  
  [1X6 [33X[0;0YProperties of digraphs[133X[101X
  
  
  [1X6.1 [33X[0;0YEdge properties[133X[101X
  
  [1X6.1-1 DigraphHasLoops[101X
  
  [33X[1;0Y[29X[2XDigraphHasLoops[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YReturns  [9Xtrue[109X  if the digraph [3Xdigraph[103X has loops, and [9Xfalse[109X if it does not. A
  loop is an edge with equal source and range.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 2], [2]]);[127X[104X
    [4X[28X<digraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr);[127X[104X
    [4X[28X[ [ 1, 1 ], [ 1, 2 ], [ 2, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphHasLoops(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 3], [1], [2]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr);[127X[104X
    [4X[28X[ [ 1, 2 ], [ 1, 3 ], [ 2, 1 ], [ 3, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XDigraphHasLoops(gr);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.1-2 IsAntisymmetricDigraph[101X
  
  [33X[1;0Y[29X[2XIsAntisymmetricDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  property is [9Xtrue[109X if the digraph [3Xdigraph[103X is antisymmetric, and [9Xfalse[109X if
  it is not.[133X
  
  [33X[0;0YA  digraph  is  [13Xantisymmetric[113X if whenever there is an edge with source [10Xu[110X and
  range  [10Xv[110X,  and  an edge with source [10Xv[110X and range [10Xu[110X, then the vertices [10Xu[110X and [10Xv[110X
  are equal.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := Digraph([[2], [1, 3], [2, 3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsAntisymmetricDigraph(gr1);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr1){[1, 2]};[127X[104X
    [4X[28X[ [ 1, 2 ], [ 2, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xgr2 := Digraph([[1, 2], [3, 3], [1]]);[127X[104X
    [4X[28X<multidigraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsAntisymmetricDigraph(gr2);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDigraphEdges(gr2);[127X[104X
    [4X[28X[ [ 1, 1 ], [ 1, 2 ], [ 2, 3 ], [ 2, 3 ], [ 3, 1 ] ][128X[104X
  [4X[32X[104X
  
  [1X6.1-3 IsBipartiteDigraph[101X
  
  [33X[1;0Y[29X[2XIsBipartiteDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  property  is [9Xtrue[109X if the digraph [3Xdigraph[103X is bipartite, and [9Xfalse[109X if it
  is not. A digraph is bipartite if and only if the vertices of [3Xdigraph[103X can be
  partitioned  into  two  non-empty sets such that the source and range of any
  edge  of  [3Xdigraph[103X lie in distinct sets. Equivalently, a digraph is bipartite
  if and only if it is 2-colorable; see [2XDigraphGreedyColouring[102X ([14X7.3-14[114X).[133X
  
  [33X[0;0YSee also [2XDigraphBicomponents[102X ([14X5.3-12[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := ChainDigraph(4);[127X[104X
    [4X[28X<digraph with 4 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsBipartiteDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := CycleDigraph(3);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsBipartiteDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.1-4 IsCompleteBipartiteDigraph[101X
  
  [33X[1;0Y[29X[2XIsCompleteBipartiteDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YReturns  [9Xtrue[109X  if  the  digraph [3Xdigraph[103X is a complete bipartite digraph, and
  [9Xfalse[109X if it is not.[133X
  
  [33X[0;0YA  digraph  is  a  [13Xcomplete  bipartite  digraph[113X  if  it  is  bipartite,  see
  [2XIsBipartiteDigraph[102X ([14X6.1-3[114X), and there exists a unique edge with source [10Xi[110X and
  range [10Xj[110X if and only if [10Xi[110X and [10Xj[110X lie in different bicomponents of [3Xdigraph[103X, see
  [2XDigraphBicomponents[102X ([14X5.3-12[114X).[133X
  
  [33X[0;0YEquivalently,  a  bipartite  digraph  with  bicomponents  of size [22Xm[122X and [22Xn[122X is
  complete precisely when it has [22X2mn[122X edges, none of which are multiple edges.[133X
  
  [33X[0;0YSee also [2XCompleteBipartiteDigraph[102X ([14X3.5-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := CycleDigraph(2);[127X[104X
    [4X[28X<digraph with 2 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsCompleteBipartiteDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := CycleDigraph(4);[127X[104X
    [4X[28X<digraph with 4 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsBipartiteDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCompleteBipartiteDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.1-5 IsCompleteDigraph[101X
  
  [33X[1;0Y[29X[2XIsCompleteDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YReturns [9Xtrue[109X if the digraph [3Xdigraph[103X is complete, and [9Xfalse[109X if it is not.[133X
  
  [33X[0;0YA  digraph  is  [13Xcomplete[113X if it has no loops, and for all [13Xdistinct[113X vertices [10Xi[110X
  and  [10Xj[110X, there is exactly one edge with source [10Xi[110X and range [10Xj[110X. Equivalently, a
  digraph with [22Xn[122X vertices is complete precisely when it has [22Xn(n - 1)[122X edges, no
  loops, and no multiple edges.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 3], [1, 3], [1, 2]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsCompleteDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 2], [1]]);[127X[104X
    [4X[28X<multidigraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsCompleteDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.1-6 IsEmptyDigraph[101X
  
  [33X[1;0Y[29X[2XIsEmptyDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsNullDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YReturns  [9Xtrue[109X  if  the  digraph  [3Xdigraph[103X is empty, and [9Xfalse[109X if it is not. A
  digraph is [13Xempty[113X if it has no edges.[133X
  
  [33X[0;0Y[10XIsNullDigraph[110X is a synonym for [10XIsEmptyDigraph[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[], []]);[127X[104X
    [4X[28X<digraph with 2 vertices, 0 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsEmptyDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsNullDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[], [1]]);[127X[104X
    [4X[28X<digraph with 2 vertices, 1 edge>[128X[104X
    [4X[25Xgap>[125X [27XIsEmptyDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsNullDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.1-7 IsFunctionalDigraph[101X
  
  [33X[1;0Y[29X[2XIsFunctionalDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis property is [9Xtrue[109X if the digraph [3Xdigraph[103X is functional.[133X
  
  [33X[0;0YA digraph is [13Xfunctional[113X if every vertex is the source of a unique edge.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := Digraph([[3], [2], [2], [1], [6], [5]]);[127X[104X
    [4X[28X<digraph with 6 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsFunctionalDigraph(gr1);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := Digraph([[1, 2], [1]]);[127X[104X
    [4X[28X<digraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsFunctionalDigraph(gr2);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr3 := Digraph(3, [1, 2, 3], [2, 3, 1]);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsFunctionalDigraph(gr3);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.1-8 IsMultiDigraph[101X
  
  [33X[1;0Y[29X[2XIsMultiDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YA  [13Xmultidigraph[113X  is  one  that  has at least two edges with equal source and
  range.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph(["a", "b", "c"], ["a", "b", "b"], ["b", "c", "a"]);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsMultiDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphFromDigraph6String("&Bug");[127X[104X
    [4X[28X<digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDuplicateFree(DigraphEdges(gr));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsMultiDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 2, 3, 2], [2, 1], [3]]);[127X[104X
    [4X[28X<multidigraph with 3 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDuplicateFree(DigraphEdges(gr));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsMultiDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.1-9 IsReflexiveDigraph[101X
  
  [33X[1;0Y[29X[2XIsReflexiveDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  property  is [9Xtrue[109X if the digraph [3Xdigraph[103X is reflexive, and [9Xfalse[109X if it
  is not. A digraph is [13Xreflexive[113X if it has a loop at every vertex.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 2], [2]]);[127X[104X
    [4X[28X<digraph with 2 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsReflexiveDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[3, 1], [4, 2], [3], [2, 1]]);[127X[104X
    [4X[28X<digraph with 4 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsReflexiveDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.1-10 IsSymmetricDigraph[101X
  
  [33X[1;0Y[29X[2XIsSymmetricDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  property  is [9Xtrue[109X if the digraph [3Xdigraph[103X is symmetric, and [9Xfalse[109X if it
  is not.[133X
  
  [33X[0;0YA [13Xsymmetric digraph[113X is one where for each non-loop edge, having source [10Xu[110X and
  range  [10Xv[110X,  there is a corresponding edge with source [10Xv[110X and range [10Xu[110X. If there
  are  [10Xn[110X edges with source [10Xu[110X and range [10Xv[110X, then there must be precisely [10Xn[110X edges
  with  source  [10Xv[110X  and  range  [10Xu[110X.  In  other  words, a symmetric digraph has a
  symmetric adjacency matrix [2XAdjacencyMatrix[102X ([14X5.2-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr1 := Digraph([[2], [1, 3], [2, 3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricDigraph(gr1);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xadj1 := AdjacencyMatrix(gr1);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay(adj1);[127X[104X
    [4X[28X[ [  0,  1,  0 ],[128X[104X
    [4X[28X  [  1,  0,  1 ],[128X[104X
    [4X[28X  [  0,  1,  1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xadj1 = TransposedMat(adj1);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr1 = DigraphReverse(gr1);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := Digraph([[2, 3], [1, 3], [2, 3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricDigraph(gr2);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xadj2 := AdjacencyMatrix(gr2);;[127X[104X
    [4X[25Xgap>[125X [27XDisplay(adj2);[127X[104X
    [4X[28X[ [  0,  1,  1 ],[128X[104X
    [4X[28X  [  1,  0,  1 ],[128X[104X
    [4X[28X  [  0,  1,  1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xadj2 = TransposedMat(adj2);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.1-11 IsTournament[101X
  
  [33X[1;0Y[29X[2XIsTournament[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  property  is [9Xtrue[109X if the digraph [3Xdigraph[103X is a tournament, and [9Xfalse[109X if
  it is not.[133X
  
  [33X[0;0YA   tournament   is   an  orientation  of  a  complete  (undirected)  graph.
  Specifically, a tournament is a digraph which has a unique directed edge (of
  some orientation) between any pair of distinct vertices, and no loops.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 3, 4], [3, 4], [4], []]);[127X[104X
    [4X[28X<digraph with 4 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsTournament(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2], [1], [3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsTournament(gr);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.1-12 IsTransitiveDigraph[101X
  
  [33X[1;0Y[29X[2XIsTransitiveDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  property is [9Xtrue[109X if the digraph [3Xdigraph[103X is transitive, and [9Xfalse[109X if it
  is  not. A digraph is [13Xtransitive[113X if whenever [10X[ i, j ][110X and [10X[ j, k ][110X are edges
  of the digraph, then [10X[ i, k ][110X is also an edge of the digraph.[133X
  
  [33X[0;0YLet  [22Xn[122X  be  the number of vertices of an arbitrary digraph, and let [22Xm[122X be the
  number  of  edges.  For general digraphs, the methods used for this property
  use  a  version of the Floyd-Warshall algorithm, and have complexity [22XO(n^3)[122X.
  However     for     digraphs     which     are     topologically    sortable
  [[2XDigraphTopologicalSort[102X ([14X5.1-7[114X)], then methods with complexity [22XO(m + n + m ⋅
  n)[122X will be used when appropriate.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 2], [3], [3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsTransitiveDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr2 := Digraph([[1, 2, 3], [3], [3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsTransitiveDigraph(gr2);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr2 = DigraphTransitiveClosure(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr3 := Digraph([[1, 2, 2, 3], [3, 3], [3]]);[127X[104X
    [4X[28X<multidigraph with 3 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsTransitiveDigraph(gr3);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.1-13 IsPreorderDigraph[101X
  
  [33X[1;0Y[29X[2XIsPreorderDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsQuasiorderDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YA  digraph  is a preorder digraph if and only if the digraph satisifies both
  [2XIsReflexiveDigraph[102X  ([14X6.1-9[114X)  and  [2XIsTransitiveDigraph[102X  ([14X6.1-12[114X).  A preorder
  digraph (or quasiorder digraph) [3Xdigraph[103X corresponds to the preorder relation
  [22X≤[122X defined by [22Xx ≤ y[122X if and only if [10X[x, y][110X is an edge of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1], [2, 3], [2, 3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsPreorderDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1 .. 4], [1 .. 4], [1 .. 4], [1 .. 4]]);[127X[104X
    [4X[28X<digraph with 4 vertices, 16 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsPreorderDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2], [3], [4], [5], [1]]);[127X[104X
    [4X[28X<digraph with 5 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsPreorderDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1], [1, 2], [2, 3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsPreorderDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.1-14 IsPartialOrderDigraph[101X
  
  [33X[1;0Y[29X[2XIsPartialOrderDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YA  digraph  is a partial order digraph if and only if the digraph satisifies
  all   of  [2XIsReflexiveDigraph[102X  ([14X6.1-9[114X),  [2XIsAntisymmetricDigraph[102X  ([14X6.1-2[114X)  and
  [2XIsTransitiveDigraph[102X  ([14X6.1-12[114X).  A  partial  order [3Xdigraph[103X corresponds to the
  partial  order  relation [22X≤[122X defined by [22Xx ≤ y[122X if and only if [10X[x, y][110X is an edge
  of [3Xdigraph[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 3], [2, 3], [3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsPartialOrderDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := CycleDigraph(5);[127X[104X
    [4X[28X<digraph with 5 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsPartialOrderDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 1], [1, 1, 2], [3], [3, 3, 4, 4]]);[127X[104X
    [4X[28X<multidigraph with 4 vertices, 10 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsPartialOrderDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.1-15 IsMeetSemilatticeDigraph[101X
  
  [33X[1;0Y[29X[2XIsMeetSemilatticeDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsJoinSemilatticeDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsLatticeDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0Y[10XIsMeetSemilatticeDigraph[110X  returns  [9Xtrue[109X  if  the  digraph  [3Xdigraph[103X is a meet
  semilattice; [10XIsJoinSemilatticeDigraph[110X returns [9Xtrue[109X if the digraph [3Xdigraph[103X is
  a join semilattice; and [10XIsLatticeDigraph[110X returns [9Xtrue[109X if the digraph [3Xdigraph[103X
  is both a meet and a join semilattice.[133X
  
  [33X[0;0YFor a partial order digraph [2XIsPartialOrderDigraph[102X ([14X6.1-14[114X) the corresponding
  partial  order  is the relation [22X≤[122X, defined by [22Xx ≤ y[122X if and only if [10X[x, y][110X is
  an  edge. A digraph is a [13Xmeet semilattice[113X if it is a partial order and every
  pair  of  vertices  has  a  greatest  lower bound (meet) with respect to the
  aforementioned  relation.  A [13Xjoin semilattice[113X is a partial order where every
  pair  of  vertices  has  a  least  upper  bound  (join)  with respect to the
  relation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 3], [2, 3], [3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsMeetSemilatticeDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsJoinSemilatticeDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLatticeDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1], [2], [1 .. 3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsJoinSemilatticeDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsMeetSemilatticeDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLatticeDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1 .. 4], [2, 4], [3, 4], [4]]);[127X[104X
    [4X[28X<digraph with 4 vertices, 9 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsMeetSemilatticeDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsJoinSemilatticeDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLatticeDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 1, 1], [1, 1, 2, 2],[127X[104X
    [4X[25X>[125X [27X                  [1, 3, 3], [1, 2, 3, 3, 4]]);[127X[104X
    [4X[28X<multidigraph with 4 vertices, 15 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsMeetSemilatticeDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsJoinSemilatticeDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsLatticeDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X6.2 [33X[0;0YRegularity[133X[101X
  
  [1X6.2-1 IsInRegularDigraph[101X
  
  [33X[1;0Y[29X[2XIsInRegularDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  property is [9Xtrue[109X if there is an integer [10Xn[110X such that for every vertex [10Xv[110X
  of  digraph  [3Xdigraph[103X  there  are  exactly [10Xn[110X edges terminating in [10Xv[110X. See also
  [2XIsOutRegularDigraph[102X ([14X6.2-2[114X) and [2XIsRegularDigraph[102X ([14X6.2-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsInRegularDigraph(CompleteDigraph(4));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsInRegularDigraph(ChainDigraph(4));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.2-2 IsOutRegularDigraph[101X
  
  [33X[1;0Y[29X[2XIsOutRegularDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  property is [9Xtrue[109X if there is an integer [10Xn[110X such that for every vertex [10Xv[110X
  of  digraph  [3Xdigraph[103X  there  are  exactly  [10Xn[110X  edges  starting at [10Xv[110X. See also
  [2XIsInRegularDigraph[102X ([14X6.2-1[114X) and [2XIsRegularDigraph[102X ([14X6.2-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsOutRegularDigraph(CompleteDigraph(4));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsOutRegularDigraph(ChainDigraph(4));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.2-3 IsRegularDigraph[101X
  
  [33X[1;0Y[29X[2XIsRegularDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  property is [9Xtrue[109X if there is an integer [10Xn[110X such that for every vertex [10Xv[110X
  of  digraph [3Xdigraph[103X there are exactly [10Xn[110X edges starting and terminating at [10Xv[110X.
  In  other  words, the property is [9Xtrue[109X if [3Xdigraph[103X is both in-regular and and
  out-regular.  See  also  [2XIsInRegularDigraph[102X  ([14X6.2-1[114X) and [2XIsOutRegularDigraph[102X
  ([14X6.2-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsRegularDigraph(CompleteDigraph(4));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsRegularDigraph(ChainDigraph(4));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.2-4 IsDistanceRegularDigraph[101X
  
  [33X[1;0Y[29X[2XIsDistanceRegularDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YIf [3Xdigraph[103X is a connected symmetric graph, this property returns [9Xtrue[109X if for
  any  two  vertices [10Xu[110X and [10Xv[110X of [3Xdigraph[103X and any two integers [10Xi[110X and [10Xj[110X between [10X0[110X
  and the diameter of [3Xdigraph[103X, the number of vertices at distance [10Xi[110X from [10Xu[110X and
  distance  [10Xj[110X from [10Xv[110X depends only on [10Xi[110X, [10Xj[110X, and the distance between vertices [10Xu[110X
  and [10Xv[110X.[133X
  
  [33X[0;0YAlternatively,  a  distance  regular  graph is a graph for which there exist
  integers  [10Xb_i[110X,  [10Xc_i[110X,  and  [10Xi[110X  such that for any two vertices [10Xu[110X, [10Xv[110X in [3Xdigraph[103X
  which  are  distance [10Xi[110X apart, there are exactly [10Xb_i[110X neighbors of [10Xv[110X which are
  at  distance [10Xi - 1[110X away from [10Xu[110X, and [10Xc_i[110X neighbors of [10Xv[110X which are at distance
  [10Xi  +  1[110X  away  from  [10Xu[110X.  This definition is used to check whether [3Xdigraph[103X is
  distance regular.[133X
  
  [33X[0;0YIn the case where [3Xdigraph[103X is not symmetric or not connected, the property is
  [9Xfalse[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphSymmetricClosure(ChainDigraph(5));;[127X[104X
    [4X[25Xgap>[125X [27XIsDistanceRegularDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 3, 4], [1, 3, 4], [1, 2, 4], [1, 2, 3]]);[127X[104X
    [4X[28X<digraph with 4 vertices, 12 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDistanceRegularDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X6.3 [33X[0;0YConnectivity and cycles[133X[101X
  
  [1X6.3-1 IsAcyclicDigraph[101X
  
  [33X[1;0Y[29X[2XIsAcyclicDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  property is [9Xtrue[109X if the digraph [3Xdigraph[103X is acyclic, and [9Xfalse[109X if it is
  not. A digraph is [13Xacyclic[113X if every directed cycle on the digraph is trivial.
  See  section  [14X1.1-1[114X for the definition of a directed cycle, and of a trivial
  directed cycle.[133X
  
  [33X[0;0YThe  method  used  in  this  operation  has complexity [22XO(m+n)[122X where [22Xm[122X is the
  number  of  edges  (counting  multiple  edges as one) and [22Xn[122X is the number of
  vertices in the digraph.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPetersen := Graph(SymmetricGroup(5), [[1, 2]], OnSets,[127X[104X
    [4X[25X>[125X [27Xfunction(x, y)[127X[104X
    [4X[25X>[125X [27X  return IsEmpty(Intersection(x, y));[127X[104X
    [4X[25X>[125X [27Xend);;[127X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph(Petersen);[127X[104X
    [4X[28X<digraph with 10 vertices, 30 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsAcyclicDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphFromDiSparse6String([127X[104X
    [4X[25X>[125X [27X".b_OGCIDBaPGkULEbQHCeRIdrHcuZMfRyDAbPhTi|zF");[127X[104X
    [4X[28X<digraph with 35 vertices, 34 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsAcyclicDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsAcyclicDigraph(ChainDigraph(10));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsAcyclicDigraph(CycleDigraph(10));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.3-2 IsChainDigraph[101X
  
  [33X[1;0Y[29X[2XIsChainDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0Y[10XIsChainDigraph[110X  returns  [9Xtrue[109X  if  the  digraph [3Xdigraph[103X is isomorphic to the
  chain  digraph  with the same number of vertices as [3Xdigraph[103X, and [9Xfalse[109X if it
  is not; see [2XChainDigraph[102X ([14X3.5-1[114X).[133X
  
  [33X[0;0YA  digraph  is  a [13Xchain[113X if and only if it is a directed tree, in which every
  vertex has out degree at most one; see [2XIsDirectedTree[102X ([14X6.3-7[114X) and [2XOutDegrees[102X
  ([14X5.2-8[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 3], [2, 3], [3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsChainDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := ChainDigraph(5);[127X[104X
    [4X[28X<digraph with 5 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsChainDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphReverse(gr);[127X[104X
    [4X[28X<digraph with 5 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsChainDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.3-3 IsConnectedDigraph[101X
  
  [33X[1;0Y[29X[2XIsConnectedDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  property  is [9Xtrue[109X if the digraph [3Xdigraph[103X is weakly connected and [9Xfalse[109X
  if  it  is  not.  A digraph [3Xdigraph[103X is [13Xweakly connected[113X if it is possible to
  travel  from  any  vertex  to any other vertex by traversing edges in either
  direction (possibly against the orientation of some of them).[133X
  
  [33X[0;0YThe  method  used  in  this  function  has  complexity [22XO(m)[122X if the digraph's
  [2XDigraphSource[102X  ([14X5.2-5[114X)  attribute is set, otherwise it has complexity [22XO(m+n)[122X
  (where  [22Xm[122X  is  the  number  of  edges and [22Xn[122X is the number of vertices of the
  digraph).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2], [3], []]);;[127X[104X
    [4X[25Xgap>[125X [27XIsConnectedDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 3], [4], [3], []]);;[127X[104X
    [4X[25Xgap>[125X [27XIsConnectedDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.3-4 IsBiconnectedDigraph[101X
  
  [33X[1;0Y[29X[2XIsBiconnectedDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YA connected digraph is [13Xbiconnected[113X if it is still connected (in the sense of
  [2XIsConnectedDigraph[102X ([14X6.3-3[114X)) when any vertex is removed. [10XIsBiconnectedDigraph[110X
  returns  [9Xtrue[109X if the digraph [3Xdigraph[103X is biconnected, and [9Xfalse[109X if it is not.
  In   particular,  [10XIsBiconnectedDigraph[110X  returns  [9Xfalse[109X  if  [3Xdigraph[103X  is  not
  connected.[133X
  
  [33X[0;0YMultiple edges and loops are ignored by this method.[133X
  
  [33X[0;0YThe  method  used  in  this  operation  has complexity [22XO(m+n)[122X where [22Xm[122X is the
  number of edges (counting multiple edges as one, and not counting loops) and
  [22Xn[122X  is  the  number  of  vertices in the digraph. See also [2XArticulationPoints[102X
  ([14X5.3-13[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsBiconnectedDigraph(Digraph([[1, 3], [2, 3], [3]]));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsBiconnectedDigraph(CycleDigraph(5));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xdigraph := Digraph([[1, 1], [1, 1, 2], [3], [3, 3, 4, 4]]);;[127X[104X
    [4X[25Xgap>[125X [27XIsBiconnectedDigraph(digraph);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.3-5 IsStronglyConnectedDigraph[101X
  
  [33X[1;0Y[29X[2XIsStronglyConnectedDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis property is [9Xtrue[109X if the digraph [3Xdigraph[103X is strongly connected and [9Xfalse[109X
  if it is not.[133X
  
  [33X[0;0YA  digraph  [3Xdigraph[103X  is  [13Xstrongly connected[113X if there is a directed path from
  every  vertex to every other vertex. See section [14X1.1-1[114X for the definition of
  a directed path.[133X
  
  [33X[0;0YThe  method used in this operation is based on Gabow's Algorithm [Gab00] and
  has  complexity  [22XO(m+n)[122X,  where  [22Xm[122X is the number of edges (counting multiple
  edges as one) and [22Xn[122X is the number of vertices in the digraph.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := CycleDigraph(250000);[127X[104X
    [4X[28X<digraph with 250000 vertices, 250000 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsStronglyConnectedDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := DigraphRemoveEdges(gr, [[250000, 1]]);[127X[104X
    [4X[28X<digraph with 250000 vertices, 249999 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsStronglyConnectedDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.3-6 IsAperiodicDigraph[101X
  
  [33X[1;0Y[29X[2XIsAperiodicDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis  property  is  [9Xtrue[109X  if  the  digraph [3Xdigraph[103X is aperiodic, i.e. if its
  [2XDigraphPeriod[102X ([14X5.3-14[114X) is equal to 1. Otherwise, the property is [9Xfalse[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[6], [1], [2], [3], [4, 4], [5]]);[127X[104X
    [4X[28X<multidigraph with 6 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsAperiodicDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2], [3, 5], [4], [5], [1, 2]]);[127X[104X
    [4X[28X<digraph with 5 vertices, 7 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsAperiodicDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.3-7 IsDirectedTree[101X
  
  [33X[1;0Y[29X[2XIsDirectedTree[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YReturns  [9Xtrue[109X  if the digraph [3Xdigraph[103X is a directed tree, and [9Xfalse[109X if it is
  not.[133X
  
  [33X[0;0YA  [13Xdirected tree[113X is an acyclic digraph with precisely 1 source, such that no
  two  vertices  share  an  out-neighbour.  Note  the  empty  digraph  is  not
  considered a directed tree as it has no source.[133X
  
  [33X[0;0YSee also [2XDigraphSources[102X ([14X5.1-6[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[], [2]]);[127X[104X
    [4X[28X<digraph with 2 vertices, 1 edge>[128X[104X
    [4X[25Xgap>[125X [27XIsDirectedTree(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[3], [3], []]);[127X[104X
    [4X[28X<digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDirectedTree(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2], [3], []]);[127X[104X
    [4X[28X<digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDirectedTree(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2, 3], [6], [4, 5], [], [], []]);[127X[104X
    [4X[28X<digraph with 6 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsDirectedTree(gr);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.3-8 IsUndirectedTree[101X
  
  [33X[1;0Y[29X[2XIsUndirectedTree[102X( [3Xdigraph[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsUndirectedForest[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThe  property  [10XIsUndirectedTree[110X  returns  [9Xtrue[109X  if the digraph [3Xdigraph[103X is an
  undirected tree, and the property [10XIsUndirectedForest[110X returns [9Xtrue[109X if [3Xdigraph[103X
  is an undirected forest; otherwise, these properties return [9Xfalse[109X.[133X
  
  [33X[0;0YAn  [13Xundirected  tree[113X  is a symmetric digraph without loops, in which for any
  pair of distinct vertices [10Xu[110X and [10Xv[110X, there is exactly one directed path from [10Xu[110X
  to  [10Xv[110X.  See [2XIsSymmetricDigraph[102X ([14X6.1-10[114X) and [2XDigraphHasLoops[102X ([14X6.1-1[114X), and see
  section  [14X1.1-1[114X  for the definition of directed path. This definition implies
  that an undirected tree has no multiple edges.[133X
  
  [33X[0;0YAn  [13Xundirected forest[113X is a digraph, each of whose connected components is an
  undirected  tree.  In  other  words, an undirected forest is isomorphic to a
  disjoint  union  of undirected trees. See [2XDigraphConnectedComponents[102X ([14X5.3-8[114X)
  and  [2XDigraphDisjointUnion[102X  ([14X3.3-26[114X). In particular, every undirected tree is
  an undirected forest.[133X
  
  [33X[0;0YPlease  note that the digraph with zero vertices is considered to be neither
  an undirected tree nor an undirected forest.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[3], [3], [1, 2]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsUndirectedTree(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricDigraph(gr) and not DigraphHasLoops(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[3], [5], [1, 4], [3], [2]]);[127X[104X
    [4X[28X<digraph with 5 vertices, 6 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsConnectedDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsUndirectedTree(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsUndirectedForest(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 2], [1], [2]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 4 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsUndirectedTree(gr) or IsUndirectedForest(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsSymmetricDigraph(gr) or not DigraphHasLoops(gr);[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.3-9 IsEulerianDigraph[101X
  
  [33X[1;0Y[29X[2XIsEulerianDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YThis property returns true if the digraph [3Xdigraph[103X is Eulerian.[133X
  
  [33X[0;0YA  digraph  is  called  [13XEulerian[113X  if  there exists a directed circuit on the
  digraph  which  includes  every edge exactly once. See section [14X1.1-1[114X for the
  definition of a directed circuit.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[]]);[127X[104X
    [4X[28X<digraph with 1 vertex, 0 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsEulerianDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2], []]);[127X[104X
    [4X[28X<digraph with 2 vertices, 1 edge>[128X[104X
    [4X[25Xgap>[125X [27XIsEulerianDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[3], [], [2]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsEulerianDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[2], [3], [1]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsEulerianDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.3-10 IsHamiltonianDigraph[101X
  
  [33X[1;0Y[29X[2XIsHamiltonianDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YIf  [3Xdigraph[103X is Hamiltonian, then this property returns [9Xtrue[109X, and [9Xfalse[109X if it
  is not.[133X
  
  [33X[0;0YA  digraph  with  [10Xn[110X  vertices  is  [13XHamiltonian[113X if it has a directed cycle of
  length [10Xn[110X. See Section [14X1.1-1[114X for the definition of a directed cycle. Note the
  empty digraphs on 0 and 1 vertices are considered to be Hamiltonian.[133X
  
  [33X[0;0YThe  method  used  in  this  operation  has  the  worst  case  complexity as
  [2XDigraphMonomorphism[102X ([14X7.3-4[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg := Digraph([[]]);[127X[104X
    [4X[28X<digraph with 1 vertex, 0 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsHamiltonianDigraph(g);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xg := Digraph([[2], [1]]);[127X[104X
    [4X[28X<digraph with 2 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsHamiltonianDigraph(g);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xg := Digraph([[3], [], [2]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 2 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsHamiltonianDigraph(g);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xg := Digraph([[2], [3], [1]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 3 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsHamiltonianDigraph(g);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X6.3-11 IsCycleDigraph[101X
  
  [33X[1;0Y[29X[2XIsCycleDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0Y[10XIsCycleDigraph[110X  returns  [9Xtrue[109X  if  the  digraph [3Xdigraph[103X is isomorphic to the
  cycle  digraph  with the same number of vertices as [3Xdigraph[103X, and [9Xfalse[109X if it
  is not; see [2XCycleDigraph[102X ([14X3.5-5[114X).[133X
  
  [33X[0;0YA  digraph  is  a  [13Xcycle[113X if and only if it is strongly connected and has the
  same number of edges as vertices.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgr := Digraph([[1, 3], [2, 3], [3]]);[127X[104X
    [4X[28X<digraph with 3 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsCycleDigraph(gr);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xgr := CycleDigraph(5);[127X[104X
    [4X[28X<digraph with 5 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27XIsCycleDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xgr := OnDigraphs(gr, (1, 2, 3));[127X[104X
    [4X[28X<digraph with 5 vertices, 5 edges>[128X[104X
    [4X[25Xgap>[125X [27Xgr = CycleDigraph(5);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsCycleDigraph(gr);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X6.4 [33X[0;0YPlanarity[133X[101X
  
  [1X6.4-1 IsPlanarDigraph[101X
  
  [33X[1;0Y[29X[2XIsPlanarDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YA  [13Xplanar[113X  digraph  is a digraph that can be embedded in the plane in such a
  way  that  its edges do not intersect. A digraph is planar if and only if it
  does not have a subdigraph that is homeomorphic to either the complete graph
  on  [10X5[110X  vertices  or the complete bipartite graph with vertex sets of sizes [10X3[110X
  and [10X3[110X.[133X
  
  [33X[0;0Y[10XIsPlanarDigraph[110X  returns  [9Xtrue[109X if the digraph [3Xdigraph[103X is planar and [9Xfalse[109X if
  it  is  not.  The  directions and multiplicities of any edges in [3Xdigraph[103X are
  ignored by [10XIsPlanarDigraph[110X.[133X
  
  [33X[0;0YSee also [2XIsOuterPlanarDigraph[102X ([14X6.4-2[114X).[133X
  
  [33X[0;0YThis      method      uses      the      reference     implementation     in
  [5Xedge-addition-planarity-suite[105X  by  John Boyer of the algorithms described in
  [BM06].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPlanarDigraph(CompleteDigraph(4));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPlanarDigraph(CompleteDigraph(5));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsPlanarDigraph(CompleteBipartiteDigraph(2, 3));[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPlanarDigraph(CompleteBipartiteDigraph(3, 3));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X6.4-2 IsOuterPlanarDigraph[101X
  
  [33X[1;0Y[29X[2XIsOuterPlanarDigraph[102X( [3Xdigraph[103X ) [32X property[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YAn  [13Xouter  planar[113X  digraph is a digraph that can be embedded in the plane in
  such  a  way that its edges do not intersect, and all vertices belong to the
  unbounded face of the embedding. A digraph is outer planar if and only if it
  does not have a subdigraph that is homeomorphic to either the complete graph
  on  [10X4[110X  vertices  or the complete bipartite graph with vertex sets of sizes [10X2[110X
  and [10X3[110X.[133X
  
  [33X[0;0Y[10XIsOuterPlanarDigraph[110X returns [9Xtrue[109X if the digraph [3Xdigraph[103X is outer planar and
  [9Xfalse[109X  if  it  is  not.  The  directions  and multiplicities of any edges in
  [3Xdigraph[103X are ignored by [10XIsPlanarDigraph[110X.[133X
  
  [33X[0;0YSee   also   [2XIsPlanarDigraph[102X   ([14X6.4-1[114X).   This  method  uses  the  reference
  implementation   in  [5Xedge-addition-planarity-suite[105X  by  John  Boyer  of  the
  algorithms described in [BM06].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsOuterPlanarDigraph(CompleteDigraph(4));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsOuterPlanarDigraph(CompleteDigraph(5));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsOuterPlanarDigraph(CompleteBipartiteDigraph(2, 3));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsOuterPlanarDigraph(CompleteBipartiteDigraph(3, 3));[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsOuterPlanarDigraph(CycleDigraph(10));[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
