  
  
                                    [1X[5XDigraphs[105X[101X
  
  
                              [1XMethods for digraphs[101X
  
  
                                  Version 0.5
  
  
                                  Jan De Beule
  
                                 Julius Jonušas
  
                               James D. Mitchell
  
                                 Michael Torpey
  
                                  Wilf Wilson
  
  
  
  Jan De Beule
      Email:    [7Xmailto:jdebeule@cage.ugent.be[107X
      Homepage: [7Xhttp://homepages.vub.ac.be/~jdbeule/[107X
  Julius Jonušas
      Email:    [7Xmailto:jj252@st-and.ac.uk[107X
      Homepage: [7Xhttp://www-circa.mcs.st-and.ac.uk/~julius[107X
  James D. Mitchell
      Email:    [7Xmailto:jdm3@st-and.ac.uk[107X
      Homepage: [7Xhttp://tinyurl.com/jdmitchell[107X
  Michael Torpey
      Email:    [7Xmailto:mct25@st-and.ac.uk[107X
      Homepage: [7Xhttp://www-circa.mcs.st-and.ac.uk/~mct25[107X
  Wilf Wilson
      Email:    [7Xmailto:waw7@st-and.ac.uk[107X
      Homepage: [7Xhttp://www-circa.mcs.st-and.ac.uk/~waw7[107X
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe  [5XDigraphs[105X  package  is  a  [5XGAP[105X  package  containing  methods for graphs,
  digraphs, and multidigraphs.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2014-16 by Julius Jonušas, J.D. Mitchell, Michael Torpey, Wilf Wilson.[133X
  
  [33X[0;0Y[5XDigraphs[105X  is  free  software; you can redistribute it and/or modify it under
  the      terms      of      the      GNU      General     Public     License
  ([7Xhttp://www.fsf.org/licenses/gpl.html[107X)  as  published  by  the Free Software
  Foundation;  either  version 3 of the License, or (at your option) any later
  version.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YWe  would  like to thank Chris Jefferson for his help in including the [5Xbliss[105X
  tool   in   the  package.  This  package's  methods  for  computing  digraph
  homomorphisms  are  based on work by Max Neunhöffer, and independently Artur
  Schäfer.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (digraphs)[101X
  
  1 [33X[0;0YThe [5XDigraphs[105X package[133X
    1.1 [33X[0;0YIntroduction[133X
  2 [33X[0;0YInstalling [5XDigraphs[105X[133X
    2.1 [33X[0;0YFor those in a hurry[133X
    2.2 [33X[0;0YOptional package dependencies[133X
      2.2-1 [33X[0;0YThe Grape package[133X
    2.3 [33X[0;0YCompiling the kernel module[133X
    2.4 [33X[0;0YRebuilding the documentation[133X
      2.4-1 DigraphsMakeDoc
    2.5 [33X[0;0YTesting your installation[133X
      2.5-1 DigraphsTestInstall
      2.5-2 DigraphsTestStandard
  3 [33X[0;0YCreating digraphs[133X
    3.1 [33X[0;0YCreating digraphs[133X
      3.1-1 IsDigraph
      3.1-2 IsCayleyDigraph
      3.1-3 IsDigraphWithAdjacencyFunction
      3.1-4 DigraphType
      3.1-5 Digraph
      3.1-6 DigraphByAdjacencyMatrix
      3.1-7 DigraphByEdges
      3.1-8 EdgeOrbitsDigraph
      3.1-9 DigraphByInNeighbours
    3.2 [33X[0;0YChanging representations[133X
      3.2-1 AsBinaryRelation
      3.2-2 AsDigraph
      3.2-3 Graph
      3.2-4 AsGraph
      3.2-5 AsTransformation
    3.3 [33X[0;0YNew digraphs from old[133X
      3.3-1 DigraphCopy
      3.3-2 InducedSubdigraph
      3.3-3 ReducedDigraph
      3.3-4 MaximalSymmetricSubdigraph
      3.3-5 QuotientDigraph
      3.3-6 DigraphReverse
      3.3-7 DigraphDual
      3.3-8 DigraphSymmetricClosure
      3.3-9 DigraphReflexiveTransitiveClosure
      3.3-10 DigraphReflexiveTransitiveReduction
      3.3-11 DigraphAddVertex
      3.3-12 DigraphAddVertices
      3.3-13 DigraphAddEdge
      3.3-14 DigraphAddEdgeOrbit
      3.3-15 DigraphAddEdges
      3.3-16 DigraphRemoveVertex
      3.3-17 DigraphRemoveVertices 
      3.3-18 DigraphRemoveEdge
      3.3-19 DigraphRemoveEdgeOrbit
      3.3-20 DigraphRemoveEdges
      3.3-21 DigraphRemoveLoops
      3.3-22 DigraphRemoveAllMultipleEdges
      3.3-23 DigraphReverseEdges
      3.3-24 DigraphDisjointUnion
      3.3-25 DigraphEdgeUnion
      3.3-26 DigraphJoin
      3.3-27 LineDigraph
      3.3-28 LineUndirectedDigraph
      3.3-29 DoubleDigraph
      3.3-30 BipartiteDoubleDigraph
      3.3-31 DigraphAddAllLoops
      3.3-32 DistanceDigraph
    3.4 [33X[0;0YRandom digraphs[133X
      3.4-1 RandomDigraph
      3.4-2 RandomMultiDigraph
      3.4-3 RandomTournament
    3.5 [33X[0;0YStandard examples[133X
      3.5-1 ChainDigraph
      3.5-2 CompleteDigraph
      3.5-3 CompleteBipartiteDigraph
      3.5-4 CycleDigraph
      3.5-5 EmptyDigraph
      3.5-6 CayleyDigraph
      3.5-7 JohnsonDigraph
  4 [33X[0;0YOperators[133X
    4.1 [33X[0;0YOperators for digraphs[133X
  5 [33X[0;0YAttributes and operations[133X
    5.1 [33X[0;0YVertices and edges[133X
      5.1-1 DigraphVertices
      5.1-2 DigraphNrVertices
      5.1-3 DigraphEdges
      5.1-4 DigraphNrEdges
      5.1-5 DigraphSinks
      5.1-6 DigraphSources
      5.1-7 DigraphTopologicalSort
      5.1-8 DigraphBicomponents
      5.1-9 DigraphVertexLabel
      5.1-10 DigraphVertexLabels
      5.1-11 DigraphInEdges
      5.1-12 DigraphOutEdges
      5.1-13 IsDigraphEdge
    5.2 [33X[0;0YNeighbours and degree[133X
      5.2-1 AdjacencyMatrix
      5.2-2 BooleanAdjacencyMatrix
      5.2-3 DigraphAdjacencyFunction
      5.2-4 DigraphRange
      5.2-5 OutNeighbours
      5.2-6 InNeighbours
      5.2-7 OutDegrees
      5.2-8 InDegrees
      5.2-9 OutDegreeOfVertex
      5.2-10 OutNeighboursOfVertex
      5.2-11 InDegreeOfVertex
      5.2-12 InNeighboursOfVertex
      5.2-13 DigraphLoops
    5.3 [33X[0;0YReachability and connectivity[133X
      5.3-1 DigraphDiameter
      5.3-2 DigraphShortestDistance
      5.3-3 DigraphShortestDistances
      5.3-4 DigraphLongestDistanceFromVertex
      5.3-5 DigraphDistanceSet
      5.3-6 DigraphGirth
      5.3-7 DigraphUndirectedGirth
      5.3-8 DigraphConnectedComponents
      5.3-9 DigraphConnectedComponent
      5.3-10 DigraphStronglyConnectedComponents
      5.3-11 DigraphStronglyConnectedComponent
      5.3-12 DigraphPeriod
      5.3-13 DigraphFloydWarshall
      5.3-14 IsReachable
      5.3-15 DigraphPath
      5.3-16 IteratorOfPaths
      5.3-17 DigraphAllSimpleCircuits
      5.3-18 DigraphLongestSimpleCircuit
      5.3-19 DigraphLayers
      5.3-20 DigraphDegeneracy
      5.3-21 DigraphDegeneracyOrdering
  6 [33X[0;0YProperties of digraphs[133X
    6.1 [33X[0;0YEdge properties[133X
      6.1-1 DigraphHasLoops
      6.1-2 IsAntisymmetricDigraph
      6.1-3 IsBipartiteDigraph
      6.1-4 IsCompleteBipartiteDigraph
      6.1-5 IsCompleteDigraph
      6.1-6 IsEmptyDigraph
      6.1-7 IsFunctionalDigraph
      6.1-8 IsMultiDigraph
      6.1-9 IsReflexiveDigraph
      6.1-10 IsSymmetricDigraph
      6.1-11 IsTournament
      6.1-12 IsTransitiveDigraph
    6.2 [33X[0;0YRegularity[133X
      6.2-1 IsInRegularDigraph
      6.2-2 IsOutRegularDigraph
      6.2-3 IsRegularDigraph
      6.2-4 IsDistanceRegularDigraph
    6.3 [33X[0;0YConnectivity and cycles[133X
      6.3-1 IsAcyclicDigraph
      6.3-2 IsConnectedDigraph
      6.3-3 IsStronglyConnectedDigraph
      6.3-4 IsAperiodicDigraph
  7 [33X[0;0YHomomorphisms[133X
    7.1 [33X[0;0YActing on digraphs[133X
      7.1-1 OnDigraphs
      7.1-2 OnMultiDigraphs
    7.2 [33X[0;0YIsomorphisms, and Canonical labellings[133X
      7.2-1 AutomorphismGroup
      7.2-2 AutomorphismGroup
      7.2-3 DigraphGroup
      7.2-4 DigraphSchreierVector
      7.2-5 DigraphOrbitReps
      7.2-6 DigraphStabilizer
      7.2-7 DigraphOrbits
      7.2-8 RepresentativeOutNeighbours
      7.2-9 DigraphCanonicalLabelling
      7.2-10 IsIsomorphicDigraph
      7.2-11 IsomorphismDigraphs
    7.3 [33X[0;0YHomomorphisms of digraphs[133X
      7.3-1 HomomorphismDigraphsFinder
      7.3-2 DigraphHomomorphism
      7.3-3 HomomorphismsDigraphs
      7.3-4 DigraphMonomorphism
      7.3-5 MonomorphismsDigraphs
      7.3-6 DigraphEpimorphism
      7.3-7 EpimorphismsDigraphs
      7.3-8 GeneratorsOfEndomorphismMonoid
      7.3-9 DigraphColoring
      7.3-10 DigraphEmbedding
  8 [33X[0;0YFinding cliques and independent sets[133X
    8.1 [33X[0;0YFinding cliques[133X
      8.1-1 IsClique
      8.1-2 CliquesFinder
      8.1-3 DigraphClique
      8.1-4 DigraphMaximalCliques
    8.2 [33X[0;0YFinding independent sets[133X
      8.2-1 IsIndependentSet
      8.2-2 DigraphIndependentSet
      8.2-3 DigraphMaximalIndependentSets
  9 [33X[0;0YVisualising and IO[133X
    9.1 [33X[0;0YVisualising a digraph[133X
      9.1-1 Splash
      9.1-2 DotDigraph
      9.1-3 DotSymmetricDigraph
    9.2 [33X[0;0YReading and writing graphs to a file[133X
      9.2-1 DigraphFromGraph6String
      9.2-2 Graph6String
      9.2-3 DigraphFile
      9.2-4 ReadDigraphs
      9.2-5 WriteDigraphs
      9.2-6 IteratorFromDigraphFile
      9.2-7 DigraphPlainTextLineEncoder
      9.2-8 TournamentLineDecoder
      9.2-9 AdjacencyMatrixUpperTriangleLineDecoder
      9.2-10 TCodeDecoder
      9.2-11 PlainTextString
      9.2-12 WritePlainTextDigraph
  A [33X[0;0YGrape to Digraphs Command Map[133X
    A.1 [33X[0;0YFunctions to construct and modify graphs[133X
    A.2 [33X[0;0YFunctions to inspect graphs, vertices and edges[133X
    A.3 [33X[0;0YFunctions to determine regularity properties of graphs[133X
    A.4 [33X[0;0YSome special vertex subsets of a graph[133X
    A.5 [33X[0;0YFunctions to construct new graphs from old[133X
    A.6 [33X[0;0YVertex-Colouring and Complete Subgraphs[133X
    A.7 [33X[0;0YAutomorphism groups and isomorphism testing for graphs[133X
  
  
  [32X
