VertexDegree in &Grape;.\n \n \n \ \n \n\n \n \n VertexD\ egrees\n \n \n \n \n (see also:\n ,\n ,\n \ ,\n ,\n )\n \n \n \n \n The operation behaves the same as\n VertexDegrees in &Grape;.\n \n \n \n \n\n \n \n IsLoopy\n \ \n \n \n \n\ \n \n The operation behaves the same as\n IsLoopy in \ &Grape;.\n \n \n \n \n\ \n \n \n IsSimpleGraph\n \ \n \n \n \n and not \n \ \n \n \n \n \n \n \n The &Grape; function IsSimpleGraph is equi\ valent to\n \n and not\ .\n \n \n <\ /Row>\n \n\n \n \n Adjacency\ \n \n \n \n \n (see also )\n \n \n \n \ \n The operation behaves the same\n as Adjacency in &Grape;.\n \ \n \n \n \n\n \n \n IsEdge\n \n \n \ \n \ \n \n \n The argum\ ents of\n \n can be a digraph and an edge, or a digraph and two vertices.\ In the\n first form the operation\n \n behaves the same as \ IsEdge in &Grape;.\n \n \n \n \ \n\n \n \n DirectedEdges\n \ \n \n \n \ \n \n \n The attri\ bute behaves the same\n as Directed\ Edges in &Grape;.\n \n \n \n \ \n\n \n \n UndirectedEdges\n \ \n \n None\n \n \n \n There is no analogous function i\ n &Digraphs;, and no plans to\n include one.\n \n \ \n \n \n\n \n \n \ Distance\n \n \n \n \n \n \n \ The operation behaves the same as\n Dis\ tance in &Grape;.\n \n \n \n \ \n\n \n \n Diameter\n \ \n \n \n \n \n \n The attribut\ e behaves the same\n as Diameter\ in &Grape;.\n \n \n \n \n\n \n \n Girth\n \ \n \n \n \n \n \n The attribut\ e behaves the same\n as G\ irth in &Grape; except that in the case that the\n argument is \ a forest, infinity is returned by &Digraphs;\n whereas -1\ is returned by &Grape;.\n \n \n \ \n \n\n \n \n IsConnectedGra\ ph\n \n \n \n \n \n \n The attribute behave\ s the\n same as IsConnectedGraph in &Grape;.\n \n \n \n \n\n \n \n IsBipartite\n \n \n \ \n \n \ and \n \n \n \ \n \n The &Grape; function\ IsBipartite is equivalent to\n and .\n \n \n \n \n\n \n \n IsNullGraph\n \n \n \ \n \n or <\ Ref Prop=\"IsEmptyDigraph\"/>\n \n \n \n \n The &Grape; function IsNullG\ raph is equivalent to\n and its \ synonym .\n \n \ \n \n \n\n \n \n \ IsCompleteGraph\n \n \n \n \n \n \n The &Grape; function IsCompleteGraph is ro\ ughly\n equivalent to\n except that in &Digraphs;\n a complete digraph has no loops.\n \n \n \n \n\n\n <\ /Table>\n \n\n \n \n Functions to determ\ ine regularity properties of graphs\n \n \n \ \n The table in this section contains more information when vie\ wed in\n html format.\n \n \n\n \n \n\n \n \n &Grape; co\ mmand\n \n \n &Digraphs; command\n \ \n \n \n Comments\n\ \n \n \n \n\n \n \n IsRegularGraph\n \n \ \n \n \n (see also\n \ \n and\n )\n \n \n \n \n \ The &Grape; function IsRegularGraph is\n equivalent to <\ Ref Prop=\"IsOutRegularDigraph\"/>.\n \n \n \ \n \n\n \n \n LocalPar\ ameters\n \n \n None\n \ \n \n \n The &Grape; functi\ on LocalParameters has no equivalent\n function in &Digraphs\ ;, although this might change in the future.\n \n \n \n \n\n \n \n GlobalParameters\n \n \n None\n \ \n \n \n The &Grap\ e; function GlobalParameters has no equivalent\n function in\ &Digraphs;, although this might change in the future.\n \n \ \n \n \n\n \n \n \ IsDistanceRegular\n \n \n <\ Ref Prop=\"IsDistanceRegularDigraph\"/>\n \n \n \ \n The &Grape; function IsDistanceRegul\ ar is\n equivalent to .\ \n \n \n \n \n\n <\ Row>\n \n CollapsedAdjacencyMat\n \ \n \n None\n \n \n \n The &Grape; function CollapsedAdjacencyMat has no equivalent\n function in &Digraphs;, although this might \ change in the future.\n \n \n \n \ \n\n \n \n OrbitalGraphColadjMats\ \n \n \n None\n \n \ \n \n The &Grape; function Or\ bitalGraphColadjMats has no equivalent\n function in &Digraphs;\ , although this might change in the future.\n \n \ \n \n \n\n \n \n \ VertexTransitiveDRGs\n \n \n None\n \ \n \n \n The &G\ rape; function VertexTransitiveDRGs has no equivalent\n func\ tion in &Digraphs;, although this might change in the future.\n \n \n \n \n\n \n\n \n\n \n \n Some special vertex subsets of a gra\ ph\n \n \n \n The table in th\ is section contains more information when viewed in\n html format.\n \ \n \n\n \n \n\n \ \n \n &Grape; command\n \n <\ Item>\n &Digraphs; command\n \n \n \ \n Comments\n \n \ \n \n \n\n \n \n \ ConnectedComponent\n \n \n \n \n \n \n When applied to a simple di/graph and a vertex,\ \n ConnectedComponent in &Grape; is equivalent to\n \ .\n can also be applied to\n non-simple graphs (\ in the language of &Grape;).\n \n \n \ \n \n\n \n \n ConnectedCompo\ nents\n \n \n \n \n \n \n When applied to a simple di/graph,\n Connected\ Components in &Grape; is equivalent to\n .\n \ can also be applied to\n non-simple graphs (in the language of &Gra\ pe;).\n \n \n \n \n\n \ \n \n Bicomponents\n \n \ \n \n \n \ \n \n When applied to a si\ mple bipartite di/graph,\n Bicomponents in &Grape; is equiva\ lent to\n .\n The def\ inition of does not require\n it\ s argument to be simple (in the language of &Grape;), i.e. the\n bi\ partite graphs in &Grape; are a proper subset of the bipartite\n di\ graphs in &Digraphs;.\n \n \n \n \ \n\n \n \n DistanceSet\n \ \n \n \n \n \ \n \n DistanceSet i\ n &Grape; is equivalent to\n in &Digraphs;.\n \ \n \n \n \n\n \ \n \n Layers\n \n \n \ \n \n \n \ \n Layers in &Grape; is equivalent to\ \n in &Digraphs;.\n \n\ \n \n \n\n \n \ \n IndependentSet\n \n \n \ \n \n \n \ \n IndependentSet in &Grape; is equiv\ alent to\n in &Digraphs;.\n \ \n \n \n \n\n \n\n \n\n \n \n Functions to construct \ new graphs from old\n \n \n \n \ The table in this section contains more information when viewed in\n \ html format.\n \n \n\n \n <\ HorLine/>\n\n \n \n &Grape; command\n <\ /Item>\n \n &Digraphs; command\n \n \ \n \n Comments\n \ \n \n \n \n\n \n \n InducedSubgraph\n \n \n \ \n \n \n \ \n in &Digr\ aphs; does not allow a third\n argument G specifying a group\ of automorphisms on the\n resultant graph. This may be included in\ a future release.\n \n \n \n \n\n \n \n DistanceSetInduced\n \ \n \n None\n \n \n \n There is no analogous function i\ n &Digraphs;, and no plans to\n include one.\n \n \ \n \n \n\n \n \n \ DistanceGraph\n \n \n \n \n \ \n \n DistanceGraph\ in &Grape; is equivalent to\n in\n &Digraphs;.\n \n \ \n \n \n\n \n \n \ ComplementGraph\n \n \n \n \n \n \n in &Digraphs; is equivalen\ t to\n ComplementGraph in &Grape; with the optional argument\ \n comploops set to true (loops/nonloops are\n \ complemented).\n \n \n \n \n\n \n \n PointGraph\n \ \n \n None\n \n \n \ \n There is no analogous function in &Digr\ aphs;, and no plans to\n include one.\n \n \n \n \n\n \n \n \ EdgeGraph\n \n \n \n \n \n \n EdgeGraph in &Grape; is equivalent to\n \ in &Digraphs;.\n \n \ \n \n \n\n \n \n \ SwitchedGraph\n \n \n None\ \n \n \n \n \ There is no analogous function in &Digraphs;, and no plans to\n inc\ lude one.\n \n \n \n \n\ \n \n \n UnderlyingGraph\n \n \n \n \ \n \n \n Underl\ yingGraph in &Grape; is equivalent to\n in &Digraphs;.\n \n \n \n \n\n \n \n QuotientGrap\ h\n \n \n \n \n \n \n \ in &Digraphs; takes a partition\n \ p of the vertices of the digraph, rather than a set of\n \ generating pairs. Hence, the equivalence might not be\n group-inva\ riant. Also note that in &Digraphs; multiple edges may be\n create\ d, whereas in &Grape; they are effectively combined into a\n single\ edge.\n \n \n \n \n\n \ \n \n BipartiteDouble\n \ \n \n \n \n \n \n Bipartite\ Double in &Grape; is equivalent to\n in &Digraphs;.\n \n \n \n\ \n\n \n \n GeodesicsGraph\n \n \n None\n \n \ \n \n There is no analogous funct\ ion in &Digraphs;, and no plans to\n include one.\n \ \n \n \n \n\n \n \n CollapsedIndependentOrbitsGraph\n \n \ \n None\n \n \n \n There is no analogous function in &Digraphs;, although \ this feature\n might be implemented in the future.\n \n \n \n \n\n \n \n CollapsedCompleteOrbitsGraph\n \n <\ Item>\n None\n \n \n \n There is no analogous function in &Digraphs;, although th\ is feature\n might be implemented in the future.\n \ \n \n \n \n\n \n \n NewGroupGraph\n \n \n \ None\n \n \n \n \ There is no analogous function in &Digraphs;. However, the same\n \ results would be possible by calling on\n \ the digraph to produce a copy gr, and then applying\n \ SetDigraphGroup(gr, G).\n \n \n \n \n\n \n\n \n\n \n \n Vertex-Colouring and Complete Subgraphs\n \n \n \n The table in this section contains more inf\ ormation when viewed in\n html format.\n \n \n\n <\ Table Align=\"|l|l|l|\">\n \n\n \n \n \ &Grape; command\n \n \n &Digraphs; \ command\n \n \n \n \ Comments\n \n \n \n \n\n \n \n VertexColouring\n \ \n \n \n \n \n \n VertexColouring in &Grape; is equivalent to in &Digra\ phs;,\n except it returns a transformation rather than a list of ve\ rtex\n colors.\n \n \n \n \ \n\n \n \n CompleteSubgraphs<\ /C>\n \n \n \ \n \n \n \n \ The functionality of CompleteSubgraphs in &Grape; is provided\n \ by and \n in &Digraphs;.\n \n \n \ \n \n\n \n \n CompleteSubgra\ phsOfGivenSize\n \n \n \n \n \n \n Much of the functionality of CompleteSubgraphsOfGivenSize in\n &Grape; is provided by and in &Digraphs;, except that \ the\n functionality relating to vertex weights and partial colourin\ gs is\n not implemented.\n \n \n \ \n \n\n \n \n\n \n \n Automorphism groups and isomorphism testing for graphs\n \n \n \n The table in this section c\ ontains more information when viewed in\n html format.\n \n \ \n\n \n \n\n \n \ \n &Grape; command\n \n \n \ &Digraphs; command\n \n \n \n Comments\n \n \n \n \n\n \n \n AutGroupGrap\ h\n \n \n \n \n \n \n AutGroupGraph in &Grape; is equivalent to\ \n in &Di\ graphs;.\n \n \n \n \n\ \n \n \n GraphIsomorphism\n \n \n \n \n \n \n \ GraphIsomorphism in &Grape; is equivalent to in &Digraphs;.\n \ Note however that GraphIsomorphism's optional argument\n \ firstunbindcanon is not applicable, since the canonical\n \ labellings of a digraph are not portable between sessions.\n \ \n \n \n \n\n \n \n IsIsomorphicGraph\n \n \n \ \n \n \n \n IsIsomorphic\ Graph in &Grape; is equivalent to in &Digraphs;.\n Note however that \ IsIsomorphicGraph's optional argument\n firstunbindcanon<\ /A> is not applicable, since the canonical\n labellings of a digrap\ h are not portable between sessions.\n \n \n \ \n \n\n \n \n GraphIs\ omorphismClassRepresentatives\n \n \n \ None\n \n \n \n \ There is no analogous function in &Digraphs;, although this feature\n \ might be implemented in the future.\n \n \ \n \n \n\n \n \n\n \ \n\n\n\n\n\n", inputorigins := [ [ 1, "/builddir/build/BUILD/digraphs-1.4.0/../pkg/digraphs-1.4.0/doc/main.\ xml", 1 ], [ 36, "/builddir/build/BUILD/digraphs-1.4.0/../pkg/digraphs-1.4.0/doc/main\ .xml", 2 ], [ 90, "/builddir/build/BUILD/digraphs-1.4.0/../pkg/digraphs-1.4.0/doc/main\ .xml", 3 ], [ 132, "/builddir/build/BUILD/digraphs-1.4.0/../pkg/digraphs-1.4.0/doc/main\ .xml", 4 ], [ 208, "/builddir/build/BUILD/digraphs-1.4.0/../pkg/digraphs-1.4.0/doc/main\ .xml", 5 ], [ 256, "/builddir/build/BUILD/digraphs-1.4.0/../pkg/digraphs-1.4.0/doc/main\ .xml", 6 ], [ 298, "/builddir/build/BUILD/digraphs-1.4.0/../pkg/digraphs-1.4.0/doc/main\ .xml", 7 ], [ 368, 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:= ~, six := [ [ "Title page", ".", [ 0, 0, 0 ], 1, 1, "title page", "X7D2C85EC87DD46E5" ], [ "Abstract", ".-2", [ 0, 0, 2 ], 56, 2, "abstract", "X7AA6C5737B711C89" ], [ "Copyright", ".-1", [ 0, 0, 1 ], 62, 2, "copyright", "X81488B807F2A1CF1" ], [ "Acknowledgements", ".-3", [ 0, 0, 3 ], 74, 2, "acknowledgements", "X82A988D47DFAFCFA" ], [ "Table of Contents", ".-4", [ 0, 0, 4 ], 81, 3, "table of contents", "X8537FEB07AF2BEC8" ], [ "\033[1X\033[33X\033[0;-2YThe \033[5XDigraphs\033[105X\033[101X\027\\ 033[1X\027 package\033[133X\033[101X", "1", [ 1, 0, 0 ], 1, 5, "the digraphs package", "X7F202ABA780E595D" ], [ "\033[1X\033[33X\033[0;-2YIntroduction\033[133X\033[101X", "1.1", [ 1, 1, 0 ], 4, 5, "introduction", "X7DFB63A97E67C0A1" ], [ "\033[1X\033[33X\033[0;-2YDefinitions\033[133X\033[101X", "1.1-1", [ 1, 1, 1 ], 75, 6, "definitions", "X84541F61810C741D" ], [ "\033[1X\033[33X\033[0;-2YInstalling \033[5XDigraphs\033[105X\033[101X\\ 027\033[1X\027\033[133X\033[101X", "2", [ 2, 0, 0 ], 1, 7, 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317, 318, 319, 320, 34, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 35, 334, 335, 36, 336, 337, 338, 37, 38, 340, 339, 341, 342, 39, 344, 343, 345, 347, 346, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 40, 374, 375, 376, 377, 378, 379, 380, 381, 383, 382, 384, 386, 385, 390, 394, 393, 392, 391, 389, 388, 387, 395, 396, 397, 399, 398, 400, 401, 402, 403, 404, 405, 406, 41, 42, 407, 408, 409, 410, 411, 417, 416, 415, 412, 413, 414, 418, 43, 420, 419, 422, 421, 428, 427, 425, 424, 423, 426, 44, 45, 429, 430, 431, 432, 433, 434, 438, 437, 436, 435, 439, 440, 441, 442, 46, 444, 443, 447, 448, 446, 445, 452, 451, 450, 449, 453, 454, 455, 456, 458, 457, 459, 460, 461, 463, 462, 465, 464, 466, 467, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57 ], start := 0, stop := 487835, toctext := "1 \n The Digraphs package\n \n 1.1\ Introduction\n\n\n   1.1-1 De\ finitions\n\n \n \n<\ a href=\"chap2_mj.html#X817088B27F90D596\">2 Installin\ g Digraphs\n\ 2.1 For those in a hurry\n\n \n 2.2 \ Optional package dependencies\n\n\n &\ nbsp; 2.2-1 The Grape package\n\n \n 2.3 Compiling th\ e kernel module\n\n \n 2.4 Rebuilding the documentation\n\n\n   2.4-1 DigraphsMakeDoc\n \n 2.5 Testing your installation\n\n\n <\ span class=\"nocss\">  2.5-1 DigraphsTestInstall\n   2.5-2 DigraphsTestStandard\n   \ 2.5-3 DigraphsTestExtreme\n \n
\n3 Cr\ eating digraphs\n 3\ .1 Creating digraphs\n\n\n &nb\ sp;3.1-1 IsDigraph\n   3.1-2 IsMutableDigraph\n   3.1-3 IsImmutableDigraph\n   3.1-4 IsCayleyDigraph\n   3.1-5 IsDigraphWithAdjacencyFunction\n   3.1-6 DigraphByOutNeighboursType\n   3.1-7 Digraph\n   3.1-8 DigraphByAdjacencyMatrix\n   3.1-9 DigraphByEdges\n \   3.1-10 EdgeOrbitsDigraph\n   3.1-11 DigraphByInNeighbours\n   3.1-12 CayleyDigraph\n \n 3.2 Changing representations\n\n\n   3.2-1 AsBinaryRelation\n   3.2\ -2 AsDigraph\n &n\ bsp; 3.2-3 Graph\n   <\ a href=\"chap3_mj.html#X7C4F13E080EC16B0\">3.2-4 AsGraph\n   3.2-5 AsTransformation\n \n 3.3 New\ digraphs from old\n\n\n   3.3-1 DigraphImmutableCopy\n   3.3-2 DigraphImmutableCopyIfImmutable\n   3.3-3 InducedSubdigraph\n   3.3-4 ReducedDigraph\n   3.3-5 MaximalSymmetricSubdigraph\n   3.3-6 MaximalAntiSymmetricSubdigraph\n   3.3-7 UndirectedSpanningForest\n   3.3-8 QuotientDigraph\n   3.3-9 DigraphReverse\n  3.3-10 DigraphDual\n   \ 3.3-11 DigraphSymmetricClosure\n   3.3-12 DigraphTransitiveClosure\n   3.3-13 DigraphTransitiveReduction\n \   3.3-14 DigraphAddVertex\n   3.3-15 DigraphAddVertices\n   3.3\ -16 DigraphAddEdge\n   3.3-17 Digr\ aphAddEdgeOrbit\n   3.3-18 Digraph\ AddEdges\n \ 3.3-19 DigraphRemoveV\ ertex\n &nb\ sp;3.3-20 DigraphRemoveVert\ ices\n &nbs\ p;3.3-21 DigraphRemoveEdge<\ /a>\n   3.3-22 DigraphRemoveEdgeOrbit<\ /a>\n   3.3-23 DigraphRemoveEdges<\ /span>\n   \ 3.3-24 DigraphRemoveLoops\n   3.3-25 DigraphRemoveAllMultipleEdges\n   3.3-26 DigraphReverseEdges<\ /span>\n   \ 3.3-27 DigraphDisjointUnion\n   3.3-28 DigraphEdgeUnion\n\   3.3-29 DigraphJoin\n   3.3-30 DigraphCartesianProduct\n   3.3-31 DigraphDirectProduct\n   3.3-32 DigraphCartesianProductProjections\n\   3.3-33 DigraphDirectProductProjections\ \n   3.3-34 LineDigraph\n   3.3-35 LineUndirectedDigraph\n   3.3-36 DoubleDigraph\n   3.3-37 BipartiteDoubleDigraph\n   3.3-38 DigraphAddAllLoops\n   3.3-39 DistanceDigraph\n\   3.3-40 DigraphClosure\n   3.3-41 DigraphMycielskian\n \n 3.4 Random digraphs\ \n\n\n   3.4-1 RandomDigraph\n   3.4-2 R\ andomMultiDigraph\n   3.4-3 Random\ Tournament\n &nbs\ p; 3.4-4 RandomLattice\ \n \n 3.\ 5 Standard examples\n\n\n   \ ;3.5-1 ChainDigraph\n   3.5-2 CompleteDigraph\n   3.5-3 CompleteBipartiteDigraph\n   3.5-4 CompleteMultipartiteDigraph\ \n   3.5-5 CycleDigraph\n   3.5-6 EmptyDigraph\n   3.5-7 JohnsonDigraph\n \   3.5-8 PetersenGraph\n   3.5\ -9 GeneralisedPetersenGraph\n \n
\n4 \ Operators\n 4.1 Operators for digraphs\n\n\n   \ ;4.1-1 IsSubdigraph\n   4.1-2 IsUndirectedSpanningTree\n \n \n5 Attributes and operations\n&nbs\ p;5.1 Vertices and edges\n\n\n   5.1-1 DigraphVertices\n   5.1-2 DigraphNrVertices\n   5.1-3 DigraphEdges\n   5.1-4 DigraphNrEdges\n   5.1-\ 5 DigraphNrLoops\n   5.1-6 Digrap\ hSinks\n &n\ bsp;5.1-7 DigraphSources\n   5.1-8 DigraphTopologicalSort\ \n   5.1-9 DigraphVertexLabel\n   5.1-10 DigraphVertexLabels\ \n   5.1-11 DigraphEdgeLabel\n   5.1-12 DigraphEdgeLabels\n   5.1-13 DigraphInEdges\n   5.1-14 DigraphOutEdges\n<\ br />  5.1-15 IsDigraphEdge\n   5.1-16 IsMatching\n   5.1-17 Di\ graphMaximalMatching\n   5.1-18 Di\ graphMaximumMatching\n \n 5.2 Neighbours and degree\ \n\n\n   5.\ 2-1 AdjacencyMatrix\n   5.2-2 Char\ acteristicPolynomial\n   5.2-3 Boo\ leanAdjacencyMatrix\n   5.2-4 Digr\ aphAdjacencyFunction\n   5.2-5 Dig\ raphRange\n \ ; 5.2-6 OutNeighbours<\ /a>\n   5.2-7 InNeighbours\ \n   5.2-8 OutDegrees\n   5.2-9 InDegrees\n  5.2-10 OutDegreeOfVertex\n <\ span class=\"nocss\">  5.2-11 OutNeighboursOfVertex\n   5.2-12 InDegreeOfVertex\n   5\ .2-13 InNeighboursOfVertex\n   5.2\ -14 DigraphLoops\n   5.2-15 Parti\ alOrderDigraphMeetOfVertices\n   5\ .2-16 DegreeMatrix\n   5.2-17 Lapl\ acianMatrix\n \n 5.3 Reachability and connectivity\ \n\n\n   5.\ 3-1 DigraphDiameter\n   5.3-2 Digr\ aphShortestDistance\n   5.3-3 Digr\ aphShortestDistances\n   5.3-4 Dig\ raphLongestDistanceFromVertex\n   \ 5.3-5 DigraphDistanceSet\n   5.3-6\ DigraphGirth\n &\ nbsp; 5.3-7 DigraphOdd\ Girth\n &nb\ sp;5.3-8 DigraphUndirectedG\ irth\n &nbs\ p;5.3-9 DigraphConnectedCom\ ponents\n &\ nbsp;5.3-10 DigraphConnecte\ dComponent\n &nbs\ p; 5.3-11 DigraphStron\ glyConnectedComponents\n   5.3-12 \ DigraphStronglyConnectedComponent\n   5.3-13 DigraphBicomponents\n   5.3-14 ArticulationPoints\n   5.3\ -15 Bridges\n &nb\ sp; 5.3-16 StrongOrien\ tation\n &n\ bsp;5.3-17 DigraphPeriod\n   5.3-18 DigraphFloydWarshall<\ /span>\n   \ 5.3-19 IsReachable\n   5.3-20 VerticesReachableFrom\n   5.3-21 DigraphPath\n   5.3-22 DigraphShortestPath\n   5.3-23 IteratorOfPaths\n  5.3-24 DigraphAllSimpleCircuits\n\   5.3-25 DigraphLongestSimpleCircuit\n   5.3-26 DigraphLayers\n   5.3-27 DigraphDegeneracy\n   5.3-28 DigraphDegeneracyOrdering\n \   5.3-29 HamiltonianPath\n   \ 5.3-30 NrSpanningTrees\n   5.3-31 \ DigraphDijkstra\n \n 5.4 Cayley graphs of groups\n<\ /span>\n\n   5.4-1\ GroupOfCayleyDigraph\n   5.4-2 Ge\ neratorsOfCayleyDigraph\n \n 5.5 Associated semigroups<\ /a>\n\n\n   5.5-1 AsSemigroup\n   5.5-2 AsSem\ igroup\n \n 5.6 Planarity\n\n\n   <\ /span>5.6-1 KuratowskiPlanarSubdig\ raph\n &nbs\ p;5.6-2 KuratowskiOuterPlan\ arSubdigraph\n &n\ bsp; 5.6-3 PlanarEmbed\ ding\n &nbs\ p;5.6-4 OuterPlanarEmbeddin\ g\n   <\ /span>5.6-5 SubdigraphHomeomorphic\ ToK23\n \n \n6 Properties of digraphs\ \n 6.1 Edge properties\n\n\n\   6.1-1 DigraphHasLoops\n   6.1-2 IsAntiSymmetricDigraph\n   6.1-3 IsBipartiteDigraph\n   6.1-4 IsCompleteBipartiteDigraph\n   6.1-5 IsCompleteDigraph\n   6.1-6 IsCompleteMultipartiteDigraph\n   6.1-7 IsEmptyDigraph\n   6.1-8 IsEquivalenceDigraph\n\   6.1-9 IsFunctionalDigraph\n   6.1-10 IsMultiDigraph\n   6.1-11 IsReflexiveDigraph\n   6.1\ -12 IsSymmetricDigraph\n   6.1-13 \ IsTournament\n &n\ bsp; 6.1-14 IsTransiti\ veDigraph\n \ ; 6.1-15 IsPreorderDig\ raph\n &nbs\ p;6.1-16 IsPartialOrderDigr\ aph\n   \ ;6.1-17 IsMeetSemilatticeDi\ graph\n \n 6.2 Regularity\n\n\n   \ 6.2-1 IsInRegularDigraph\n   6.2-2 IsOutRegularDigraph\n   6.2-3 IsRegularDigraph\n<\ span class=\"ContSS\">   6.2-4 IsDistanceRegularDigraph\ \n \n 6.3 Connectivity and cycles\n\n\n   6.3-1 IsAcyclicDigraph\n   6.3-2 IsChainDigraph\n   6.3-3 IsConnectedDigraph\n   6.3-4 IsBiconnectedDigraph\n   6.3-5 IsBridgelessDigraph\n   6.3-6 IsStronglyConnectedDigraph\n   6.3-7 IsAperiodicDigraph\n   6.3-8 IsDirectedTree\n  6.3-9 IsUndirectedTree\n   6.3-10 IsEulerianDigraph\n   6.\ 3-11 IsHamiltonianDigraph\n   6.3-\ 12 IsCycleDigraph\n \n 6.4 Planarity\n\n\n &n\ bsp; 6.4-1 IsPlanarDig\ raph\n &nbs\ p;6.4-2 IsOuterPlanarDigrap\ h\n \n 6\ .5 Homomorphisms and transformations\n\n\n   6.5-1 IsD\ igraphCore\n &nbs\ p; 6.5-2 IsEdgeTransit\ ive\n   \ ;6.5-3 IsVertexTransitive\n \n
\n7 Homomorphisms\n 7.1 Acti\ ng on digraphs\n\n\n   7.1-1 OnDigraphs\n  7.1-2 OnMultiDigraphs\n \n 7.2 Isomorphisms and can\ onical labellings\n\n\n   7.2-1 DigraphsUseNauty\n   7.2-2 AutomorphismGroup\n\   7.2-3 BlissAutomorphismGroup\n<\ br />  7.2-4 NautyAutomorphismGroup\n  7.2-5 AutomorphismGroup\n   7.2-6 AutomorphismGroup\n   7\ .2-7 BlissCanonicalLabelling\n   7\ .2-8 BlissCanonicalLabelling\n   7\ .2-9 BlissCanonicalDigraph\n   7.2\ -10 DigraphGroup\n   7.2-11 Digra\ phOrbits\n \ 7.2-12 DigraphOrbitRe\ ps\n   \ 7.2-13 DigraphSchreierVecto\ r\n   <\ /span>7.2-14 DigraphStabilizer\ \n   7.2-15 IsIsomorphicDigraph\n   7.2-16 IsIsomorphicDigraph\n   7.2-17 IsomorphismDigraphs\n\   7.2-18 IsomorphismDigraphs\n   7.2-19 RepresentativeOutNeighbours\ \n   7.2-20 IsDigraphIsomorphism\n\   7.2-21 IsDigraphColouring\n \n 7.3 Homomorphisms of digraphs\n\n\n   \ 7.3-1 HomomorphismDigraphsFinder\n   7.3-2 DigraphHomomorphism\n   <\ a href=\"chap7_mj.html#X8653EDA183E06D05\">7.3-3 HomomorphismsDigraphs\n   7.3-4 DigraphMonomorphism\ \n   7.3-5 MonomorphismsDigraphs\n\   7.3-6 DigraphEpimorphism\n   7.3-7 EpimorphismsDigraphs\n   7.3-8 DigraphEmbedding\n   7.3-9 EmbeddingsDigraphs\n   7.3-10 IsDigraphHomomorphism\n   7.3-11 IsDigraphEmbedding\n<\ br />  7.3-12 DigraphsRespectsColouring\n   7.3-13 GeneratorsOfEndomorphismMonoid\n   7.3-14 DigraphColouring\n   7.3-15 DigraphGreedyColouring\n   7.3-16 DigraphWelshPowellOrder\n   7.3-17 ChromaticNumber\n<\ br />  7.3-18 DigraphCore\n \n \n8 Finding cliques and independent sets\n 8.1 Finding cliques\ \n\n\n   8.\ 1-1 IsClique\n &n\ bsp; 8.1-2 CliquesFind\ er\n   \ 8.1-3 DigraphClique\n   8.1-4 DigraphMaximalCliques\n   8.1-5 CliqueNumber\n \n&n\ bsp;8.2 Finding independent sets\n\n\ \n   8.2-1 IsIndependentSet\n   8.2-2 DigraphIndependentSet\n   8.2-3 DigraphMaximalIndependentSets\n<\ /div> \n \n9 Visualising and IO\n 9.1 Visualising\ a digraph\n\n\n   9.1-1 Splash\n   9.1-2 DotDigraph\n   9.1-3 DotS\ ymmetricDigraph\n   9.1-4 DotParti\ alOrderDigraph\n \   9.1-5 DotPreord\ erDigraph\n \ ; 9.1-6 DotHighlighted\ Digraph\n \n 9.2 Reading and writing digraphs to a file\ \n\n\n   9.2-1 String\n   9.2-2 DigraphF\ romGraph6String\n   9.2-3 Graph6St\ ring\n &nbs\ p;9.2-4 DigraphFile\n   9.2-5 ReadDigraphs\n   9.2-6 WriteDigraphs\n   9.2-7 IteratorFromDigraphFile\n   9.2-8 DigraphPlainTextLineEncoder\n   9.2-9 TournamentLineDecoder\n   9.2-10 AdjacencyMatrixUpperTriangleLineDecoder\n   9.2-11 TCodeDecoder\n   9.2-12 PlainTextString\n   9.2-13 WritePlainTextDigraph\n   9.2-14 WriteDIMACSDigraph\n \n\nA \n Grape to Digraphs Command Map\n \n A.1 \n \ Functions to construct and modify graphs\n \n\n \n A.2 \n \ Functions to inspect graphs, vertices and edges\n \n\ \n \n A.3 \n Functions to determine regularity properties of graphs\n \ \n\n \n\ A.4 \n Some special vertex subsets of a graph\n \ \n\n \n A.5 \n Functions to construct new graphs from\ old\n \n\n \n A.6 \n Vertex-Colouring and Complete S\ ubgraphs\n \n\n \n A.7 \n Automorphism groups and iso\ morphism testing for graphs\n \n\n \n \nReference\ s \nIndex \n1 \n The Digraphs package\n \n\ 1.1 Introduction\n\ \n\n   1.1-\ 1 Definitions\n\n \n\n2 Installing Digraphs\n&nbs\ p;2.1 For those in a hurry\n\n \n<\ span class=\"tocline\"> 2.2 Optional package dependenc\ ies\n\n\n<\ br />  2.2-1 The Grape package\n\n \n 2.3 Compiling the kernel module\n\n \n 2.4 Rebui\ lding the documentation\n\n\n   2.4-1 DigraphsMakeDoc\n\ \n \ 2.5 Testing your installation\n\n\n<\ span class=\"ContSS\">   2.5-1 DigraphsTestInstall\n   2.5-2 DigraphsTestStandard\n   2.5-3 DigraphsTestExtreme\n \n \n3 Creating digraphs\n 3.1 Creating digraphs\n\n\n   3.1-1 IsDigraph\n   3.1-2 IsMu\ tableDigraph\n &n\ bsp; 3.1-3 IsImmutable\ Digraph\n &\ nbsp;3.1-4 IsCayleyDigraph<\ /a>\n   3.1-5 IsDigraphWithAdjacencyFu\ nction\n &n\ bsp;3.1-6 DigraphByOutNeigh\ boursType\n \ ; 3.1-7 Digraph\n   3.1-8 DigraphByAdjacencyMatrix\n   <\ a href=\"chap3_mj.html#X7F37B6768349E269\">3.1-9 DigraphByEdges\n   3.1-10 EdgeOrbitsDigraph\n   3.1-11 DigraphByInNeighbours\n   3.1-12 CayleyDigraph\n \n 3.2 Chan\ ging representations\n\n\n   3.2-1 AsBinaryRelation\n   3.2-2 AsDigraph\n   3.2-3 Graph\n   3.2-4 AsGr\ aph\n   \ ;3.2-5 AsTransformation\ \n \n 3.3 New digraphs from old\n\n\n   \ ;3.3-1 DigraphImmutableCopy\ \n   3.3-2 DigraphImmutableCopyIfI\ mmutable\n \ 3.3-3 InducedSubdigra\ ph\n   \ 3.3-4 ReducedDigraph\n   3.3-5 MaximalSymmetricSubdigraph\ \n   3.3-6 MaximalAntiSymmetricSubdigr\ aph\n   \ ;3.3-7 UndirectedSpanningFo\ rest\n &nbs\ p;3.3-8 QuotientDigraph\ \n   3.3-9 DigraphReverse\n\   3.3-10 DigraphDual\n   3.3-11 DigraphSymmetricClosure\n   3.3-12 DigraphTransitiveClosure\n   3.3-13 DigraphTransitiveReduction\n   3.3-14 DigraphAddVertex\n   3.3-15 DigraphAddVertices\n   3.3-16 DigraphAddEdge\n   3.3-17 DigraphAddEdgeOrbit\n  3.3-18 DigraphAddEdges\n   3.3-19 DigraphRemoveVertex\n   3.3-20 DigraphRemoveVertices\n   \ 3.3-21 DigraphRemoveEdge\n   3.3-2\ 2 DigraphRemoveEdgeOrbit\n   3.3-2\ 3 DigraphRemoveEdges\n   3.3-24 Di\ graphRemoveLoops\n   3.3-25 Digra\ phRemoveAllMultipleEdges\n   3.3-2\ 6 DigraphReverseEdges\n   3.3-27 D\ igraphDisjointUnion\n   3.3-28 Dig\ raphEdgeUnion\n &\ nbsp; 3.3-29 DigraphJo\ in\n   \ 3.3-30 DigraphCartesianProd\ uct\n   \ ;3.3-31 DigraphDirectProduc\ t\n   <\ /span>3.3-32 DigraphCartesianProdu\ ctProjections\n &\ nbsp; 3.3-33 DigraphDi\ rectProductProjections\n   3.3-34 \ LineDigraph\n &nb\ sp; 3.3-35 LineUndirec\ tedDigraph\n &nbs\ p; 3.3-36 DoubleDigrap\ h\n   <\ /span>3.3-37 BipartiteDoubleDigrap\ h\n   <\ /span>3.3-38 DigraphAddAllLoops\n   3.3-39 DistanceDigraph\n   3.3-40 DigraphClosure\n   3.3-41 DigraphMycielskian\n \n&nb\ sp;3.4 Random digraphs\n\n\n   3.4-1 RandomDigraph\n   3.4-2 RandomMultiDigraph\n\   3.4-3 RandomTournament\n <\ span class=\"nocss\">  3.4-4 RandomLattice\n \n 3.5 Standard examples\n\n\n   3\ .5-1 ChainDigraph\n   3.5-2 Comple\ teDigraph\n \ ; 3.5-3 CompleteBipart\ iteDigraph\n &nbs\ p; 3.5-4 CompleteMulti\ partiteDigraph\n \   3.5-5 CycleDigr\ aph\n   \ ;3.5-6 EmptyDigraph\n   3.5-7 JohnsonDigraph\n   3.5-8 PetersenGraph\n   3.5-9 GeneralisedPetersenGraph\n\ \n \n\ 4 Operators\n 4.1 Operators for digraphs\n\n\n   \ 4.1-1 IsSubdigraph\n   4.1-2 IsUnd\ irectedSpanningTree\n \n \n\ 5 Attribut\ es and operations\n\ 5.1 Vertices and edges\n\n\n &\ nbsp;5.1-1 DigraphVertices<\ /a>\n   5.1-2 DigraphNrVertices\n   5.1-3 DigraphEdges\n   5.1-4 DigraphNrEdges\n   5.1-5 DigraphNrLoops\n\   5.1-6 DigraphSinks\n   5.1-7 DigraphSources\n   5.1-8\ DigraphTopologicalSort\n   5.1-9\ DigraphVertexLabel\n   5.1-10 Dig\ raphVertexLabels\n   5.1-11 Digra\ phEdgeLabel\n &nb\ sp; 5.1-12 DigraphEdge\ Labels\n &n\ bsp;5.1-13 DigraphInEdges\n   5.1-14 DigraphOutEdges\n   5.1-15 IsDigraphEdge\n   5.1-16 IsMatching\n   5.1-17 DigraphMaximalMatching\n   5.1-18 DigraphMaximumMatching\n \n 5.2 Neig\ hbours and degree\n\n\n   5.2-1 AdjacencyMatrix\n   5.2-2 CharacteristicPolynomial\n   5.2-3 BooleanAdjacencyMatrix\n   5.2-4 DigraphAdjacencyFunction\n   5.2-5 DigraphRange\n   5.2-6 OutNeighbours\n   5.\ 2-7 InNeighbours\n   5.2-8 OutDeg\ rees\n &nbs\ p;5.2-9 InDegrees\n   5.2-10 OutDegreeOfVertex\n   5.2-11 OutNeighboursOfVertex\n   5.2-12 InDegreeOfVertex\n   5.2-13 InNeighboursOfVertex\n   5.2-14 DigraphLoops\n   5.2-15 PartialOrderDigraphMeetOfVertices\n   5.2-16 DegreeMatrix\n   5.2-17 LaplacianMatrix\n \n 5.3 Reachability\ and connectivity\n\n\n   5.3-1 DigraphDiameter\n   5.3-2 DigraphShortestDistance\n   5.3-3 DigraphShortestDistances\n   5.3-4 DigraphLongestDistanceFromVertex\n   5.3-5 DigraphDistanceSet\n   5.3-6 DigraphGirth\n\   5.3-7 DigraphOddGirth\n   5.3-8 DigraphUndirectedGirth\n   5.3-9 DigraphConnectedComponents\n \   5.3-10 DigraphConnectedComponent\n  5.3-11 DigraphStronglyConnectedComponents\n   5.3-12 DigraphStronglyConnectedComponent\n   5.3-13 DigraphBicomponents\n   5.3-14 ArticulationPoints\n   5.3-15 Bridges\n  5.3-16 StrongOrientation\n <\ span class=\"nocss\">  5.3-17 DigraphPeriod\n   5.3\ -18 DigraphFloydWarshall\n   5.3-1\ 9 IsReachable\n &\ nbsp; 5.3-20 VerticesR\ eachableFrom\n &n\ bsp; 5.3-21 DigraphPat\ h\n   <\ /span>5.3-22 DigraphShortestPath\n   5.3-23 IteratorOfPaths\n   5.3-24 DigraphAllSimpleCircuits\n   5.3-25 DigraphLongestSimpleCircuit\n   5.3-26 DigraphLayers\ \n   5.3-27 DigraphDegeneracy\n   5.3-28 DigraphDegeneracyOrdering\n\   5.3-29 HamiltonianPath\n   5.3-30 NrSpanningTrees\n   5.3-31 DigraphDijkstra\n \n 5.4 Cayley g\ raphs of groups\n\n\n   5.4-1 GroupOfCayleyDigraph\n   5.4-2 GeneratorsOfCayleyDigraph\n\ \n \ 5.5 Associated semigroups\n\n\n   5.5-1 AsSemigroup\n   5.5-2 AsSemigroup\n \n 5.6 Planarity\ \n\n\n   5.6-1 KuratowskiPlanarSubdigraph\n <\ span class=\"nocss\">  5.6-2 KuratowskiOuterPlanarSubdigraph\n   5.6-3 PlanarEmbedding\n <\ span class=\"nocss\">  5.6-4 OuterPlanarEmbedding\n   5.6-5 SubdigraphHomeomorphicToK23\n \n \n6 Properties of digraphs\n 6.1 Edge properties\n\ \n\n   6.1-1 Digra\ phHasLoops\n &nbs\ p; 6.1-2 IsAntiSymmetr\ icDigraph\n \ ; 6.1-3 IsBipartiteDig\ raph\n &nbs\ p;6.1-4 IsCompleteBipartite\ Digraph\n &\ nbsp;6.1-5 IsCompleteDigrap\ h\n   <\ /span>6.1-6 IsCompleteMultipartite\ Digraph\n &\ nbsp;6.1-7 IsEmptyDigraph\n   6.1-8 IsEquivalenceDigraph<\ /span>\n   \ 6.1-9 IsFunctionalDigraph\n   6.1-10 IsMultiDigraph\n   6.1-11 IsReflexiveDigraph\n   6.1-12 IsSymmetricDigraph\n   6.1-13 IsTournament\n   6.1-14 IsTransitiveDigraph\n  6.1-15 IsPreorderDigraph\n <\ span class=\"nocss\">  6.1-16 IsPartialOrderDigraph\n   6.1-17 IsMeetSemilatticeDigraph\n \n 6.2 Regularity\n\n\n   6.2-1 IsInRegularDigraph\n   6.\ 2-2 IsOutRegularDigraph\n   6.2-3\ IsRegularDigraph\n   6.2-4 IsDist\ anceRegularDigraph\n \n 6.3 Connectivity and cycles\ \n\n\n   6.\ 3-1 IsAcyclicDigraph\n   6.3-2 IsC\ hainDigraph\n &nb\ sp; 6.3-3 IsConnectedD\ igraph\n &n\ bsp;6.3-4 IsBiconnectedDigr\ aph\n   \ ;6.3-5 IsBridgelessDigraph<\ /a>\n   6.3-6 IsStronglyConnectedDigra\ ph\n   \ 6.3-7 IsAperiodicDigraph\n   6.3-8 IsDirectedTree\ \n   6.3-9 IsUndirectedTree\n   6.3-10 IsEulerianDigraph\n   6.3-11 IsHamiltonianDigraph\n   6.3-12 IsCycleDigraph\n \n <\ a href=\"chap6_mj.html#X7E2305528492DDC0\">6.4 Planari\ ty\n\n\n  6.4-1 IsPlanarDigraph\n   6.4-2 IsOuterPlanarDigraph\n \n 6.5 Homomorphisms and tran\ sformations\n\n\n   6.5-1 IsDigraphCore\n  6.5-2 IsEdgeTransitive\n   6.5-3 IsVertexTransitive\n \n \n7 \ Homomorphisms\n 7.1\ Acting on digraphs\n\n\n   \ ;7.1-1 OnDigraphs\n   7.1-2 OnMultiDigraphs\n \n 7.2 Isomorphisms and canonical labellings\n\n\n &nb\ sp;7.2-1 DigraphsUseNauty\n   7.2-2 AutomorphismGroup\n   7.2-3 BlissAutomorphismGroup\n   7.2-4 NautyAutomorphismGroup\n   7.2-5 AutomorphismGroup\n   7.2-6 AutomorphismGroup\n   7.2-7 BlissCanonicalLabelling\n   7.2-8 BlissCanonicalLabelling\n   7.2-9 BlissCanonicalDigraph\n   7.2-10 DigraphGroup\n   7.2-11 DigraphOrbits\n \   7.2-12 DigraphOrbitReps\n   7.2-13 DigraphSchreierVector\n   \ 7.2-14 DigraphStabilizer\n   7.2-1\ 5 IsIsomorphicDigraph\n   7.2-16 I\ sIsomorphicDigraph\n   7.2-17 Isom\ orphismDigraphs\n   7.2-18 Isomorp\ hismDigraphs\n &n\ bsp; 7.2-19 Representa\ tiveOutNeighbours\n   7.2-20 IsDig\ raphIsomorphism\n   7.2-21 IsDigra\ phColouring\n \n 7.3 Homomorphisms of digraphs\n\n\n   7.3-1 H\ omomorphismDigraphsFinder\n   7.3-\ 2 DigraphHomomorphism\n   7.3-3 Ho\ momorphismsDigraphs\n   7.3-4 Digr\ aphMonomorphism\n   7.3-5 Monomorp\ hismsDigraphs\n &\ nbsp; 7.3-6 DigraphEpi\ morphism\n \ 7.3-7 EpimorphismsDig\ raphs\n &nb\ sp;7.3-8 DigraphEmbedding\n   7.3-9 EmbeddingsDigraphs\n   7.3-10 IsDigraphHomomorphism\n   7.3-11 IsDigraphEmbedding\ \n   7.3-12 DigraphsRespectsColouring\n   7.3-13 GeneratorsOfEndomorphismMonoid\ \n   7.3-14 DigraphColouring\n   <\ a href=\"chap7_mj.html#X7AB7200D831013C1\">7.3-15 DigraphGreedyColouring\n   <\ a href=\"chap7_mj.html#X7F9CB3B27B9590DB\">7.3-16 DigraphWelshPowellOrder<\ /span>\n   \ 7.3-17 ChromaticNumber\ \n   7.3-18 DigraphCore\n \n \n8 Finding cliques and independent sets\n\ 8.1 F\ inding cliques\n\n\n   8.1-1 IsClique\n   8.1-2 CliquesFinder\n   8\ .1-3 DigraphClique\n   8.1-4 Digra\ phMaximalCliques\n   8.1-5 Clique\ Number\n \n 8.2 Finding independent sets\n\n\ \n   8.2-1 IsInde\ pendentSet\n &nbs\ p; 8.2-2 DigraphIndepe\ ndentSet\n \ 8.2-3 DigraphMaximalI\ ndependentSets\n \n \n9 Visualising a\ nd IO\n 9.1 Visualising a digraph\n\n\n   9.1-1 Splash\n   9.1-2 DotDigraph\n   9.1-3 DotSymmetricDigraph\n\   9.1-4 DotPartialOrderDigraph\n<\ br />  9.1-5 DotPreorderDigraph\n \   9.1-6 DotHighlightedDigraph\n \n 9.2 Reading and writin\ g digraphs to a file\n\n\n   9.2-1 String\n<\ br />  9.2-2 DigraphFromGraph6String\n<\ br />  9.2-3 Graph6String\n   9.2-4 DigraphFile\n   9.2-5 ReadD\ igraphs\n &\ nbsp;9.2-6 WriteDigraphs\n   9.2-7 IteratorFromDigraphFile\n   9.2-8 DigraphPlainTextLineEncode\ r\n   <\ /span>9.2-9 TournamentLineDecoder<\ /a>\n   9.2-10 AdjacencyMatrixUpperTri\ angleLineDecoder\n   9.2-11 TCode\ Decoder\n &\ nbsp;9.2-12 PlainTextString\ \n   9.2-13 WritePlainTextDigraph<\ /a>\n   9.2-14 WriteDIMACSDigraph<\ /span>\n \n \nA \n Grape to Digraphs Command \ Map\n \n A.1 \n Functions to construct and modify graphs\n \n\n \n A.2 \ \n Functions to inspect graphs, vertices and edge\ s\n \n\n \n A.3 \n Functions to determine regularity \ properties of graphs\n \n\n \n A.4 \n Some special v\ ertex subsets of a graph\n \n\n \n A.5 \n Functions t\ o construct new graphs from old\n \n\n \n A.6 \n Vert\ ex-Colouring and Complete Subgraphs\n \n\n \n A.7 \n \ Automorphism groups and isomorphism testing for graphs\n \n\n \n \nReferences \nIndex\n" ) gap> gap> ───────────────────────────────────────────────────────────────────────────── Loading datastructures 0.2.5 (datastructures - GAP Data Structures) by Markus Pfeiffer (http://www.morphism.de/~markusp), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/), and Steve Linton (http://sl4.host.cs.st-andrews.ac.uk/). Homepage: https://gap-packages.github.io/datastructures Report issues at https://github.com/gap-packages/datastructures/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading NautyTracesInterface 0.2 (An interface to nauty) by Sebastian Gutsche (https://sebasguts.github.io), Alice Niemeyer (http://www.maths.uwa.edu.au/~alice/), and Pascal Schweitzer (https://lii.rwth-aachen.de/~schweitzer). Homepage: http://TODO/ ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading orb 4.8.3 (Methods to enumerate orbits) by Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef), and Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske). maintained by: Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske), and Max Horn (https://www.quendi.de/math). Homepage: https://gap-packages.github.io/orb Report issues at https://github.com/gap-packages/orb/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.8.3 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~lsoicher/). Homepage: https://gap-packages.github.io/grape Report issues at https://github.com/gap-packages/grape/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading Digraphs 1.4.0 (Digraphs - Methods for digraphs) by Jan De Beule (http://homepages.vub.ac.be/~jdbeule/), Julius Jonusas (http://julius.jonusas.work), James Mitchell (https://jdbm.me), Michael Torpey (https://mtorpey.github.io), Maria Tsalakou (https://mariatsalakou.github.io/), and Wilf A. Wilson (https://wilf.me). with contributions by: Stuart Burrell (sb235@st-andrews.ac.uk), Reinis Cirpons (rc234@st-andrews.ac.uk), Luke Elliott (le27@st-andrews.ac.uk), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk), Markus Pfeiffer (https://www.morphism.de/~markusp), Christopher Russell (cr66@st-andrews.ac.uk), Finn Smith (fls3@st-andrews.ac.uk), and Murray Whyte (mw231@st-andrews.ac.uk). maintained by: James Mitchell (https://jdbm.me) and Wilf A. Wilson (https://wilf.me). Homepage: https://digraphs.github.io/Digraphs Report issues at https://github.com/digraphs/Digraphs/issues ───────────────────────────────────────────────────────────────────────────── #I File: /builddir/build/BUILD/digraphs-1.4.0/../pkg/digraphs-1.4.0/doc/manual.lab written. gap> gap> Executing(%install): /bin/sh -e /var/tmp/rpm-tmp.Jcbf90 explicitly decompress any DWARF compressed ELF sections in /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/lib/gap/pkg/digraphs-1.4.0/bin/x86_64-redhat-linux-gnu-default64-kv7/digraphs.so extracting debug info from /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/lib/gap/pkg/digraphs-1.4.0/bin/x86_64-redhat-linux-gnu-default64-kv7/digraphs.so + rm -fr ../pkg + RPM_EC=0 ++ jobs -p + exit 0 + umask 022 + cd /builddir/build/BUILD + '[' /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64 '!=' / ']' + rm -rf /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64 ++ dirname /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64 + mkdir -p /builddir/build/BUILDROOT + mkdir /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64 + cd digraphs-1.4.0 + mkdir -p /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/lib/gap/pkg/digraphs-1.4.0/doc + cp -a bin data gap tst VERSIONS PackageInfo.g init.g makedoc.g read.g /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/lib/gap/pkg/digraphs-1.4.0 + cp -p doc/digraphs.bib doc/lefttoc.css doc/manual.css doc/nocolorprompt.css doc/ragged.css doc/times.css doc/toggless.css doc/chap0.html doc/chap0_mj.html doc/chap1.html doc/chap1_mj.html doc/chap2.html doc/chap2_mj.html doc/chap3.html doc/chap3_mj.html doc/chap4.html doc/chap4_mj.html doc/chap5.html doc/chap5_mj.html doc/chap6.html doc/chap6_mj.html doc/chap7.html doc/chap7_mj.html doc/chap8.html doc/chap8_mj.html doc/chap9.html doc/chap9_mj.html doc/chapA.html doc/chapA_mj.html doc/chapBib.html doc/chapBib_mj.html doc/chapInd.html doc/chapInd_mj.html doc/chooser.html doc/manual.js doc/rainbow.js doc/toggless.js doc/manual.lab doc/manual.pdf doc/manual.six doc/chap0.txt doc/chap1.txt doc/chap2.txt doc/chap3.txt doc/chap4.txt doc/chap5.txt doc/chap6.txt doc/chap7.txt doc/chap8.txt doc/chap9.txt doc/chapA.txt doc/chapBib.txt doc/chapInd.txt doc/attr.xml doc/cliques.xml doc/constructors.xml doc/digraph.xml doc/display.xml doc/examples.xml doc/grahom.xml doc/grape.xml doc/io.xml doc/isomorph.xml doc/labels.xml doc/main.xml doc/oper.xml doc/orbits.xml doc/planar.xml doc/prop.xml doc/utils.xml doc/z-appA.xml doc/z-chap0.xml doc/z-chap1.xml doc/z-chap2.xml doc/z-chap3.xml doc/z-chap4.xml doc/z-chap5.xml doc/z-chap6.xml doc/z-chap8.xml doc/z-chap9.xml doc/z-title.xml /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/lib/gap/pkg/digraphs-1.4.0/doc + /usr/lib/rpm/find-debuginfo.sh -j2 --strict-build-id -m -i --build-id-seed 1.4.0-1.fc34 --unique-debug-suffix -1.4.0-1.fc34.x86_64 --unique-debug-src-base gap-pkg-digraphs-1.4.0-1.fc34.x86_64 --run-dwz --dwz-low-mem-die-limit 10000000 --dwz-max-die-limit 110000000 -S debugsourcefiles.list /builddir/build/BUILD/digraphs-1.4.0 original debug info size: 2664kB, size after compression: 2516kB /usr/lib/rpm/sepdebugcrcfix: Updated 1 CRC32s, 0 CRC32s did match. 1036 blocks + /usr/lib/rpm/check-buildroot + /usr/lib/rpm/redhat/brp-ldconfig + /usr/lib/rpm/brp-compress + /usr/lib/rpm/redhat/brp-strip-lto /usr/bin/strip + /usr/lib/rpm/brp-strip-static-archive /usr/bin/strip + /usr/lib/rpm/redhat/brp-python-bytecompile '' 1 0 + /usr/lib/rpm/brp-python-hardlink + /usr/lib/rpm/redhat/brp-mangle-shebangs *** WARNING: ./usr/src/debug/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/extern/bliss-0.73/graph.cc is executable but has no shebang, removing executable bit Executing(%check): /bin/sh -e /var/tmp/rpm-tmp.8Usw7z + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.4.0 + export LC_ALL=C.UTF-8 + LC_ALL=C.UTF-8 + cd tst + gap -l '/builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/lib/gap;/usr/lib/gap' ┌───────┐ GAP 4.11.0 of 29-Feb-2020 │ GAP │ https://www.gap-system.org └───────┘ Architecture: x86_64-redhat-linux-gnu-default64-kv7 Configuration: gmp 6.2.0, GASMAN, readline Loading the library and packages ... #I autpgrp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I alnuth package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I crisp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I ctbllib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I factint package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I fga package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I irredsol package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I laguna package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polenta package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polycyclic package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I resclasses package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I sophus package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I tomlib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') Packages: GAPDoc 1.6.4, IO 4.7.0, PrimGrp 3.4.1, SmallGrp 1.4.2, TransGrp 3.0 Try '??help' for help. See also '?copyright', '?cite' and '?authors' gap> ───────────────────────────────────────────────────────────────────────────── Loading datastructures 0.2.5 (datastructures - GAP Data Structures) by Markus Pfeiffer (http://www.morphism.de/~markusp), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/), and Steve Linton (http://sl4.host.cs.st-andrews.ac.uk/). Homepage: https://gap-packages.github.io/datastructures Report issues at https://github.com/gap-packages/datastructures/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading NautyTracesInterface 0.2 (An interface to nauty) by Sebastian Gutsche (https://sebasguts.github.io), Alice Niemeyer (http://www.maths.uwa.edu.au/~alice/), and Pascal Schweitzer (https://lii.rwth-aachen.de/~schweitzer). Homepage: http://TODO/ ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading orb 4.8.3 (Methods to enumerate orbits) by Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef), and Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske). maintained by: Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske), and Max Horn (https://www.quendi.de/math). Homepage: https://gap-packages.github.io/orb Report issues at https://github.com/gap-packages/orb/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.8.3 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~lsoicher/). Homepage: https://gap-packages.github.io/grape Report issues at https://github.com/gap-packages/grape/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading Digraphs 1.4.0 (Digraphs - Methods for digraphs) by Jan De Beule (http://homepages.vub.ac.be/~jdbeule/), Julius Jonusas (http://julius.jonusas.work), James Mitchell (https://jdbm.me), Michael Torpey (https://mtorpey.github.io), Maria Tsalakou (https://mariatsalakou.github.io/), and Wilf A. Wilson (https://wilf.me). with contributions by: Stuart Burrell (sb235@st-andrews.ac.uk), Reinis Cirpons (rc234@st-andrews.ac.uk), Luke Elliott (le27@st-andrews.ac.uk), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk), Markus Pfeiffer (https://www.morphism.de/~markusp), Christopher Russell (cr66@st-andrews.ac.uk), Finn Smith (fls3@st-andrews.ac.uk), and Murray Whyte (mw231@st-andrews.ac.uk). maintained by: James Mitchell (https://jdbm.me) and Wilf A. Wilson (https://wilf.me). Homepage: https://digraphs.github.io/Digraphs Report issues at https://github.com/digraphs/Digraphs/issues ───────────────────────────────────────────────────────────────────────────── true gap> Digraphs package: testinstall.tst msecs: 293 true gap> Digraphs package: standard/attr.tst msecs: 2194 true gap> Digraphs package: standard/cliques.tst msecs: 522 true gap> Digraphs package: standard/constructors.tst msecs: 33 true gap> Digraphs package: standard/display.tst msecs: 17 true gap> Digraphs package: standard/examples.tst msecs: 11 true gap> Digraphs package: standard/grahom.tst msecs: 430 true gap> Digraphs package: standard/grape.tst msecs: 10 true gap> Digraphs package: standard/io.tst msecs: 2319 true gap> Digraphs package: standard/isomorph.tst msecs: 342 true gap> Digraphs package: standard/labels.tst msecs: 4 true gap> Digraphs package: standard/oper.tst msecs: 507 true gap> Digraphs package: standard/orbits.tst msecs: 2 true gap> Digraphs package: standard/planar.tst msecs: 31 true gap> Digraphs package: standard/prop.tst msecs: 2320 true gap> gap> + gap -l '/builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/lib/gap;/usr/lib/gap' ┌───────┐ GAP 4.11.0 of 29-Feb-2020 │ GAP │ https://www.gap-system.org └───────┘ Architecture: x86_64-redhat-linux-gnu-default64-kv7 Configuration: gmp 6.2.0, GASMAN, readline Loading the library and packages ... #I autpgrp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I alnuth package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I crisp package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I ctbllib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I factint package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I fga package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I irredsol package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I laguna package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polenta package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I polycyclic package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I resclasses package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I sophus package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') #I tomlib package is not available. Check that the name is correct #I and it is present in one of the GAP root directories (see '??RootPaths') Packages: GAPDoc 1.6.4, IO 4.7.0, PrimGrp 3.4.1, SmallGrp 1.4.2, TransGrp 3.0 Try '??help' for help. See also '?copyright', '?cite' and '?authors' gap> ───────────────────────────────────────────────────────────────────────────── Loading datastructures 0.2.5 (datastructures - GAP Data Structures) by Markus Pfeiffer (http://www.morphism.de/~markusp), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk/), and Steve Linton (http://sl4.host.cs.st-andrews.ac.uk/). Homepage: https://gap-packages.github.io/datastructures Report issues at https://github.com/gap-packages/datastructures/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading NautyTracesInterface 0.2 (An interface to nauty) by Sebastian Gutsche (https://sebasguts.github.io), Alice Niemeyer (http://www.maths.uwa.edu.au/~alice/), and Pascal Schweitzer (https://lii.rwth-aachen.de/~schweitzer). Homepage: http://TODO/ ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading orb 4.8.3 (Methods to enumerate orbits) by Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Max Neunhöffer (http://www-groups.mcs.st-and.ac.uk/~neunhoef), and Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske). maintained by: Juergen Mueller (http://www.math.rwth-aachen.de/~Juergen.Mueller), Felix Noeske (http://www.math.rwth-aachen.de/~Felix.Noeske), and Max Horn (https://www.quendi.de/math). Homepage: https://gap-packages.github.io/orb Report issues at https://github.com/gap-packages/orb/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading GRAPE 4.8.3 (GRaph Algorithms using PErmutation groups) by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~lsoicher/). Homepage: https://gap-packages.github.io/grape Report issues at https://github.com/gap-packages/grape/issues ───────────────────────────────────────────────────────────────────────────── ───────────────────────────────────────────────────────────────────────────── Loading Digraphs 1.4.0 (Digraphs - Methods for digraphs) by Jan De Beule (http://homepages.vub.ac.be/~jdbeule/), Julius Jonusas (http://julius.jonusas.work), James Mitchell (https://jdbm.me), Michael Torpey (https://mtorpey.github.io), Maria Tsalakou (https://mariatsalakou.github.io/), and Wilf A. Wilson (https://wilf.me). with contributions by: Stuart Burrell (sb235@st-andrews.ac.uk), Reinis Cirpons (rc234@st-andrews.ac.uk), Luke Elliott (le27@st-andrews.ac.uk), Max Horn (https://www.quendi.de/math), Christopher Jefferson (http://caj.host.cs.st-andrews.ac.uk), Markus Pfeiffer (https://www.morphism.de/~markusp), Christopher Russell (cr66@st-andrews.ac.uk), Finn Smith (fls3@st-andrews.ac.uk), and Murray Whyte (mw231@st-andrews.ac.uk). maintained by: James Mitchell (https://jdbm.me) and Wilf A. Wilson (https://wilf.me). Homepage: https://digraphs.github.io/Digraphs Report issues at https://github.com/digraphs/Digraphs/issues ───────────────────────────────────────────────────────────────────────────── true gap> Digraphs package: standard/digraph.tst msecs: 4359 gap> + find /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/lib/gap/pkg/digraphs-1.4.0 -size 0 -delete + RPM_EC=0 ++ jobs -p + exit 0 Processing files: gap-pkg-digraphs-1.4.0-1.fc34.x86_64 Executing(%doc): /bin/sh -e /var/tmp/rpm-tmp.0gEaYY + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.4.0 + DOCDIR=/builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/share/doc/gap-pkg-digraphs + export LC_ALL=C + LC_ALL=C + export DOCDIR + /usr/bin/mkdir -p /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/share/doc/gap-pkg-digraphs + cp -pr CHANGELOG.md /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/share/doc/gap-pkg-digraphs + cp -pr README.md /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/share/doc/gap-pkg-digraphs + RPM_EC=0 ++ jobs -p + exit 0 Executing(%license): /bin/sh -e /var/tmp/rpm-tmp.h0ysFt + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.4.0 + LICENSEDIR=/builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/share/licenses/gap-pkg-digraphs + export LC_ALL=C + LC_ALL=C + export LICENSEDIR + /usr/bin/mkdir -p /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/share/licenses/gap-pkg-digraphs + cp -pr GPL /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/share/licenses/gap-pkg-digraphs + cp -pr LICENSE /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64/usr/share/licenses/gap-pkg-digraphs + RPM_EC=0 ++ jobs -p + exit 0 Provides: bundled(bliss) = 0.73 gap-pkg-digraphs = 1.4.0-1.fc34 gap-pkg-digraphs(x86-64) = 1.4.0-1.fc34 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Requires: libc.so.6()(64bit) libc.so.6(GLIBC_2.11)(64bit) libc.so.6(GLIBC_2.14)(64bit) libc.so.6(GLIBC_2.2.5)(64bit) libc.so.6(GLIBC_2.3.4)(64bit) libc.so.6(GLIBC_2.4)(64bit) libc.so.6(GLIBC_2.7)(64bit) libgcc_s.so.1()(64bit) libgcc_s.so.1(GCC_3.0)(64bit) libgcc_s.so.1(GCC_3.3.1)(64bit) libm.so.6()(64bit) libplanarity.so.0()(64bit) libpthread.so.0()(64bit) libstdc++.so.6()(64bit) libstdc++.so.6(CXXABI_1.3)(64bit) libstdc++.so.6(CXXABI_1.3.9)(64bit) libstdc++.so.6(GLIBCXX_3.4)(64bit) libstdc++.so.6(GLIBCXX_3.4.29)(64bit) rtld(GNU_HASH) Recommends: gap-pkg-grape(x86-64) gap-pkg-nautytracesinterface(x86-64) Processing files: gap-pkg-digraphs-doc-1.4.0-1.fc34.noarch Provides: gap-pkg-digraphs-doc = 1.4.0-1.fc34 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Processing files: gap-pkg-digraphs-debugsource-1.4.0-1.fc34.x86_64 Provides: gap-pkg-digraphs-debugsource = 1.4.0-1.fc34 gap-pkg-digraphs-debugsource(x86-64) = 1.4.0-1.fc34 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Processing files: gap-pkg-digraphs-debuginfo-1.4.0-1.fc34.x86_64 Provides: debuginfo(build-id) = b49b939219840361f967d35c243057a9e5040aa3 gap-pkg-digraphs-debuginfo = 1.4.0-1.fc34 gap-pkg-digraphs-debuginfo(x86-64) = 1.4.0-1.fc34 Requires(rpmlib): rpmlib(CompressedFileNames) <= 3.0.4-1 rpmlib(FileDigests) <= 4.6.0-1 rpmlib(PayloadFilesHavePrefix) <= 4.0-1 Recommends: gap-pkg-digraphs-debugsource(x86-64) = 1.4.0-1.fc34 Checking for unpackaged file(s): /usr/lib/rpm/check-files /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64 Wrote: /builddir/build/RPMS/gap-pkg-digraphs-debuginfo-1.4.0-1.fc34.x86_64.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-doc-1.4.0-1.fc34.noarch.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-1.4.0-1.fc34.x86_64.rpm Wrote: /builddir/build/RPMS/gap-pkg-digraphs-debugsource-1.4.0-1.fc34.x86_64.rpm Executing(%clean): /bin/sh -e /var/tmp/rpm-tmp.bLY2b5 + umask 022 + cd /builddir/build/BUILD + cd digraphs-1.4.0 + /usr/bin/rm -rf /builddir/build/BUILDROOT/gap-pkg-digraphs-1.4.0-1.fc34.x86_64 + RPM_EC=0 ++ jobs -p + exit 0 Finish: rpmbuild gap-pkg-digraphs-1.4.0-1.fc34.src.rpm Finish: build phase for gap-pkg-digraphs-1.4.0-1.fc34.src.rpm INFO: chroot_scan: 3 files copied to /var/lib/copr-rpmbuild/results/chroot_scan INFO: /var/lib/mock/fedora-rawhide-x86_64-1611958039.900791/root/var/log/dnf.rpm.log /var/lib/mock/fedora-rawhide-x86_64-1611958039.900791/root/var/log/dnf.librepo.log /var/lib/mock/fedora-rawhide-x86_64-1611958039.900791/root/var/log/dnf.log INFO: Done(/var/lib/copr-rpmbuild/results/gap-pkg-digraphs-1.4.0-1.fc34.src.rpm) Config(child) 3 minutes 8 seconds INFO: Results and/or logs in: /var/lib/copr-rpmbuild/results INFO: Cleaning up build root ('cleanup_on_success=True') Start: clean chroot INFO: unmounting tmpfs. Finish: clean chroot Finish: run INFO: unmounting tmpfs.