attr.xml

#############################################################################
##
#W  attr.xml
#Y  Copyright (C) 2014-19                               James D. Mitchell
##
##  Licensing information can be found in the README file of this package.
##
#############################################################################
##

<#GAPDoc Label="DigraphVertices">
<ManSection>
  <Attr Name="DigraphVertices" Arg="digraph"/>
  <Returns>A list of positive integers.</Returns>
  <Description>
    Returns the vertices of the digraph <A>digraph</A>. <P/>

    Note that the vertices of a digraph are always the range of
    positive integers from <C>1</C> to the number of vertices of the
    graph, <Ref Attr="DigraphNrVertices"/>.
    Arbitrary <E>labels</E> can be assigned to the vertices of a digraph;
    see <Ref Oper="DigraphVertexLabels"/> for more information about this.

    <Example><![CDATA[
gap> gr := Digraph(["a", "b", "c"],
>                  ["a", "b", "b"],
>                  ["b", "c", "a"]);
<immutable digraph with 3 vertices, 3 edges>
gap> DigraphVertices(gr);
[ 1 .. 3 ]
gap> gr := Digraph([1, 2, 3, 4, 5, 7],
>                  [1, 2, 2, 4, 4],
>                  [2, 7, 5, 3, 7]);
<immutable digraph with 6 vertices, 5 edges>
gap> DigraphVertices(gr);
[ 1 .. 6 ]
gap> DigraphVertices(RandomDigraph(100));
[ 1 .. 100 ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphVertices(D);
[ 1 .. 3 ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphNrVertices">
<ManSection>
  <Attr Name="DigraphNrVertices" Arg="digraph"/>
  <Returns>An non-negative integer.</Returns>
  <Description>
    Returns the number of vertices of the digraph <A>digraph</A>. <P/>

    <Example><![CDATA[
gap> gr := Digraph(["a", "b", "c"],
>                  ["a", "b", "b"],
>                  ["b", "c", "a"]);
<immutable digraph with 3 vertices, 3 edges>
gap> DigraphNrVertices(gr);
3
gap> gr := Digraph([1, 2, 3, 4, 5, 7],
>                  [1, 2, 2, 4, 4],
>                  [2, 7, 5, 3, 7]);
<immutable digraph with 6 vertices, 5 edges>
gap> DigraphNrVertices(gr);
6
gap> DigraphNrVertices(RandomDigraph(100));
100
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphNrVertices(D);
3
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphEdges">
<ManSection>
  <Attr Name="DigraphEdges" Arg="digraph"/>
  <Returns>A list of lists of two positive integers.</Returns>
  <Description>
    Returns a list of edges of the digraph <A>digraph</A>,
    where each edge is a pair of elements of <Ref Attr="DigraphVertices"/> of
    the form <C>[source,range]</C>.
    <P/>

    <!-- TODO remove this paragraph when multidigraphs are removed -->
    The entries of <C>DigraphEdges(</C><A>digraph</A><C>)</C> are in one-to-one
    correspondence with the edges of <A>digraph</A>.  Hence
    <C>DigraphEdges(</C><A>digraph</A><C>)</C> is duplicate-free if and only if
    <A>digraph</A> contains no multiple edges. <P/>

    The entries of <C>DigraphEdges</C> are guaranteed to be sorted by their
    first component (i.e. by the source of each edge), but they are not
    necessarily then sorted by the second component.
    <Example><![CDATA[
gap> gr := DigraphFromDiSparse6String(".DaXbOe?EAM@G~");
<immutable multidigraph with 5 vertices, 16 edges>
gap> edges := ShallowCopy(DigraphEdges(gr));; Sort(edges);
gap> edges;
[ [ 1, 1 ], [ 1, 3 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 2, 1 ], 
  [ 2, 2 ], [ 2, 3 ], [ 2, 5 ], [ 3, 2 ], [ 3, 4 ], [ 3, 5 ], 
  [ 4, 2 ], [ 4, 4 ], [ 4, 5 ], [ 5, 1 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphEdges(D);
[ [ 1, 2 ], [ 2, 3 ], [ 3, 1 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphNrEdges">
<ManSection>
  <Attr Name="DigraphNrEdges" Arg="digraph"/>
  <Returns>An integer.</Returns>
  <Description>
    Returns the number of edges of the digraph <A>digraph</A>.
    <Example><![CDATA[
gap> gr := Digraph([
> [1, 3, 4, 5], [1, 2, 3, 5], [2, 4, 5], [2, 4, 5], [1]]);;
gap> DigraphNrEdges(gr);
15
gap> gr := Digraph(["a", "b", "c"],
>                  ["a", "b", "b"],
>                  ["b", "a", "a"]);
<immutable multidigraph with 3 vertices, 3 edges>
gap> DigraphNrEdges(gr);
3
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphNrEdges(D);
3
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphNrAdjacencies">
<ManSection>
  <Attr Name="DigraphNrAdjacencies" Arg="digraph" />
  <Returns>An integer.</Returns>
  <Description> Returns the number of sets <M>\{u, v\}</M> of vertices of the digraph <A>digraph</A>, such that
    either <M>(u, v)</M> or <M>(v, u)</M> is an edge. The following equality holds for
    any digraph <C>D</C> with no multiple edges: <C>DigraphNrAdjacencies(D) * 2 - DigraphNrLoops(D)
    = DigraphNrEdges(DigraphSymmetricClosure(D))</C>.
    <Example><![CDATA[
gap> gr := Digraph([
> [1, 3, 4, 5], [1, 2, 3, 5], [2, 4, 5], [2, 4, 5], [1]]);;
gap> DigraphNrAdjacencies(gr);
13
gap> DigraphNrAdjacencies(gr) * 2 - DigraphNrLoops(gr) =
> DigraphNrEdges(DigraphSymmetricClosure(gr));
true
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphNrAdjacenciesWithoutLoops">
<ManSection>
  <Attr Name="DigraphNrAdjacenciesWithoutLoops" Arg="digraph" />
  <Returns>An integer.</Returns>
  <Description> Returns the number of sets <M>\{u, v\}</M> of vertices of the digraph <A>digraph</A>, such that
    <M>u \neq v</M> and either <M>(u, v)</M> or <M>(v, u)</M> is an edge. The following equality holds for
    any digraph <C>D</C> with no multiple edges: <C>DigraphNrAdjacenciesWithoutLoops(D) * 2 + DigraphNrLoops(D)
    = DigraphNrEdges(DigraphSymmetricClosure(D))</C>.
    <Example><![CDATA[
gap> gr := Digraph([
> [1, 3, 4, 5], [1, 2, 3, 5], [2, 4, 5], [2, 4, 5], [1]]);;
gap> DigraphNrAdjacenciesWithoutLoops(gr);
10
gap> DigraphNrAdjacenciesWithoutLoops(gr) * 2 + DigraphNrLoops(gr) =
> DigraphNrEdges(DigraphSymmetricClosure(gr));
true
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphNrLoops">
<ManSection>
  <Attr Name="DigraphNrLoops" Arg="digraph"/>
  <Returns>An integer.</Returns>
  <Description>
    This function returns the number of loops of the digraph <A>digraph</A>. See <Ref Prop="DigraphHasLoops"/>. <P/>
    <Example><![CDATA[
gap> D := Digraph([[2, 3], [1, 4], [3, 3, 5], [], [2, 5]]);
<immutable multidigraph with 5 vertices, 9 edges>
gap> DigraphNrLoops(D);
3
gap> D := EmptyDigraph(5);
<immutable empty digraph with 5 vertices>
gap> DigraphNrLoops(D);
0
gap> D := CompleteDigraph(5);
<immutable complete digraph with 5 vertices>
gap> DigraphNrLoops(D);
0
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="RangeSourceDigraph">
<ManSection>
  <Attr Name="DigraphRange" Arg="digraph"/>
  <Attr Name="DigraphSource" Arg="digraph"/>
  <Returns>A list of positive integers.</Returns>
  <Description>
    <C>DigraphRange</C> and <C>DigraphSource</C> return the range and source of
    the digraph <A>digraph</A>. More precisely, position <C>i</C> in
    <C>DigraphSource(<A>digraph</A>)</C> and
    <C>DigraphRange(<A>digraph</A>)</C> give, respectively, the source and 
    range of the <C>i</C>th edge of <A>digraph</A>.

    <Example><![CDATA[
gap> gr := Digraph([
> [2, 1, 3, 5], [1, 3, 4], [2, 3], [2], [1, 2, 3, 4]]);
<immutable digraph with 5 vertices, 14 edges>
gap> DigraphSource(gr);
[ 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 5, 5 ]
gap> DigraphRange(gr);
[ 2, 1, 3, 5, 1, 3, 4, 2, 3, 2, 1, 2, 3, 4 ]
gap> DigraphEdges(gr);
[ [ 1, 2 ], [ 1, 1 ], [ 1, 3 ], [ 1, 5 ], [ 2, 1 ], [ 2, 3 ], 
  [ 2, 4 ], [ 3, 2 ], [ 3, 3 ], [ 4, 2 ], [ 5, 1 ], [ 5, 2 ], 
  [ 5, 3 ], [ 5, 4 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphRange(D);
[ 2, 3, 1 ]
gap> DigraphSource(D);
[ 1, 2, 3 ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="OutNeighbours">
<ManSection>
  <Attr Name="OutNeighbours" Arg="digraph"/>
  <Attr Name="OutNeighbors" Arg="digraph"/>
  <Oper Name="OutNeighboursMutableCopy" Arg="digraph"/>
  <Oper Name="OutNeighborsMutableCopy" Arg="digraph"/>
  <Returns>The adjacencies of a digraph.</Returns>
  <Description>
    <C>OutNeighbours</C> returns the list <C>out</C> of out-neighbours of
    each vertex of the digraph <A>digraph</A>.
    <!-- TODO update the following sentence once multidigraphs are gone -->
    More specifically, a vertex <C>j</C> appears in <C>out[i]</C> each time
    there exists the edge <C>[i, j]</C> in <A>digraph</A>. <P/>

    The function <C>OutNeighbours</C> returns an immutable list of
    lists, whereas the function <C>OutNeighboursMutableCopy</C> returns a copy
    of <C>OutNeighbours</C> which is a mutable list of mutable lists. <P/>

    Note that the entries of <C>out</C> are not guaranteed to be sorted in any
    particular order.

    <Example><![CDATA[
gap> gr := Digraph(["a", "b", "c"],
>                  ["a", "b", "b"],
>                  ["b", "a", "c"]);
<immutable digraph with 3 vertices, 3 edges>
gap> OutNeighbours(gr);
[ [ 2 ], [ 1, 3 ], [  ] ]
gap> gr := Digraph([[1, 2, 3], [2, 1], [3]]);
<immutable digraph with 3 vertices, 6 edges>
gap> OutNeighbours(gr);
[ [ 1, 2, 3 ], [ 2, 1 ], [ 3 ] ]
gap> gr := DigraphByAdjacencyMatrix([
>  [1, 2, 1],
>  [1, 1, 0],
>  [0, 0, 1]]);
<immutable multidigraph with 3 vertices, 7 edges>
gap> OutNeighbours(gr);
[ [ 1, 2, 2, 3 ], [ 1, 2 ], [ 3 ] ]
gap> OutNeighboursMutableCopy(gr);
[ [ 1, 2, 2, 3 ], [ 1, 2 ], [ 3 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> OutNeighbours(D);
[ [ 2 ], [ 3 ], [ 1 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="InNeighbours">
<ManSection>
  <Attr Name="InNeighbours" Arg="digraph"/>
  <Attr Name="InNeighbors" Arg="digraph"/>
  <Oper Name="InNeighboursMutableCopy" Arg="digraph"/>
  <Oper Name="InNeighborsMutableCopy" Arg="digraph"/>
  <Returns>A list of lists of vertices.</Returns>
  <Description>
    <C>InNeighbours</C> returns the list <C>inn</C> of in-neighbours of each
    vertex of the digraph <A>digraph</A>.
    <!-- TODO update the following sentence once multidigraphs are gone -->
    More specifically, a vertex <C>j</C> appears in <C>inn[i]</C> each time
    there exists an edge <C>[j,i]</C> in <A>digraph</A>. <P/>

    The function <C>InNeighbours</C> returns an immutable list of
    lists, whereas the function <C>InNeighboursMutableCopy</C> returns a copy
    of <C>InNeighbours</C> which is a mutable list of mutable lists. <P/>

    Note that the entries of <C>inn</C> are not necessarily sorted into
    ascending order, particularly if <A>digraph</A> was constructed via
    <Ref Oper="DigraphByInNeighbours"/>.

    <Example><![CDATA[
gap> gr := Digraph(["a", "b", "c"],
>                  ["a", "b", "b"],
>                  ["b", "a", "c"]);
<immutable digraph with 3 vertices, 3 edges>
gap> InNeighbours(gr);
[ [ 2 ], [ 1 ], [ 2 ] ]
gap> gr := Digraph([[1, 2, 3], [2, 1], [3]]);
<immutable digraph with 3 vertices, 6 edges>
gap> InNeighbours(gr);
[ [ 1, 2 ], [ 1, 2 ], [ 1, 3 ] ]
gap> gr := DigraphByAdjacencyMatrix([
>  [1, 2, 1],
>  [1, 1, 0],
>  [0, 0, 1]]);
<immutable multidigraph with 3 vertices, 7 edges>
gap> InNeighbours(gr);
[ [ 1, 2 ], [ 1, 1, 2 ], [ 1, 3 ] ]
gap> InNeighboursMutableCopy(gr);
[ [ 1, 2 ], [ 1, 1, 2 ], [ 1, 3 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> InNeighbours(D);
[ [ 3 ], [ 1 ], [ 2 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphAdjacencyFunction">
<ManSection>
  <Attr Name="DigraphAdjacencyFunction" Arg="digraph"/>
  <Returns>A function.</Returns>
  <Description>
    If <A>digraph</A> is a digraph, then <C>DigraphAdjacencyFunction</C> returns
    a function which takes two integer parameters <C>x, y</C> and returns
    <K>true</K> if there exists an edge from vertex <C>x</C> to vertex <C>y</C>
    in <A>digraph</A> and <K>false</K> if not.

    <Example><![CDATA[
gap> digraph := Digraph([[1, 2], [3], []]);
<immutable digraph with 3 vertices, 3 edges>
gap> foo := DigraphAdjacencyFunction(digraph);
function( u, v ) ... end
gap> foo(1, 1);
true
gap> foo(1, 2);
true
gap> foo(1, 3);
false
gap> foo(3, 1);
false
gap> gr := Digraph(["a", "b", "c"],
>                  ["a", "b", "b"],
>                  ["b", "a", "a"]);
<immutable multidigraph with 3 vertices, 3 edges>
gap> foo := DigraphAdjacencyFunction(gr);
function( u, v ) ... end
gap> foo(1, 2);
true
gap> foo(3, 2);
false
gap> foo(3, 1);
false
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> foo := DigraphAdjacencyFunction(D);
function( u, v ) ... end
gap> foo(1, 2);
true
gap> foo(2, 1);
false
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="AdjacencyMatrix">
<ManSection>
  <Attr Name="AdjacencyMatrix" Arg="digraph"/>
  <Oper Name="AdjacencyMatrixMutableCopy" Arg="digraph"/>
  <Returns>A square matrix of non-negative integers.</Returns>
  <Description>
    This function returns the adjacency matrix <C>mat</C> of the digraph
    <A>digraph</A>.
    <!-- Update the following sentence when multidigraphs are gone -->
    The value of the matrix entry <C>mat[i][j]</C> is the number of edges
    in <A>digraph</A> with source <C>i</C> and range <C>j</C>. If <A>digraph</A>
    has no vertices, then the empty list is returned. <P/>

    The function <C>AdjacencyMatrix</C> returns an immutable list of
    lists, whereas the function <C>AdjacencyMatrixMutableCopy</C> returns a copy
    of <C>AdjacencyMatrix</C> that is a mutable list of mutable lists. <P/>

    <Example><![CDATA[
gap> gr := Digraph([
> [2, 2, 2], [1, 3, 6, 8, 9, 10], [4, 6, 8],
> [1, 2, 3, 9], [3, 3], [3, 5, 6, 10], [1, 2, 7],
> [1, 2, 3, 10, 5, 6, 10], [1, 3, 4, 5, 8, 10],
> [2, 3, 4, 6, 7, 10]]);
<immutable multidigraph with 10 vertices, 44 edges>
gap> mat := AdjacencyMatrix(gr);;
gap> Display(mat);
[ [  0,  3,  0,  0,  0,  0,  0,  0,  0,  0 ],
  [  1,  0,  1,  0,  0,  1,  0,  1,  1,  1 ],
  [  0,  0,  0,  1,  0,  1,  0,  1,  0,  0 ],
  [  1,  1,  1,  0,  0,  0,  0,  0,  1,  0 ],
  [  0,  0,  2,  0,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  1,  0,  1,  1,  0,  0,  0,  1 ],
  [  1,  1,  0,  0,  0,  0,  1,  0,  0,  0 ],
  [  1,  1,  1,  0,  1,  1,  0,  0,  0,  2 ],
  [  1,  0,  1,  1,  1,  0,  0,  1,  0,  1 ],
  [  0,  1,  1,  1,  0,  1,  1,  0,  0,  1 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> Display(AdjacencyMatrix(D));
[ [  0,  1,  0 ],
  [  0,  0,  1 ],
  [  1,  0,  0 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="BooleanAdjacencyMatrix">
<ManSection>
  <Attr Name="BooleanAdjacencyMatrix" Arg="digraph"/>
  <Oper Name="BooleanAdjacencyMatrixMutableCopy" Arg="digraph"/>
  <Returns>A square matrix of booleans.</Returns>
  <Description>
    If <A>digraph</A> is a digraph with a positive number of vertices
    <C>n</C>, then <C>BooleanAdjacencyMatrix(</C><A>digraph</A><C>)</C>
    returns the boolean adjacency matrix <C>mat</C> of <A>digraph</A>.
    <!-- TODO Update the following sentence when multidigraphs are gone -->
    The
    value of the matrix entry <C>mat[j][i]</C> is <K>true</K> if and only if
    there exists an edge in <A>digraph</A> with source <C>j</C> and range
    <C>i</C>.  If <A>digraph</A> has no vertices, then the empty list is
    returned. <P/>

    <!-- TODO Remove this sentence when multidigraphs are gone -->
    Note that the boolean adjacency matrix loses information about multiple
    edges.  <P/>

    The attribute <C>BooleanAdjacencyMatrix</C> returns an immutable list of
    immutable lists, whereas the function
    <C>BooleanAdjacencyMatrixMutableCopy</C> returns a copy of the
    <C>BooleanAdjacencyMatrix</C> that is a mutable list of mutable lists. <P/>
    <Example><![CDATA[
gap> gr := Digraph([[3, 4], [2, 3], [1, 2, 4], [4]]);
<immutable digraph with 4 vertices, 8 edges>
gap> PrintArray(BooleanAdjacencyMatrix(gr));
[ [  false,  false,   true,   true ],
  [  false,   true,   true,  false ],
  [   true,   true,  false,   true ],
  [  false,  false,  false,   true ] ]
gap> gr := CycleDigraph(4);;
gap> PrintArray(BooleanAdjacencyMatrix(gr));
[ [  false,   true,  false,  false ],
  [  false,  false,   true,  false ],
  [  false,  false,  false,   true ],
  [   true,  false,  false,  false ] ]
gap> BooleanAdjacencyMatrix(EmptyDigraph(0));
[  ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> PrintArray(BooleanAdjacencyMatrix(D));
[ [  false,   true,  false ],
  [  false,  false,   true ],
  [   true,  false,  false ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DegreeMatrix">
<ManSection>
  <Attr Name="DegreeMatrix" Arg="digraph"/>
  <Returns>A square matrix of non-negative integers.</Returns>
  <Description>
    Returns the out-degree matrix <C>mat</C> of the digraph
    <A>digraph</A>. The value of the <C>i</C>th diagonal matrix entry is the
    out-degree of the vertex <C>i</C> in <A>digraph</A>. All off-diagonal
    entries are <C>0</C>.
    If <A>digraph</A> has no vertices, then the empty list is returned. <P/>

    See <Ref Attr="OutDegrees"/> for more information.
    <Example><![CDATA[
gap> D := Digraph([[1, 2, 3], [4], [1, 3, 4], []]);
<immutable digraph with 4 vertices, 7 edges>
gap> PrintArray(DegreeMatrix(D));
[ [  3,  0,  0,  0 ],
  [  0,  1,  0,  0 ],
  [  0,  0,  3,  0 ],
  [  0,  0,  0,  0 ] ]
gap> D := CycleDigraph(5);;
gap> PrintArray(DegreeMatrix(D));
[ [  1,  0,  0,  0,  0 ],
  [  0,  1,  0,  0,  0 ],
  [  0,  0,  1,  0,  0 ],
  [  0,  0,  0,  1,  0 ],
  [  0,  0,  0,  0,  1 ] ]
gap> DegreeMatrix(EmptyDigraph(0));
[  ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> Display(DegreeMatrix(D));
[ [  1,  0,  0 ],
  [  0,  1,  0 ],
  [  0,  0,  1 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="LaplacianMatrix">
<ManSection>
  <Attr Name="LaplacianMatrix" Arg="digraph"/>
  <Returns>A square matrix of integers.</Returns>
  <Description>
    Returns the out-degree Laplacian matrix <C>mat</C> of the
    digraph <A>digraph</A>. The out-degree Laplacian matrix is defined as
    <C>DegreeMatrix(digraph) - AdjacencyMatrix(digraph)</C>.  If
    <A>digraph</A> has no vertices, then the empty list is returned. <P/>

    See <Ref Attr="DegreeMatrix"/> and <Ref Attr="AdjacencyMatrix"/> for more
    information.
    <Example><![CDATA[
gap> gr := Digraph([[1, 2, 3], [4], [1, 3, 4], []]);
<immutable digraph with 4 vertices, 7 edges>
gap> PrintArray(LaplacianMatrix(gr));
[ [   2,  -1,  -1,   0 ],
  [   0,   1,   0,  -1 ],
  [  -1,   0,   2,  -1 ],
  [   0,   0,   0,   0 ] ]
gap> LaplacianMatrix(EmptyDigraph(0));
[  ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> Display(LaplacianMatrix(D));
[ [   1,  -1,   0 ],
  [   0,   1,  -1 ],
  [  -1,   0,   1 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="OutDegrees">
<ManSection>
  <Attr Name="OutDegrees" Arg="digraph"/>
  <Attr Name="OutDegreeSequence" Arg="digraph"/>
  <Attr Name="OutDegreeSet" Arg="digraph"/>
  <Returns>A list of non-negative integers.</Returns>
  <Description>
    Given a digraph <A>digraph</A> with <M>n</M> vertices, the function
    <C>OutDegrees</C> returns an immutable list <C>out</C> of length <M>n</M>, such that
    for a vertex <C>i</C> in <A>digraph</A>, the value of <C>out[i]</C> is the
    out-degree of vertex <C>i</C>.
    See <Ref Oper="OutDegreeOfVertex"/>. <P/>

    The function <C>OutDegreeSequence</C> returns the same list,
    after it has been sorted into non-increasing order. <P/>

    The function <C>OutDegreeSet</C> returns the same list, sorted into
    increasing order with duplicate entries removed. <P/>

    <Example><![CDATA[
gap> D := Digraph([[1, 3, 2, 2], [], [2, 1], []]);
<immutable multidigraph with 4 vertices, 6 edges>
gap> OutDegrees(D);
[ 4, 0, 2, 0 ]
gap> OutDegreeSequence(D);
[ 4, 2, 0, 0 ]
gap> OutDegreeSet(D);
[ 0, 2, 4 ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> OutDegrees(D);
[ 1, 1, 1 ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="InDegrees">
<ManSection>
  <Attr Name="InDegrees" Arg="digraph"/>
  <Attr Name="InDegreeSequence" Arg="digraph"/>
  <Attr Name="InDegreeSet" Arg="digraph"/>
  <Returns>A list of non-negative integers.</Returns>
  <Description>

    Given a digraph <A>digraph</A> with <M>n</M> vertices, the function
    <C>InDegrees</C> returns an immutable list <C>inn</C> of length <M>n</M>, such that
    for a vertex <C>i</C> in <A>digraph</A>, the value of <C>inn[i]</C> is
    the in-degree of vertex <C>i</C>.
    See <Ref Oper="InDegreeOfVertex"/>. <P/>

    The function <C>InDegreeSequence</C> returns the same list,
    after it has been sorted into non-increasing order. <P/>

    The function <C>InDegreeSet</C> returns the same list, sorted into
    increasing order with duplicate entries removed. <P/>

    <Example><![CDATA[
gap> D := Digraph([[1, 3, 2, 2, 4], [], [2, 1, 4], []]);
<immutable multidigraph with 4 vertices, 8 edges>
gap> InDegrees(D);
[ 2, 3, 1, 2 ]
gap> InDegreeSequence(D);
[ 3, 2, 2, 1 ]
gap> InDegreeSet(D);
[ 1, 2, 3 ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> InDegrees(D);
[ 1, 1, 1 ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphSources">
<ManSection>
  <Attr Name="DigraphSources" Arg="digraph"/>
  <Returns>A list of vertices.</Returns>
  <Description>
    This function returns an immutable list of the sources of the digraph
    <A>digraph</A>.
    A source of a digraph is a vertex with in-degree zero.
    See <Ref Oper="InDegreeOfVertex"/>.
    <Example><![CDATA[
gap> gr := Digraph([[3, 5, 2, 2], [3], [], [5, 2, 5, 3], []]);
<immutable multidigraph with 5 vertices, 9 edges>
gap> DigraphSources(gr);
[ 1, 4 ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphSources(D);
[  ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphSinks">
<ManSection>
  <Attr Name="DigraphSinks" Arg="digraph"/>
  <Returns>A list of vertices.</Returns>
  <Description>
    This function returns a list of the sinks of the digraph
    <A>digraph</A>.
    A sink of a digraph is a vertex with out-degree zero.
    See <Ref Oper="OutDegreeOfVertex"/>.
    <Example><![CDATA[
gap> gr := Digraph([[3, 5, 2, 2], [3], [], [5, 2, 5, 3], []]);
<immutable multidigraph with 5 vertices, 9 edges>
gap> DigraphSinks(gr);
[ 3, 5 ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphSinks(D);
[  ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphStronglyConnectedComponents">
<ManSection>
  <Attr Name="DigraphStronglyConnectedComponents" Arg="digraph"/>
  <Attr Name="DigraphNrStronglyConnectedComponents" Arg="digraph"/>
  <Returns>A record.</Returns>
  <Description>
    This function returns the record <C>scc</C> corresponding to the strongly
    connected components of the digraph <A>digraph</A>.  Two vertices of
    <A>digraph</A> are in the same strongly connected component whenever they
    are equal, or there is a directed path from each vertex to the other.  The
    set of strongly connected components is a partition of the vertex set of
    <A>digraph</A>.
    <P/>

    The record <C>scc</C> has 2 components: <C>comps</C> and <C>id</C>.
    The component <C>comps</C> is a list of the strongly connected components
    of <A>digraph</A> (each of which is a list of vertices).
    The component <C>id</C> is a list such that the vertex <C>i</C> is an
    element of the strongly connected component <C>comps[id[i]]</C>. <P/>

    The method used in this function is a non-recursive version of Gabow's
    Algorithm <Cite Key="Gab00"/> and has complexity <M>O(m+n)</M> where
    <M>m</M> is the number of edges (counting multiple edges as one) and
    <M>n</M> is the number of vertices in the digraph. <P/>

    <C>DigraphNrStronglyConnectedComponents(<A>digraph</A>)</C> is simply a
    shortcut for
    <C>Length(DigraphStronglyConnectedComponents(<A>digraph</A>).comps)</C>,
    and is no more efficient.

    <Example><![CDATA[
gap> gr := Digraph([[2], [3, 1], []]);
<immutable digraph with 3 vertices, 3 edges>
gap> DigraphStronglyConnectedComponents(gr);
rec( comps := [ [ 3 ], [ 1, 2 ] ], id := [ 2, 2, 1 ] )
gap> DigraphNrStronglyConnectedComponents(gr);
2
gap> D := DigraphDisjointUnion(CycleDigraph(4), CycleDigraph(5));
<immutable digraph with 9 vertices, 9 edges>
gap> DigraphStronglyConnectedComponents(D);
rec( comps := [ [ 1, 2, 3, 4 ], [ 5, 6, 7, 8, 9 ] ], 
  id := [ 1, 1, 1, 1, 2, 2, 2, 2, 2 ] )
gap> DigraphNrStronglyConnectedComponents(D);
2
gap> D := CycleDigraph(IsMutableDigraph, 2);
<mutable digraph with 2 vertices, 2 edges>
gap> G := CycleDigraph(3);
<immutable cycle digraph with 3 vertices>
gap> DigraphDisjointUnion(D, G);
<mutable digraph with 5 vertices, 5 edges>
gap> DigraphStronglyConnectedComponents(D);
rec( comps := [ [ 1, 2 ], [ 3, 4, 5 ] ], id := [ 1, 1, 2, 2, 2 ] )
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphConnectedComponents">
<ManSection>
  <Attr Name="DigraphConnectedComponents" Arg="digraph"/>
  <Attr Name="DigraphNrConnectedComponents" Arg="digraph"/>
  <Returns>A record.</Returns>
  <Description>
    This function returns the record <C>wcc</C> corresponding to the weakly
    connected components of the digraph <A>digraph</A>.  Two vertices of
    <A>digraph</A> are in the same weakly connected component whenever they are
    equal, or there exists a directed path (ignoring the orientation of edges)
    between them.  More formally, two vertices are in the same weakly connected
    component of <A>digraph</A> if and only if they are in the same strongly
    connected component (see <Ref Attr="DigraphStronglyConnectedComponents"/>)
    of the <Ref Oper="DigraphSymmetricClosure"/> of <A>digraph</A>.  <P/>

    The set of weakly connected components is a partition of
    the vertex set of <A>digraph</A>. <P/>

    The record <C>wcc</C> has 2 components: <C>comps</C> and <C>id</C>.
    The component <C>comps</C> is a list of the weakly connected components
    of  <A>digraph</A> (each of which is a list of vertices).
    The component <C>id</C> is a list such that the vertex <C>i</C> is an
    element of the weakly connected component <C>comps[id[i]]</C>. <P/>

    The method used in this function has complexity  <M>O(m+n)</M>, where
    <M>m</M> is the number of edges and
    <M>n</M> is the number of vertices in the digraph. <P/>

    <C>DigraphNrConnectedComponents(<A>digraph</A>)</C> is simply a shortcut
    for <C>Length(DigraphConnectedComponents(<A>digraph</A>).comps)</C>,
    and is no more efficient.

    <Example><![CDATA[
gap> gr := Digraph([[2], [3, 1], []]);
<immutable digraph with 3 vertices, 3 edges>
gap> DigraphConnectedComponents(gr);
rec( comps := [ [ 1, 2, 3 ] ], id := [ 1, 1, 1 ] )
gap> gr := Digraph([[1], [1, 2], []]);
<immutable digraph with 3 vertices, 3 edges>
gap> DigraphConnectedComponents(gr);
rec( comps := [ [ 1, 2 ], [ 3 ] ], id := [ 1, 1, 2 ] )
gap> DigraphNrConnectedComponents(gr);
2
gap> gr := EmptyDigraph(0);
<immutable empty digraph with 0 vertices>
gap> DigraphConnectedComponents(gr);
rec( comps := [  ], id := [  ] )
gap> D := CycleDigraph(IsMutableDigraph, 2);
<mutable digraph with 2 vertices, 2 edges>
gap> G := CycleDigraph(3);
<immutable cycle digraph with 3 vertices>
gap> DigraphDisjointUnion(D, G);
<mutable digraph with 5 vertices, 5 edges>
gap> DigraphConnectedComponents(D);
rec( comps := [ [ 1, 2 ], [ 3, 4, 5 ] ], id := [ 1, 1, 2, 2, 2 ] )
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphTopologicalSort">
<ManSection>
  <Attr Name="DigraphTopologicalSort" Arg="digraph"/>
  <Returns>A list of positive integers, or <K>fail</K>.</Returns>
  <Description>
    If <A>digraph</A> is a digraph whose only directed cycles are loops, then
    <C>DigraphTopologicalSort</C> returns the vertices of <A>digraph</A> ordered
    so that every edge's source appears no earlier in the list than its range.
    If the digraph <A>digraph</A> contains directed cycles of length greater
    than <M>1</M>, then this operation returns <K>fail</K>.
    <P/>

    See Section <Ref Subsect="Definitions" Style="Number" /> for the definition
    of a directed cycle, and the definition of a loop.

    <P/>

    The method used for this attribute has complexity <M>O(m+n)</M> where
    <M>m</M> is the number of edges (counting multiple edges as one) and
    <M>n</M> is the number of vertices in the digraph. <P/>
    <Example><![CDATA[
gap> D := Digraph([
> [2, 3], [], [4, 6], [5], [], [7, 8, 9], [], [], []]);
<immutable digraph with 9 vertices, 8 edges>
gap> DigraphTopologicalSort(D);
[ 2, 5, 4, 7, 8, 9, 6, 3, 1 ]
gap> D := Digraph(IsMutableDigraph, [[2, 3], [3], [4], []]);
<mutable digraph with 4 vertices, 4 edges>
gap> DigraphTopologicalSort(D);
[ 4, 3, 2, 1 ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphDegeneracy">
<ManSection>
  <Attr Name="DigraphDegeneracy" Arg="digraph"/>
  <Returns>A non-negative integer, or <K>fail</K>.</Returns>
  <Description>
    If <A>digraph</A> is a symmetric digraph without multiple edges - see
    <Ref Prop="IsSymmetricDigraph"/> and <Ref Prop="IsMultiDigraph"/> - then
    this attribute returns the degeneracy of <A>digraph</A>. <P/>

    The degeneracy of a digraph is the least integer <C>k</C> such
    that every induced of <A>digraph</A> contains a vertex whose number of
    neighbours (excluding itself) is at most <C>k</C>. Note that this means
    that loops are ignored.<P/>

    If <A>digraph</A> is not symmetric or has multiple edges then this
    attribute returns <K>fail</K>.
    <Example><![CDATA[
gap> D := DigraphSymmetricClosure(ChainDigraph(5));;
gap> DigraphDegeneracy(D);
1
gap> D := CompleteDigraph(5);;
gap> DigraphDegeneracy(D);
4
gap> D := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4], [2], []]);
<immutable digraph with 6 vertices, 10 edges>
gap> DigraphDegeneracy(D);
1
gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 10, 3);
<mutable digraph with 20 vertices, 60 edges>
gap> DigraphDegeneracy(D);
3
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphDegeneracyOrdering">
<ManSection>
  <Attr Name="DigraphDegeneracyOrdering" Arg="digraph"/>
  <Returns>A list of integers, or <K>fail</K>.</Returns>
  <Description>
    If <A>digraph</A> is a digraph for which
    <C>DigraphDegeneracy(</C><A>digraph</A><C>)</C> is a non-negative integer
    <C>k</C> - see <Ref Attr="DigraphDegeneracy"/> - then this attribute
    returns a degeneracy ordering of the vertices of the vertices of
    <A>digraph</A>.<P/>

    A degeneracy ordering of <A>digraph</A> is a list <C>ordering</C> of the
    vertices of <A>digraph</A> ordered such that for any
    position <C>i</C> of the list, the vertex <C>ordering[i]</C> has at most
    <C>k</C> neighbours in later position of the list.<P/>

    If <C>DigraphDegeneracy(</C><A>digraph</A><C>)</C> returns <K>fail</K>,
    then this attribute returns <K>fail</K>.
    <Example><![CDATA[
gap> D := DigraphSymmetricClosure(ChainDigraph(5));;
gap> DigraphDegeneracyOrdering(D);
[ 5, 4, 3, 2, 1 ]
gap> D := CompleteDigraph(5);;
gap> DigraphDegeneracyOrdering(D);
[ 5, 4, 3, 2, 1 ]
gap> D := Digraph([[1], [2, 4, 5], [3, 4], [2, 3, 4], [2], []]);
<immutable digraph with 6 vertices, 10 edges>
gap> DigraphDegeneracyOrdering(D);
[ 1, 6, 5, 2, 4, 3 ]
gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 3, 1);
<mutable digraph with 6 vertices, 18 edges>
gap> DigraphDegeneracyOrdering(D);
[ 6, 5, 4, 1, 3, 2 ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphShortestDistances">
<ManSection>
  <Attr Name="DigraphShortestDistances" Arg="digraph"/>
  <Returns>A square matrix.</Returns>
  <Description>
    If <A>digraph</A> is a digraph with <M>n</M> vertices, then this
    function returns an <M>n \times n</M> matrix <C>mat</C>, where each entry is
    either a non-negative integer, or <K>fail</K>.  If <M>n = 0</M>, then an
    empty list is returned. <P/>

    If there is a directed path from vertex <C>i</C> to vertex <C>j</C>, then
    the value of <C>mat[i][j]</C> is the length of the shortest such directed
    path. If no such directed path exists, then the value of <C>mat[i][j]</C> is
    <C>fail</C>.  We use the convention that the distance from every vertex to
    itself is <C>0</C>, i.e. <C>mat[i][i] = 0</C> for all vertices <C>i</C>.
    <P/>

    The method used in this function is a version of the Floyd-Warshall
    algorithm, and has complexity <M>O(n^3)</M>.

    <Example><![CDATA[
gap> D := Digraph([[1, 2], [3], [1, 2], [4]]);
<immutable digraph with 4 vertices, 6 edges>
gap> mat := DigraphShortestDistances(D);;
gap> PrintArray(mat);
[ [     0,     1,     2,  fail ],
  [     2,     0,     1,  fail ],
  [     1,     1,     0,  fail ],
  [  fail,  fail,  fail,     0 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphShortestDistances(D);
[ [ 0, 1, 2 ], [ 2, 0, 1 ], [ 1, 2, 0 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphDiameter">
<ManSection>
  <Attr Name="DigraphDiameter" Arg="digraph"/>
  <Returns>An integer or <C>fail</C>.</Returns>
  <Description>
    This function returns the diameter of the digraph <A>digraph</A>.
    <P/>

    If a digraph <A>digraph</A> is strongly connected and has at least 1
    vertex, then the <E>diameter</E> is the maximum shortest distance between
    any pair of distinct vertices. Otherwise then the diameter of
    <A>digraph</A> is undefined, and this function returns the value
    <C>fail</C>. <P/>

    See <Ref Attr="DigraphShortestDistances"/>. <P/>
    <Example><![CDATA[
gap> D := Digraph([[2], [3], [4, 5], [5], [1, 2, 3, 4, 5]]);
<immutable digraph with 5 vertices, 10 edges>
gap> DigraphDiameter(D);
3
gap> D := ChainDigraph(2);
<immutable chain digraph with 2 vertices>
gap> DigraphDiameter(D);
fail
gap> IsStronglyConnectedDigraph(D);
false
gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 6, 2);
<mutable digraph with 12 vertices, 36 edges>
gap> DigraphDiameter(D);
4
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphGirth">
<ManSection>
  <Attr Name="DigraphGirth" Arg="digraph"/>
  <Returns>An integer, or <K>infinity</K>.</Returns>
  <Description>
    This attribute returns the <E>girth</E> of the digraph <A>digraph</A>.
    The <E>girth</E> of a digraph is the length of its shortest simple circuit.
    See Section <Ref Subsect="Definitions" Style="Number" /> for the definitions
    of simple circuit, directed cycle, and loop.
    <P/>

    If <A>digraph</A> has no directed cycles, then this function will return
    <K>infinity</K>.  If <A>digraph</A> contains a loop, then this function will
    return <C>1</C>.
    <P/>

    In the worst case, the method used in this function is a version of the
    Floyd-Warshall algorithm, and has complexity <C>O(<A>n</A> ^ 3)</C>, where
    <A>n</A> is the number of vertices in <A>digraph</A>.  If the digraph has
    known automorphisms [see <Ref Attr="DigraphGroup"/>], then the performance
    is likely to be better.
    <P/>

    For symmetric digraphs, see also <Ref Attr="DigraphUndirectedGirth"/>. <P/>
    <Example><![CDATA[
gap> D := Digraph([[1], [1]]);
<immutable digraph with 2 vertices, 2 edges>
gap> DigraphGirth(D);
1
gap> D := Digraph([[2, 3], [3], [4], []]);
<immutable digraph with 4 vertices, 4 edges>
gap> DigraphGirth(D);
infinity
gap> D := Digraph([[2, 3], [3], [4], [1]]);
<immutable digraph with 4 vertices, 5 edges>
gap> DigraphGirth(D);
3
gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 6, 2);
<mutable digraph with 12 vertices, 36 edges>
gap> DigraphGirth(D);
2
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphOddGirth">
<ManSection>
  <Attr Name="DigraphOddGirth" Arg="digraph"/>
  <Returns>An integer, or <K>infinity</K>.</Returns>
  <Description>
    This attribute returns the <E>odd girth</E> of the digraph <A>digraph</A>.
    The <E>odd girth</E> of a digraph is the length of its shortest simple
    circuit of odd length.  See Section <Ref Subsect="Definitions"
    Style="Number" /> for the definitions of simple circuit, directed
    cycle, and loop.
    <P/>

    If <A>digraph</A> has no directed cycles of odd length, then this function
    will return <K>infinity</K>, even if <A>digraph</A> has a directed cycle
    of even length.  If <A>digraph</A> contains a loop, then this function
    will return <C>1</C>.
    <P/>

    See also <Ref Attr="DigraphGirth"/>. <P/>
    <Example><![CDATA[
gap> D := Digraph([[2], [3, 1], [1]]);
<immutable digraph with 3 vertices, 4 edges>
gap> DigraphOddGirth(D);
3
gap> D := CycleDigraph(4);
<immutable cycle digraph with 4 vertices>
gap> DigraphOddGirth(D);
infinity
gap> D := Digraph([[2], [3], [], [3], [4]]);
<immutable digraph with 5 vertices, 4 edges>
gap> DigraphOddGirth(D);
infinity
gap> D := CycleDigraph(IsMutableDigraph, 4);
<mutable digraph with 4 vertices, 4 edges>
gap> DigraphDisjointUnion(D, CycleDigraph(5));
<mutable digraph with 9 vertices, 9 edges>
gap> DigraphOddGirth(D);
5
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphUndirectedGirth">
<ManSection>
  <Attr Name="DigraphUndirectedGirth" Arg="digraph"/>
  <Returns>An integer or <K>infinity</K>.</Returns>
  <Description>
    If <A>digraph</A> is a symmetric digraph, then this function returns the
    girth of <A>digraph</A> when treated as an undirected graph (i.e. each pair
    of edges <M>[i, j]</M> and <M>[j, i]</M> is treated as a single edge between
    <M>i</M> and <M>j</M>). <P/>

    The <E>girth</E> of an undirected graph is the length of its shortest simple
    cycle, i.e. the shortest non-trivial path starting and ending at the same
    vertex and passing through no vertex or edge more than once. <P/>

    If <A>digraph</A> has no cycles, then this function will return
    <K>infinity</K>. <P/>
    <Example><![CDATA[
gap> D := Digraph([[2, 4], [1, 3], [2, 4], [1, 3]]);
<immutable digraph with 4 vertices, 8 edges>
gap> DigraphUndirectedGirth(D);
4
gap> D := Digraph([[2], [1, 3], [2]]);
<immutable digraph with 3 vertices, 4 edges>
gap> DigraphUndirectedGirth(D);
infinity
gap> D := Digraph([[1], [], [4], [3]]);
<immutable digraph with 4 vertices, 3 edges>
gap> DigraphUndirectedGirth(D);
1
gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);
<mutable digraph with 18 vertices, 54 edges>
gap> DigraphUndirectedGirth(D);
5
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="ArticulationPoints">
<ManSection>
  <Attr Name="ArticulationPoints" Arg="digraph"/>
  <Returns>A list of vertices.</Returns>
  <Description>
    A connected digraph is <E>biconnected</E> if it is still connected (in the
    sense of <Ref Prop="IsConnectedDigraph"/>) when any vertex is removed. 
    If the digraph <A>digraph</A> is not biconnected but is connected, then any
    vertex <C>v</C> of <A>digraph</A> whose removal makes the resulting digraph
    disconnected is called an <E>articulation point</E>.<P/>

    <C>ArticulationPoints</C> returns a list of the articulation points of
    <A>digraph</A>, if any, and, in particular, returns the empty list if
    <A>digraph</A> is not connected. <P/>

    Multiple edges are ignored by this method. <P/>

    The method used in this operation has complexity <M>O(m+n)</M> where
    <M>m</M> is the number of edges and <M>n</M> is the number of vertices in
    the digraph.<P/>
      
    If <A>D</A> has a bridge (see <Ref Attr="Bridges"/>), then a node incident
    to the bridge is an articulation point if and only if it has degree at
    least <M>2</M>.  It follows that if <A>D</A> has a bridge and at least
    <M>3</M> nodes, then at least one of the nodes incident to the bridge is
    an articulation point.  The converse does not hold, there are digraphs
    with articulation points, but no bridges.
    <P/>

    See also <Ref Prop="IsBiconnectedDigraph"/> and 
    <Ref Prop="IsBridgelessDigraph"/>.
<Example><![CDATA[
gap> ArticulationPoints(CycleDigraph(5));
[  ]
gap> D := Digraph([[2, 7], [3, 5], [4], [2], [6], [1], []]);;
gap> ArticulationPoints(D);
[ 1, 2 ]
gap> ArticulationPoints(ChainDigraph(5));
[ 2, 3, 4 ]
gap> ArticulationPoints(NullDigraph(5));
[  ]
gap> D := ChainDigraph(IsMutableDigraph, 4);
<mutable digraph with 4 vertices, 3 edges>
gap> ArticulationPoints(D);
[ 2, 3 ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>


<#GAPDoc Label="MinimalCyclicEdgeCut">
<ManSection>
  <Attr Name="MinimalCyclicEdgeCut" Arg="digraph"/>
  <Returns>A list of edges or <K>fail</K>.</Returns>
  <Description>
    A cyclic edge cut of <A>digraph</A> is a set of edges such that deleting
    these edges results in at least two connected components having a cycle.
    This method computes the cyclic edge cut with minimal cardinality for cubic 
    graphs with at least 8 vertices. The cyclic edge cut is returned as a 
    list of undirected edges.<P/>

    If the given digraph is not cubic, not connected, has less than 8 vertices or 
    does not have a cyclic edge cut, the method returns <K>fail</K>.
    Multiple edges of <A>digraph</A> are ignored by this method and note 
    that <A>digraph</A> is identified as an undirected graph. <P/>

    The method used in this method has complexity <M>O(n^2*log^2(n)) </M> where
    <M>n</M> is the number of vertices in the digraph.<P/>
      
<Example><![CDATA[
gap> MinimalCyclicEdgeCut(HypercubeGraph(3));
[ [ 1, 5 ], [ 2, 6 ], [ 4, 8 ], [ 3, 7 ] ]
gap> MinimalCyclicEdgeCut(CompleteDigraph(4));
fail
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>


<#GAPDoc Label="DigraphAbsorptionProbabilities">
<ManSection>
  <Attr Name="DigraphAbsorptionProbabilities" Arg="digraph"/>
  <Returns>A matrix of rational numbers.</Returns>
  <Description>
    A random walk of infinite length through a finite digraph may pass through several
    different strongly connected components (SCCs).  However, with probability 1
    it must eventually reach an SCC which it can never leave, because the SCC
    has no out-edges leading to other SCCs.  We may say that such an SCC has
    <E>absorbed</E> the random walk, and we can calculate the probability of
    each SCC absorbing a walk starting at a given vertex.
    <P/>
    
    <C>DigraphAbsorptionProbabilities</C> returns an <M>m \times n</M> matrix
    <C>mat</C>, where <M>m</M> is the number of vertices in <C>digraph</C> and
    <M>n</M> is the number of strongly connected components.  Each entry
    <C>mat[i][j]</C> is a rational number representing the probability that an
    unbounded random walk starting at vertex <C>i</C> will be absorbed by
    strongly connected component <C>j</C> – that is, the probability that it
    will reach the component and never leave.
    <P/>

    Strongly connected components are indexed in the order given by
    <Ref Attr="DigraphStronglyConnectedComponents"/>. See also
    <Ref Oper="DigraphRandomWalk"/> and
    <Ref Attr="DigraphAbsorptionExpectedSteps"/>.
    <P/>
<Example><![CDATA[
gap> gr := Digraph([[2, 3, 4], [3], [2], []]);
<immutable digraph with 4 vertices, 5 edges>
gap> DigraphStronglyConnectedComponents(gr).comps;
[ [ 2, 3 ], [ 4 ], [ 1 ] ]
gap> DigraphAbsorptionProbabilities(gr);
[ [ 2/3, 1/3, 0 ], [ 1, 0, 0 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphAbsorptionExpectedSteps">
<ManSection>
  <Attr Name="DigraphAbsorptionExpectedSteps" Arg="digraph"/>
  <Returns>A list of rational numbers.</Returns>
  <Description>
    A random walk of unbounded length through a finite digraph may pass through
    several different strongly connected components (SCCs).  However, with
    probability 1 it must eventually reach an SCC which it can never leave,
    because the SCC has no out-edges leading to other SCCs.  When this happens,
    we say the walk has been <E>absorbed</E>.
    <P/>
    
    <C>DigraphAbsorptionExpectedSteps</C> returns a list <C>L</C> with length
    equal to the number of vertices of <A>digraph</A>, where <C>L[v]</C>
    is the expected number of steps a random walk starting at vertex <C>v</C>
    should take before it is absorbed – that is, the expected number of steps to
    reach an SCC that it can never leave.
    <P/>

    See also <Ref Oper="DigraphRandomWalk"/> and
    <Ref Attr="DigraphAbsorptionProbabilities"/>.
    <P/>
<Example><![CDATA[
gap> gr := Digraph([[2], [3, 4], [], [2]]);
<immutable digraph with 4 vertices, 4 edges>
gap> DigraphAbsorptionExpectedSteps(gr);
[ 4, 3, 0, 4 ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="StrongOrientation">
<ManSection>
  <Oper Name="StrongOrientation" Arg="D"/>
  <Attr Name="StrongOrientationAttr" Arg="D"/>
  <Returns>A digraph or <K>fail</K>.</Returns>
  <Description>
    A <E>strong orientation</E> of a connected symmetric digraph <A>D</A> (if
    it exists) is a strongly connected subdigraph <C>C</C> of <A>D</A> such
    that for every edge <C>[u, v]</C> of <A>D</A> either <C>[u, v]</C> or
    <C>[v, u]</C> is an edge of <C>C</C> but not both.  Robbin's Theorem states
    that a digraph admits a strong orientation if and only if it is
    bridgeless (see <Ref Prop="IsBridgelessDigraph"/>). 
    <P/>

    This operation returns a strong orientation of the digraph <A>D</A> if 
    <A>D</A> is symmetric and <A>D</A> admits a strong orientation. If <A>D</A>
    is symmetric but does not admit a strong orientation, then <K>fail</K> is
    returned. If <A>D</A> is not symmetric, then an error is given. 
    <P/>

    If <A>D</A> is immutable, <C>StrongOrientation(<A>D</A>)</C> returns an
    immutable digraph, and if <A>D</A> is mutable, then
    <C>StrongOrientation(<A>D</A>)</C> returns a mutable digraph. <P/>

    The method used in this operation has complexity <M>O(m+n)</M> where
    <M>m</M> is the number of edges and <M>n</M> is the number of vertices in
    the digraph.
<Example><![CDATA[
gap> StrongOrientation(DigraphSymmetricClosure(CycleDigraph(5))) 
> = CycleDigraph(5);
true
gap> D := DigraphSymmetricClosure(Digraph(
> [[2, 7], [3, 5], [4], [2], [6], [1], []]));;
gap> IsBridgelessDigraph(D);
false
gap> StrongOrientation(D);
fail
gap> StrongOrientation(NullDigraph(0));
<immutable empty digraph with 0 vertices>
gap> StrongOrientation(DigraphDisjointUnion(CompleteDigraph(3), 
>                                           CompleteDigraph(3)));
fail
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="Bridges">
<ManSection>
  <Attr Name="Bridges" Arg="D"/>
  <Returns>A (possibly empty) list of edges.</Returns>
  <Description>
    A connected digraph is <E>2-edge-connected</E> if it is still connected (in
    the sense of <Ref Prop="IsConnectedDigraph"/>) when any edge is
    removed.  If the digraph <A>D</A> is not 2-edge-connected but is
    connected, then any edge <C>[u, v]</C> of <A>D</A> whose removal
    makes the resulting digraph disconnected is called a <E>bridge</E>.<P/>

    <C>Bridges</C> returns a list of the bridges of <A>D</A>, if any, and, in
    particular, returns the empty list if <A>D</A> is not connected. <P/>

    Multiple edges are ignored by this method. <P/>

    The method used in this operation has complexity <M>O(m+n)</M> where
    <M>m</M> is the number of edges and <M>n</M> is the number of vertices in
    the digraph.
    <P/>

    If <A>D</A> has a bridge, then a node incident to the bridge is an
    articulation point (see <Ref Attr="ArticulationPoints"/>) if and only if
    it has degree at least <M>2</M>.  It follows that if <A>D</A> has a
    bridge and at least <M>3</M> nodes, then at least one of the nodes
    incident to the bridge is an articulation point. The converse does not
    hold, there are digraphs with articulation points, but no bridges.
    <P/>

    See also <Ref Prop="IsBiconnectedDigraph"/> and 
    <Ref Prop="IsBridgelessDigraph"/>.
<Example><![CDATA[
gap> D := Digraph([[2, 5], [1, 3, 4, 5], [2, 4], [2, 3], [1, 2]]);
<immutable digraph with 5 vertices, 12 edges>
gap> Bridges(D);
[  ]
gap> D := Digraph([[2], [3], [4], [2]]);
<immutable digraph with 4 vertices, 4 edges>
gap> Bridges(D);
[ [ 1, 2 ] ]
gap> Bridges(ChainDigraph(2));
[ [ 1, 2 ] ]
gap> ArticulationPoints(ChainDigraph(2));
[  ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphAllSimpleCircuits">
<ManSection>
  <Attr Name="DigraphAllSimpleCircuits" Arg="digraph"/>
  <Returns>A list of lists of vertices.</Returns>
  <Description>
    If <A>digraph</A> is a digraph, then <C>DigraphAllSimpleCircuits</C>
    returns a list of the <E>simple circuits</E> in <A>digraph</A>. <P/>

    See Section <Ref Subsect="Definitions" Style="Number" /> for the definition
    of a simple circuit, and related notions. Note that a loop is a simple
    circuit. <P/>

    For a digraph without multiple edges, a simple circuit is uniquely
    determined by its subsequence of vertices. However this is not the case for
    a multidigraph.  The attribute <C>DigraphAllSimpleCircuits</C> ignores
    multiple edges, and identifies a simple circuit using only its subsequence
    of vertices. For example, although the simple circuits <M>(v, e, v)</M> and
    <M>(v, e', v)</M> (for distinct edges <M>e</M> and <M>e'</M>) are
    mathematically distinct, <C>DigraphAllSimpleCircuits</C> considers them to
    be the same. <P/>

    With this approach, a directed circuit of length <C>n</C> can be determined
    by a list of its first <C>n</C> vertices. Thus a simple circuit <M>(v_1,
      e_1, v_2, e_2, ..., e_{n-1}, v_n, e_{n+1}, v_1)</M> can be represented as
    the list <M>[v_1, \ldots, v_n]</M>, or any cyclic permutation thereof.  For
    each simple circuit of <A>digraph</A>,
    <C>DigraphAllSimpleCircuits(<A>digraph</A>)</C> includes precisely one such
    list to represent the circuit.  <P/>

    <Example><![CDATA[
gap> D := Digraph([[], [3], [2, 4], [5, 4], [4]]);
<immutable digraph with 5 vertices, 6 edges>
gap> DigraphAllSimpleCircuits(D);
[ [ 4 ], [ 4, 5 ], [ 2, 3 ] ]
gap> D := ChainDigraph(10);;
gap> DigraphAllSimpleCircuits(D);
[  ]
gap> D := Digraph([[3], [1], [1]]);
<immutable digraph with 3 vertices, 3 edges>
gap> DigraphAllSimpleCircuits(D);
[ [ 1, 3 ] ]
gap> D := Digraph([[1, 1]]);
<immutable multidigraph with 1 vertex, 2 edges>
gap> DigraphAllSimpleCircuits(D);
[ [ 1 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphAllSimpleCircuits(D);
[ [ 1, 2, 3 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphLongestSimpleCircuit">
<ManSection>
  <Attr Name="DigraphLongestSimpleCircuit" Arg="digraph"/>
  <Returns>A list of vertices, or <K>fail</K>.</Returns>
  <Description>
    If <A>digraph</A> is a digraph, then <C>DigraphLongestSimpleCircuit</C>
    returns the longest <E>simple circuit</E> in <A>digraph</A>. See Section
    <Ref Subsect="Definitions" Style="Number" /> for the definition of simple
    circuit, and the definition of length for a simple circuit.<P/>

    This attribute computes
    <C>DigraphAllSimpleCircuits(</C><A>digraph</A><C>)</C> to find all the
    simple circuits of <A>digraph</A>, and returns one of maximal length.  A
    simple circuit is represented as a list of vertices, in the same way as
    described in <Ref Attr="DigraphAllSimpleCircuits"/>.<P/>

    If <A>digraph</A> has no simple circuits, then this attribute returns
    <K>fail</K>.  If <A>digraph</A> has multiple simple circuits of maximal
    length, then this attribute returns one of them.<P/>

    <Example><![CDATA[
gap> D := Digraph([[], [3], [2, 4], [5, 4], [4]]);;
gap> DigraphLongestSimpleCircuit(D);
[ 4, 5 ]
gap> D := ChainDigraph(10);;
gap> DigraphLongestSimpleCircuit(D);
fail
gap> D := Digraph([[3], [1], [1, 4], [1, 1]]);;
gap> DigraphLongestSimpleCircuit(D);
[ 1, 3, 4 ]
gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 4, 1);
<mutable digraph with 8 vertices, 24 edges>
gap> DigraphLongestSimpleCircuit(D);
[ 1, 2, 3, 4, 8, 7, 6, 5 ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphAllUndirectedSimpleCircuits">
<ManSection>
  <Attr Name="DigraphAllUndirectedSimpleCircuits" Arg="digraph"/>
  <Returns>A list of lists of vertices.</Returns>
  <Description>
    If <A>digraph</A> is a digraph, then <C>DigraphAllUndirectedSimpleCircuits</C>
    returns a list of the <E> undirected simple circuits</E> in <A>digraph</A>. <P/>

    See Section <Ref Subsect="Definitions" Style="Number" /> for the definition
    of a simple circuit, and related notions. Note that a loop is a simple
    circuit. A simple circuit is undirected if the orientation of the edges
    in the circuit does not matter.<P/>

    The attribute <C>DigraphAllUndirectedSimpleCircuitss</C> ignores
    multiple edges, and identifies a undirected simple circuit using only its subsequence
    of vertices.
    See Section <Ref Subsect="DigraphAllSimpleCircuits" Style="Number" /> for more details
    on simple circuits.<P/>

    <Example><![CDATA[
gap> D := ChainDigraph(10);;
gap> DigraphAllUndirectedSimpleCircuits(D);
[  ]
gap> D := Digraph([[1, 1]]);
<immutable multidigraph with 1 vertex, 2 edges>
gap> DigraphAllUndirectedSimpleCircuits(D);
[ [ 1 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphAllUndirectedSimpleCircuits(D);
[ [ 1, 2, 3 ] ]
gap> D := DigraphSymmetricClosure(CycleDigraph(IsMutableDigraph, 3));
<mutable digraph with 3 vertices, 6 edges>
gap> DigraphAllUndirectedSimpleCircuits(D);
[ [ 1, 2, 3 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphAllChordlessCycles">
<ManSection>
  <Attr Name="DigraphAllChordlessCycles" Arg="digraph"/>
  <Attr Name="DigraphAllChordlessCyclesOfMaximalLength" Arg="digraph, maxLength"/>
  <Returns>A list of lists of vertices.</Returns>
  <Description>
    If <A>digraph</A> is a digraph, then <C>DigraphAllChordlessCycles</C> 
    returns a list of the <E>chordless cycles</E> in <A>digraph</A>. 
    The method <C>DigraphAllChordlessCyclesOfMaximalLength</C> returns
    a list of all <E>chordless cycles</E> in <A>digraph</A> of length at
    most <A>maxLength</A><P/>

    A chordless cycle <M>C</M> is a undirected simple circuit (see Section <Ref Subsect="Definitions" Style="Number" /> ) 
    where each pair of vertices in <M>C</M> are not connected by an edge not in <M>C</M>. 
    Here, cycles of length two are ignored.<P/>

    For a digraph without multiple edges, a simple circuit is uniquely
    determined by its subsequence of vertices. However this is not the case for
    a multidigraph.  The attribute <C>DigraphAllChordlessCycles</C> ignores
    multiple edges, and identifies a simple circuit using only its subsequence
    of vertices. See Section <Ref Subsect="DigraphAllSimpleCircuits" Style="Number" /> for more details.<P/>

    This method uses the algorithms described in <Cite Key="US14"/>.<P/>
    <Example><![CDATA[
gap> D := ChainDigraph(10);;
gap> DigraphAllChordlessCycles(D);
[  ]
gap> D := Digraph([[1, 1]]);
<immutable multidigraph with 1 vertex, 2 edges>
gap> DigraphAllChordlessCycles(D);
[  ]
gap> D := CycleDigraph(3);;
gap> DigraphAllChordlessCycles(D);
[ [ 2, 1, 3 ] ]
gap> D := CompleteDigraph(4);
<immutable complete digraph with 4 vertices>
gap> DigraphAllChordlessCycles(D);
[ [ 2, 1, 3 ], [ 2, 1, 4 ], [ 3, 1, 4 ], [ 3, 2, 4 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="FacialWalks">
<ManSection>
  <Oper Name="FacialWalks" Arg="digraph, list"/>
  <Returns>A list of lists of vertices.</Returns>
  <Description>
    If <A>digraph</A> is an Eulerian digraph and <A>list</A> is a rotation system of <A>digraph</A>, 
    then <C>FacialWalks</C> returns a list of the <E>facial walks</E> in <A>digraph</A>. <P/>

    A rotation system defines for each vertex the ordering of the out-neighbours.
    For example, the method <Ref Subsect="PlanarEmbedding" Style="Number" /> computes for a
    planar digraph <A>D</A> the rotation system of a planar embedding of <A>D</A>.
    The facial walks of <A>digraph</A> are closed walks and they are defined by the rotation system <A>list</A>. 
    They describe the boundaries of the faces of the embedding of <A>digraph</A> given
    by the rotation system <A>list</A>.
    
    The operation <C>FacialWalks</C> ignores
    multiple edges and loops.<P/>
    Here are some examples for planar embeddings:
    <Example><![CDATA[
gap> D1 := CycleDigraph(4);;
gap> planar := PlanarEmbedding(D1);
[ [ 2 ], [ 3 ], [ 4 ], [ 1 ] ]
gap> FacialWalks(D1, planar);
[ [ 1, 2, 3, 4 ] ]
gap> nonPlanar := [[2, 4], [1, 3], [2, 4], [1, 3]];;
gap> FacialWalks(D1, nonPlanar);
[ [ 1, 2, 3, 4 ] ]
gap> D2 := CompleteMultipartiteDigraph([2, 2, 2]);;
gap> rotationSystem := PlanarEmbedding(D2);
[ [ 3, 5, 4, 6 ], [ 6, 4, 5, 3 ], [ 6, 2, 5, 1 ], [ 1, 5, 2, 6 ],
  [ 1, 3, 2, 4 ], [ 1, 4, 2, 3 ] ]
gap> FacialWalks(D2, rotationSystem);
[ [ 1, 3, 6 ], [ 1, 4, 5 ], [ 1, 5, 3 ], [ 1, 6, 4 ], [ 2, 3, 5 ],
  [ 2, 4, 6 ], [ 2, 5, 4 ], [ 2, 6, 3 ] ]
]]></Example>
    Here is an example of a non-planar digraph with a corresponding rotation system:
    <Example><![CDATA[
gap> D3 := CompleteMultipartiteDigraph([3, 3]);;
gap> rot := [[6, 5, 4], [6, 5, 4], [6, 5, 4], [1, 2, 3],
>            [1, 2, 3], [1, 2, 3]];
[ [ 6, 5, 4 ], [ 6, 5, 4 ], [ 6, 5, 4 ], [ 1, 2, 3 ], [ 1, 2, 3 ],
  [ 1, 2, 3 ] ]
gap> FacialWalks(D3, rot);
[ [ 1, 4, 2, 6, 3, 5 ], [ 1, 5, 2, 4, 3, 6 ], [ 1, 6, 2, 5, 3, 4 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="HamiltonianPath">
<ManSection>
  <Attr Name="HamiltonianPath" Arg="digraph"/>
  <Returns>A list or <K>fail</K>.</Returns>
  <Description>
    Returns a Hamiltonian path if one exists, <K>fail</K> if not.<P/>

    A <E>Hamiltonian path</E> of a digraph with <C>n</C> vertices is directed
    cycle of length <C>n</C>. If <A>digraph</A> is a digraph that contains a
    Hamiltonian path, then this function returns one, described in the form
    used by <Ref Attr="DigraphAllSimpleCircuits"/>. Note if <A>digraph</A> has
    0 or 1 vertices, then <C>HamiltonianPath</C> returns <C>[]</C> or
    <C>[1]</C>, respectively.<P/>

    The method used in this attribute has the same worst case complexity as
    <Ref Oper="DigraphMonomorphism"/>.

    <Example><![CDATA[
gap> D := Digraph([[]]);
<immutable empty digraph with 1 vertex>
gap> HamiltonianCycle(D);
[ 1 ]
gap> D := Digraph([[2], [1]]);
<immutable digraph with 2 vertices, 2 edges>
gap> HamiltonianCycle(D);
[ 1, 2 ]
gap> D := Digraph([[3], [], [2]]);
<immutable digraph with 3 vertices, 2 edges>
gap> HamiltonianCycle(D);
fail
gap> D := Digraph([[2], [3], [1]]);
<immutable digraph with 3 vertices, 3 edges>
gap> HamiltonianCycle(D);
[ 1, 2, 3 ]
gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 5, 2);
<mutable digraph with 10 vertices, 30 edges>
gap> HamiltonianCycle(D);
fail
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphPeriod">
<ManSection>
  <Attr Name="DigraphPeriod" Arg="digraph"/>
  <Returns>An integer.</Returns>
  <Description>
    This function returns the period of the digraph <A>digraph</A>.<P/>

    If a digraph <A>digraph</A> has at least one directed cycle, then the period
    is the greatest positive integer which divides the lengths of all directed
    cycles of <A>digraph</A>.  If <A>digraph</A> has no directed cycles, then
    this function returns <M>0</M>.  See Section <Ref Subsect="Definitions"
      Style="Number" /> for the definition of a directed cycle. <P/>

    A digraph with a period of <M>1</M> is said to be <E>aperiodic</E>.  See
    <Ref Prop="IsAperiodicDigraph"/>. <P/>
    <Example><![CDATA[
gap> D := Digraph([[6], [1], [2], [3], [4, 4], [5]]);
<immutable multidigraph with 6 vertices, 7 edges>
gap> DigraphPeriod(D);
6
gap> D := Digraph([[2], [3, 5], [4], [5], [1, 2]]);
<immutable digraph with 5 vertices, 7 edges>
gap> DigraphPeriod(D);
1
gap> D := ChainDigraph(2);
<immutable chain digraph with 2 vertices>
gap> DigraphPeriod(D);
0
gap> IsAcyclicDigraph(D);
true
gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 5, 2);
<mutable digraph with 10 vertices, 30 edges>
gap> DigraphPeriod(D);
1
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphLoops">
<ManSection>
  <Attr Name="DigraphLoops" Arg="digraph"/>
  <Returns>A list of vertices.</Returns>
  <Description>
    If <A>digraph</A> is a digraph, then <C>DigraphLoops</C> returns the list
    consisting of the <Ref Attr="DigraphVertices"/> of <A>digraph</A>
    at which there is a loop. See <Ref Prop="DigraphHasLoops"/>. <P/>

    <Example><![CDATA[
gap> D := Digraph([[2], [3], []]);
<immutable digraph with 3 vertices, 2 edges>
gap> DigraphHasLoops(D);
false
gap> DigraphLoops(D);
[  ]
gap> D := Digraph([[3, 5], [1], [2, 4, 3], [4], [2, 1]]);
<immutable digraph with 5 vertices, 9 edges>
gap> DigraphLoops(D);
[ 3, 4 ]
gap> D := Digraph(IsMutableDigraph, [[1], [1]]);
<mutable digraph with 2 vertices, 2 edges>
gap> DigraphLoops(D);
[ 1 ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="ChromaticNumber">
<ManSection>
  <Attr Name="ChromaticNumber" Arg="digraph"/>
  <Returns> A non-negative integer.</Returns>
  <Description>
    A <E>proper colouring</E> of a digraph is a labelling of its
    vertices in such a way that adjacent vertices have different labels.
    Equivalently, a proper digraph colouring can be defined to be a <Ref
      Oper="DigraphEpimorphism"/> from a digraph onto a complete digraph. <P/>

    If <A>digraph</A> is a digraph without loops (see <Ref
      Prop="DigraphHasLoops"/>, then <C>ChromaticNumber</C> returns the least
    non-negative integer <C>n</C> such that there is a proper colouring of
    <A>digraph</A> with <C>n</C> colours.  In other words, for a digraph with at
    least one vertex, <C>ChromaticNumber</C> returns the least number <C>n</C>
    such that <C>DigraphColouring(<A>digraph</A>, n)</C> does not return
    <K>fail</K>. See <Ref Oper="DigraphColouring"
      Label="for a digraph and a number of colours"/>. <P/>

    It is possible to select the algorithm to compute the chromatic number via
    the use of value options. The permitted algorithms and values to
    pass as options are:
    <List>
      <Item><C>lawler</C> - Lawler's Algorithm 
      <Cite Key="Law1976"/></Item>
      <Item><C>byskov</C> - Byskov's Algorithm 
      <Cite Key="Bys2002"/></Item>
      <Item><C>zykov</C> - Zykov's Algorithm 
      <Cite Key="Corneil1973"/></Item>
      <Item><C>christofides</C> - Christofides's Algorithm 
      <Cite Key="Wang1974"/></Item>
    </List>

    <Example><![CDATA[
gap> ChromaticNumber(NullDigraph(10));
1
gap> ChromaticNumber(CompleteDigraph(10));
10
gap> ChromaticNumber(CompleteBipartiteDigraph(5, 5));
2
gap> ChromaticNumber(Digraph([[], [3], [5], [2, 3], [4]]));
3
gap> ChromaticNumber(NullDigraph(0));
0
gap> D := PetersenGraph(IsMutableDigraph);
<mutable digraph with 10 vertices, 30 edges>
gap> ChromaticNumber(D);
3
gap> ChromaticNumber(CompleteDigraph(10) : lawler);
10
gap> ChromaticNumber(CompleteDigraph(10) : byskov);
10
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphBicomponents">
<ManSection>
  <Attr Name="DigraphBicomponents" Arg="digraph"/>
  <Returns>A pair of lists of vertices, or <K>fail</K>.</Returns>
  <Description>
    If <A>digraph</A> is a bipartite digraph, i.e. if it satisfies <Ref
      Prop="IsBipartiteDigraph"/>, then <C>DigraphBicomponents</C> returns a
    pair of bicomponents of <A>digraph</A>. Otherwise,
    <C>DigraphBicomponents</C> returns <K>fail</K>.<P/>

    For a bipartite digraph, the vertices can be partitioned into two non-empty
    sets such that the source and range of any edge are in distinct sets. The
    parts of this partition are called <E>bicomponents</E> of <A>digraph</A>.
    Equivalently, a pair of bicomponents of <A>digraph</A> consists of the
    color-classes of a 2-coloring of <A>digraph</A>. <P/>

    For a bipartite digraph with at least 3 vertices, there is a unique pair of
    bicomponents of bipartite if and only if the digraph is connected. See <Ref
      Prop="IsConnectedDigraph"/> for more information.
    <Example><![CDATA[
gap> D := CycleDigraph(3);
<immutable cycle digraph with 3 vertices>
gap> DigraphBicomponents(D);
fail
gap> D := ChainDigraph(5);
<immutable chain digraph with 5 vertices>
gap> DigraphBicomponents(D);
[ [ 1, 3, 5 ], [ 2, 4 ] ]
gap> D := Digraph([[5], [1, 4], [5], [5], []]);
<immutable digraph with 5 vertices, 5 edges>
gap> DigraphBicomponents(D);
[ [ 1, 3, 4 ], [ 2, 5 ] ]
gap> D := CompleteBipartiteDigraph(IsMutableDigraph, 2, 3);
<mutable digraph with 5 vertices, 12 edges>
gap> DigraphBicomponents(D);
[ [ 1, 2 ], [ 3, 4, 5 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphCore">
  <ManSection>
    <Attr Name = "DigraphCore" Arg = "D"/>
    <Returns>A list of positive integers.</Returns>
    <Description>
      If <A>D</A> is a digraph, then <C>DigraphCore</C> returns a list of vertices
      corresponding to the <C>core</C> of <A>D</A>. In particular, the subdigraph
      of <A>D</A> induced by this list is isomorphic to the core of <A>D</A>.<P/>

    The <E>core</E> of a digraph <C>D</C> is the minimal subdigraph <A>C</A> of
    <C>D</C> which is a homomorphic image of <C>D</C>. The core of a digraph 
    is unique up to isomorphism.
      <Example><![CDATA[
gap> D := DigraphSymmetricClosure(CycleDigraph(8));
<immutable symmetric digraph with 8 vertices, 16 edges>
gap> DigraphCore(D);
[ 1, 2 ]
gap> D := PetersenGraph();
<immutable digraph with 10 vertices, 30 edges>
gap> DigraphCore(D);
[ 1 .. 10 ]
gap> D := Digraph(IsMutableDigraph, [[3], [3], [4], [5], [2]]);
<mutable digraph with 5 vertices, 5 edges>
gap> DigraphCore(D);
[ 2, 3, 4, 5 ]
]]></Example>
    </Description>
  </ManSection>
<#/GAPDoc>

<#GAPDoc Label="CharacteristicPolynomial">
<ManSection>
  <Attr Name="CharacteristicPolynomial" Arg="digraph"/>
  <Returns>A polynomial with integer coefficients.</Returns>
  <Description>
    This function returns the characteristic polynomial of the digraph
     <A>digraph</A>. That is it returns the characteristic polynomial
      of the adjacency matrix of the digraph <A>digraph</A>
    <Example><![CDATA[
gap> D := Digraph([
> [2, 2, 2], [1, 3, 6, 8, 9, 10], [4, 6, 8],
> [1, 2, 3, 9], [3, 3], [3, 5, 6, 10], [1, 2, 7],
> [1, 2, 3, 10, 5, 6, 10], [1, 3, 4, 5, 8, 10],
> [2, 3, 4, 6, 7, 10]]);
<immutable multidigraph with 10 vertices, 44 edges>
gap> CharacteristicPolynomial(D);
x_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\
+18*x_1-30
gap> D := CompleteDigraph(5);
<immutable complete digraph with 5 vertices>
gap> CharacteristicPolynomial(D);
x_1^5-10*x_1^3-20*x_1^2-15*x_1-4
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> CharacteristicPolynomial(D);
x_1^3-1
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="NrSpanningTrees">
<ManSection>
  <Attr Name="NrSpanningTrees" Arg="digraph"/>
  <Returns>An integer.</Returns>
  <Description>
    Returns the number of spanning trees of the symmetric digraph
    <A>digraph</A>.
    <C>NrSpanningTrees</C> will return an error if <A>digraph</A> is not a
    symmetric digraph. <P/> 

    See <Ref Prop="IsSymmetricDigraph"/> and
    <Ref Oper="IsUndirectedSpanningTree" /> for more information. 

    <Example><![CDATA[
gap> D := CompleteDigraph(5);
<immutable complete digraph with 5 vertices>
gap> NrSpanningTrees(D);
125
gap> D := DigraphSymmetricClosure(CycleDigraph(24));;
gap> NrSpanningTrees(D);
24
gap> NrSpanningTrees(EmptyDigraph(0));
0
gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);
<mutable digraph with 18 vertices, 54 edges>
gap> NrSpanningTrees(D);
1134225
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="AsGraph">
<ManSection>
  <Attr Name="AsGraph" Arg="digraph"/>
  <Returns>A &GRAPE; package graph.</Returns>
  <Description>
    If <A>digraph</A> is a digraph, then this method returns the same as
    <Ref Oper="Graph"/>, except that if <A>digraph</A> is immutable, then the
    result will be stored as a mutable attribute of <A>digraph</A>.
    In this latter case, when <C>AsGraph(</C><A>digraph</A><C>)</C> is called
    subsequently, the same &GAP; object will be returned as before.

    <Example><![CDATA[
gap> D := Digraph([[1, 2], [3], []]);
<immutable digraph with 3 vertices, 3 edges>
gap> G := AsGraph(D);
rec( adjacencies := [ [ 1, 2 ], [ 3 ], [  ] ], group := Group(()), 
  isGraph := true, names := [ 1 .. 3 ], order := 3, 
  representatives := [ 1, 2, 3 ], schreierVector := [ -1, -2, -3 ] )
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="AsTransformation">
<ManSection>
  <Attr Name="AsTransformation" Arg="digraph"/>
  <Returns>A transformation, or <K>fail</K></Returns>
  <Description>
    If <A>digraph</A> is a functional digraph, then <C>AsTransformation</C>
    returns the transformation which is defined by <A>digraph</A>.  See <Ref
      Prop="IsFunctionalDigraph"/>. Otherwise,
    <C>AsTransformation(</C><A>digraph</A><C>)</C> returns <K>fail</K>.  <P/>

    If <A>digraph</A> is a functional digraph with <M>n</M> vertices, then
    <C>AsTransformation(</C><A>digraph</A><C>)</C> will return the
    transformation <C>f</C> of degree at most <M>n</M> where for each <M>1
      \leq i \leq n</M>, <C>i ^ f</C> is equal to the unique out-neighbour of
    vertex <C>i</C> in <A>digraph</A>.

    <Example><![CDATA[
gap> D := Digraph([[1], [3], [2]]);
<immutable digraph with 3 vertices, 3 edges>
gap> AsTransformation(D);
Transformation( [ 1, 3, 2 ] )
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> AsTransformation(D);
Transformation( [ 2, 3, 1 ] )
gap> AsPermutation(last);
(1,2,3)
gap> D := Digraph([[2, 3], [], []]);
<immutable digraph with 3 vertices, 2 edges>
gap> AsTransformation(D);
fail]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphReverse">
<ManSection>
  <Oper Name="DigraphReverse" Arg="digraph"/>
  <Attr Name="DigraphReverseAttr" Arg="digraph"/>
  <Returns>A digraph.</Returns>
  <Description>
    The reverse of a digraph is the digraph formed by reversing the orientation
    of each of its edges, i.e. for every edge <C>[i, j]</C> of a digraph, the
    reverse contains the corresponding edge <C>[j, i]</C>.<P/>

    <C>DigraphReverse</C> returns the reverse of the digraph <A>digraph</A>. If
    <A>digraph</A> is immutable, then a new immutable digraph is returned. Otherwise,
    the mutable digraph <A>digraph</A> is changed in-place into its reverse.
    <Example><![CDATA[
gap> D := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);
<immutable digraph with 5 vertices, 11 edges>
gap> DigraphReverse(D);
<immutable digraph with 5 vertices, 11 edges>
gap> OutNeighbours(last);
[ [ 2, 3, 4 ], [ 4, 5 ], [ 1, 2, 5 ], [ 4 ], [ 2, 5 ] ]
gap> D := Digraph([[2, 4], [1], [4], [3, 4]]);
<immutable digraph with 4 vertices, 6 edges>
gap> DigraphEdges(D);
[ [ 1, 2 ], [ 1, 4 ], [ 2, 1 ], [ 3, 4 ], [ 4, 3 ], [ 4, 4 ] ]
gap> DigraphEdges(DigraphReverse(D));
[ [ 1, 2 ], [ 2, 1 ], [ 3, 4 ], [ 4, 1 ], [ 4, 3 ], [ 4, 4 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> OutNeighbours(D);
[ [ 2 ], [ 3 ], [ 1 ] ]
gap> DigraphReverse(D);
<mutable digraph with 3 vertices, 3 edges>
gap> OutNeighbours(D);
[ [ 3 ], [ 1 ], [ 2 ] ]
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphDual">
<ManSection>
  <Oper Name="DigraphDual" Arg="digraph"/>
  <Attr Name="DigraphDualAttr" Arg="digraph"/>
  <Returns>A digraph.</Returns>
  <Description>
    The <E>dual</E> of <A>digraph</A> has the same vertices as <A>digraph</A>,
    and there is an edge in the dual from <C>i</C> to <C>j</C>
    whenever there is no edge from <C>i</C> to <C>j</C> in <A>digraph</A>.
    The <E>dual</E> is sometimes called the <E>complement</E>.<P/>

    <C>DigraphDual</C> returns the dual of the digraph <A>digraph</A>. If
    <A>digraph</A> is an immutable digraph, then a new immutable digraph is
    returned. Otherwise, the mutable digraph <A>digraph</A> is changed in-place
    into its dual.
    <Example><![CDATA[
gap> D := Digraph([[2, 3], [], [4, 6], [5], [],
> [7, 8, 9], [], [], []]);
<immutable digraph with 9 vertices, 8 edges>
gap> DigraphDual(D);
<immutable digraph with 9 vertices, 73 edges>
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphDual(D);
<mutable digraph with 3 vertices, 6 edges>
gap> D;
<mutable digraph with 3 vertices, 6 edges>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="ReducedDigraph">
<ManSection>
  <Oper Name="ReducedDigraph" Arg="digraph"/>
  <Attr Name="ReducedDigraphAttr" Arg="digraph"/>
  <Returns>A digraph.</Returns>
  <Description>
    This function returns a digraph isomorphic to the subdigraph of
    <A>digraph</A> induced by the set of non-isolated vertices,
    i.e. the set of those vertices of <A>digraph</A> which are the source or
    range of some edge in <A>digraph</A>.
    See <Ref Oper="InducedSubdigraph"/>.
    <P/>

    The ordering of the remaining vertices of <A>digraph</A> is preserved, as are
    the labels of the remaining vertices and edges;
    see <Ref Oper="DigraphVertexLabels"/> and <Ref Oper="DigraphEdgeLabels"/>.
    This can allow one to match a vertex in the reduced digraph to the
    corresponding vertex in <A>digraph</A>.
    <P/>

    If <A>digraph</A> is immutable, then a new immutable digraph is returned.
    Otherwise, the isolated vertices of the mutable digraph <A>digraph</A> are
    removed in-place.
    <Example><![CDATA[
gap> D := Digraph([[1, 2], [], [], [1, 4], []]);
<immutable digraph with 5 vertices, 4 edges>
gap> R := ReducedDigraph(D);
<immutable digraph with 3 vertices, 4 edges>
gap> OutNeighbours(R);
[ [ 1, 2 ], [  ], [ 1, 3 ] ]
gap> DigraphEdges(D);
[ [ 1, 1 ], [ 1, 2 ], [ 4, 1 ], [ 4, 4 ] ]
gap> DigraphEdges(R);
[ [ 1, 1 ], [ 1, 2 ], [ 3, 1 ], [ 3, 3 ] ]
gap> DigraphVertexLabel(R, 3);
4
gap> DigraphVertexLabel(R, 2);
2
gap> D := Digraph(IsMutableDigraph, [[], [3], [2]]);
<mutable digraph with 3 vertices, 2 edges>
gap> ReducedDigraph(D);
<mutable digraph with 2 vertices, 2 edges>
gap> D;
<mutable digraph with 2 vertices, 2 edges>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphRemoveAllMultipleEdges">
<ManSection>
  <Oper Name="DigraphRemoveAllMultipleEdges" Arg="digraph"/>
  <Attr Name="DigraphRemoveAllMultipleEdgesAttr" Arg="digraph"/>
  <Returns>A digraph.</Returns>
  <Description>
    If <A>digraph</A> is a digraph, then this operation returns a
    digraph  constructed from <A>digraph</A> by removing all multiple edges.
    The result is the largest subdigraph of <A>digraph</A> which does not
    contain multiple edges.<P/>

    If <A>digraph</A> is immutable, then a new immutable digraph is returned.
    Otherwise, the multiple edges of the mutable digraph <A>digraph</A> are
    removed in-place.
    <Example><![CDATA[
gap> D1 := Digraph([[1, 2, 3, 2], [1, 1, 3], [2, 2, 2]]);
<immutable multidigraph with 3 vertices, 10 edges>
gap> D2 := DigraphRemoveAllMultipleEdges(D1);
<immutable digraph with 3 vertices, 6 edges>
gap> OutNeighbours(D2);
[ [ 1, 2, 3 ], [ 1, 3 ], [ 2 ] ]
gap> D := Digraph(IsMutableDigraph, [[2, 2], [1]]);
<mutable multidigraph with 2 vertices, 3 edges>
gap> DigraphRemoveAllMultipleEdges(D);
<mutable digraph with 2 vertices, 2 edges>
gap> D;
<mutable digraph with 2 vertices, 2 edges>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphRemoveAllEdges">
<ManSection>
  <Oper Name="DigraphRemoveAllEdges" Arg="digraph"/>
  <Attr Name="DigraphRemoveAllEdgesAttr" Arg="digraph"/>
  <Returns>An empty digraph.</Returns>
  <Description>
    This operation returns a digraph constructed from <A>digraph</A>. If <A>digraph</A> is mutable, 
    its list of neighbours will be replaced with an empty 2D list of the same length. 
    <P/>

    If <A>digraph</A> is immutable, then a new, empty immutable digraph with the same number of vertices is returned.

    <Example><![CDATA[
gap> D3 := Digraph(IsMutableDigraph, [[2], [1], [2]]);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphRemoveAllEdges(D3);
<mutable empty digraph with 3 vertices>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphAddAllLoops">
<ManSection>
  <Oper Name="DigraphAddAllLoops" Arg="digraph"/>
  <Attr Name="DigraphAddAllLoopsAttr" Arg="digraph"/>
  <Returns>A digraph.</Returns>
  <Description>
    For a digraph <A>digraph</A> this operation returns a new digraph constructed
    from <A>digraph</A>, such that a loop is added for every vertex which did not
    have a loop in <A>digraph</A>.<P/>

    If <A>digraph</A> is immutable, then a new immutable digraph is returned.
    Otherwise, the loops are added to the loopless vertices of the mutable digraph
    <A>digraph</A> in-place.
    <Example><![CDATA[
gap> D := EmptyDigraph(13);
<immutable empty digraph with 13 vertices>
gap> D := DigraphAddAllLoops(D);
<immutable reflexive digraph with 13 vertices, 13 edges>
gap> OutNeighbours(D);
[ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 5 ], [ 6 ], [ 7 ], [ 8 ], [ 9 ], 
  [ 10 ], [ 11 ], [ 12 ], [ 13 ] ]
gap> D := Digraph([[1, 2, 3], [1, 3], [1]]);
<immutable digraph with 3 vertices, 6 edges>
gap> D := DigraphAddAllLoops(D);
<immutable reflexive digraph with 3 vertices, 8 edges>
gap> OutNeighbours(D);
[ [ 1, 2, 3 ], [ 1, 3, 2 ], [ 1, 3 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphAddAllLoops(D);
<mutable digraph with 3 vertices, 6 edges>
gap> D;
<mutable digraph with 3 vertices, 6 edges>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphRemoveLoops">
<ManSection>
  <Oper Name="DigraphRemoveLoops" Arg="digraph"/>
  <Attr Name="DigraphRemoveLoopsAttr" Arg="digraph"/>
  <Returns>A digraph.</Returns>
  <Description>
    If <A>digraph</A> is a digraph, then this operation returns a
    digraph  constructed from <A>digraph</A> by removing every loop. A loop
    is an edge with equal source and range.<P/>

    If <A>digraph</A> is immutable, then a new immutable digraph is returned.
    Otherwise, the loops are removed from the mutable digraph <A>digraph</A>
    in-place.
    <Example><![CDATA[
gap> D := Digraph([[1, 2, 4], [1, 4], [3, 4], [1, 4, 5], [1, 5]]);
<immutable digraph with 5 vertices, 12 edges>
gap> DigraphRemoveLoops(D);
<immutable digraph with 5 vertices, 8 edges>
gap> D := Digraph(IsMutableDigraph, [[1, 2], [1]]);
<mutable digraph with 2 vertices, 3 edges>
gap> DigraphRemoveLoops(D);
<mutable digraph with 2 vertices, 2 edges>
gap> D;
<mutable digraph with 2 vertices, 2 edges>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphSymmetricClosure">
<ManSection>
  <Oper Name="DigraphSymmetricClosure" Arg="digraph"/>
  <Attr Name="DigraphSymmetricClosureAttr" Arg="digraph"/>
  <Returns>A digraph.</Returns>
  <Description>
    If <A>digraph</A> is a digraph, then this attribute gives the minimal
    symmetric digraph which has the same vertices and contains all the edges of
    <A>digraph</A>.<P/>
    
    A digraph is <E>symmetric</E> if its adjacency matrix 
    <Ref Attr="AdjacencyMatrix"/> is symmetric. For a digraph with multiple
    edges this means that there are the same number of edges from a vertex
    <C>u</C> to a vertex <C>v</C> as there are from <C>v</C> to <C>u</C>; 
    see <Ref Prop="IsSymmetricDigraph"/>.<P/>

    If <A>digraph</A> is immutable, then a new immutable digraph is returned.
    Otherwise, the mutable digraph <A>digraph</A> is changed in-place into its
    symmetric closure.
<Example><![CDATA[
gap> D := Digraph([[1, 2, 3], [2, 4], [1], [3, 4]]);
<immutable digraph with 4 vertices, 8 edges>
gap> D := DigraphSymmetricClosure(D);
<immutable symmetric digraph with 4 vertices, 11 edges>
gap> IsSymmetricDigraph(D);
true
gap> List(OutNeighbours(D), AsSet);
[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 4 ], [ 2, 3, 4 ] ]
gap> D := Digraph([[2, 2], [1]]);
<immutable multidigraph with 2 vertices, 3 edges>
gap> D := DigraphSymmetricClosure(D);
<immutable symmetric multidigraph with 2 vertices, 4 edges>
gap> OutNeighbours(D);
[ [ 2, 2 ], [ 1, 1 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphSymmetricClosure(D);
<mutable digraph with 3 vertices, 6 edges>
gap> D;
<mutable digraph with 3 vertices, 6 edges>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="MaximalSymmetricSubdigraph">
<ManSection>
  <Oper Name="MaximalSymmetricSubdigraph" Arg="digraph"/>
  <Attr Name="MaximalSymmetricSubdigraphAttr" Arg="digraph"/>
  <Oper Name="MaximalSymmetricSubdigraphWithoutLoops" Arg="digraph"/>
  <Attr Name="MaximalSymmetricSubdigraphWithoutLoopsAttr" Arg="digraph"/>
  <Returns>A digraph.</Returns>
  <Description>
    If <A>digraph</A> is a digraph, then <C>MaximalSymmetricSubdigraph</C>
    returns a symmetric digraph without multiple edges which has the same
    vertex set as <A>digraph</A>, and whose edge list is formed from
    <A>digraph</A> by ignoring the multiplicity of edges, and by ignoring
    edges <C>[u,v]</C> for which there does not exist an edge <C>[v,u]</C>.
    <P/>

    The digraph returned by <C>MaximalSymmetricSubdigraphWithoutLoops</C> is
    the same, except that loops are removed.<P/>
    
    If <A>digraph</A> is immutable, then a new immutable digraph is returned.
    Otherwise, the mutable digraph <A>digraph</A> is changed in-place into such
    a digraph described above.<P/>

    See <Ref Prop="IsSymmetricDigraph"/>, <Ref Prop="IsMultiDigraph"/>, and
    <Ref Prop="DigraphHasLoops"/> for more information.
    <Example><![CDATA[
gap> D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);
<immutable multidigraph with 4 vertices, 7 edges>
gap> not IsSymmetricDigraph(D) and IsMultiDigraph(D);
true
gap> OutNeighbours(D);
[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]
gap> S := MaximalSymmetricSubdigraph(D);
<immutable symmetric digraph with 4 vertices, 4 edges>
gap> IsSymmetricDigraph(S) and not IsMultiDigraph(S);
true
gap> OutNeighbours(S);
[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> MaximalSymmetricSubdigraph(D);
<mutable empty digraph with 3 vertices>
gap> D;
<mutable empty digraph with 3 vertices>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="MaximalAntiSymmetricSubdigraph">
<ManSection>
  <Oper Name="MaximalAntiSymmetricSubdigraph" Arg="digraph"/>
  <Attr Name="MaximalAntiSymmetricSubdigraphAttr" Arg="digraph"/>
  <Returns>A digraph.</Returns>
  <Description>
    If <A>digraph</A> is a digraph, then <C>MaximalAntiSymmetricSubdigraph</C>
    returns an anti-symmetric subdigraph of <A>digraph</A> formed by
    retaining the vertices of <A>digraph</A>, discarding any duplicate edges,
    and discarding any edge <C>[i,j]</C> of <A>digraph</A> where <C>i > j</C>
    and the reverse edge <C>[j,i]</C> is an edge of <A>digraph</A>.
    In other words, for every
    symmetric pair of edges <C>[i,j]</C> and <C>[j,i]</C> in <A>digraph</A>, where 
    <C>i</C> and <C>j</C> are distinct, it discards the edge
    <M>[\max(i,j),\min(i,j)]</M>.
    <P/>

    If <A>digraph</A> is immutable, then a new immutable digraph is returned.
    Otherwise, the mutable digraph <A>digraph</A> is changed in-place. <P/>

    See <Ref Prop="IsAntisymmetricDigraph"/> for more information.
    <Example><![CDATA[
gap> D := Digraph([[2, 2], [1, 3], [4], [3, 1]]);
<immutable multidigraph with 4 vertices, 7 edges>
gap> not IsAntiSymmetricDigraph(D) and IsMultiDigraph(D);
true
gap> OutNeighbours(D);
[ [ 2, 2 ], [ 1, 3 ], [ 4 ], [ 3, 1 ] ]
gap> D := MaximalAntiSymmetricSubdigraph(D);
<immutable antisymmetric digraph with 4 vertices, 4 edges>
gap> IsAntiSymmetricDigraph(D) and not IsMultiDigraph(D);
true
gap> OutNeighbours(D);
[ [ 2 ], [ 3 ], [ 4 ], [ 1 ] ]
gap> D := Digraph(IsMutableDigraph, [[2], [1]]);
<mutable digraph with 2 vertices, 2 edges>
gap> MaximalAntiSymmetricSubdigraph(D);
<mutable digraph with 2 vertices, 1 edge>
gap> D;
<mutable digraph with 2 vertices, 1 edge>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphReflexiveTransitiveClosure">
<ManSection>
  <Oper Name="DigraphTransitiveClosure" Arg="digraph"/>
  <Attr Name="DigraphTransitiveClosureAttr" Arg="digraph"/>
  <Oper Name="DigraphReflexiveTransitiveClosure" Arg="digraph"/>
  <Attr Name="DigraphReflexiveTransitiveClosureAttr" Arg="digraph"/>
  <Returns>A digraph.</Returns>
  <Description>
    If <A>digraph</A> is a digraph with no multiple edges, then these
    attributes return the (reflexive) transitive closure of <A>digraph</A>.<P/>

    A digraph is <E>reflexive</E> if it has a loop at every vertex, and it is
    <E>transitive</E> if whenever <C>[i,j]</C> and <C>[j,k]</C> are edges of
    <A>digraph</A>, <C>[i,k]</C> is also an edge.  The <E>(reflexive)
      transitive closure</E> of a digraph <A>digraph</A> is the least
    (reflexive and) transitive digraph containing <A>digraph</A>.<P/>

    If <A>digraph</A> is immutable, then a new immutable digraph is returned.
    Otherwise, the mutable digraph <A>digraph</A> is changed in-place into its
    (reflexive) transitive closure.<P/>

    Let <M>n</M> be the number of vertices of <A>digraph</A>, and let
    <M>m</M> be the number of edges.  For an arbitrary digraph, these
    attributes will use a version of the Floyd-Warshall algorithm, with
    complexity <M>O(n^3)</M>.

    However, for a topologically sortable digraph [see <Ref
      Attr="DigraphTopologicalSort"/>], these attributes will use methods
    with complexity <M>O(m + n + m \cdot n)</M> when this is faster.<P/>

    <Example><![CDATA[
gap> D := DigraphFromDiSparse6String(".H`eOWR`Ul^");
<immutable digraph with 9 vertices, 8 edges>
gap> IsReflexiveDigraph(D) or IsTransitiveDigraph(D);
false
gap> OutNeighbours(D);
[ [ 4, 6 ], [ 1, 3 ], [  ], [ 5 ], [  ], [ 7, 8, 9 ], [  ], [  ], 
  [  ] ]
gap> T := DigraphTransitiveClosure(D);
<immutable transitive digraph with 9 vertices, 18 edges>
gap> OutNeighbours(T);
[ [ 4, 6, 5, 7, 8, 9 ], [ 1, 3, 4, 5, 6, 7, 8, 9 ], [  ], [ 5 ], 
  [  ], [ 7, 8, 9 ], [  ], [  ], [  ] ]
gap> RT := DigraphReflexiveTransitiveClosure(D);
<immutable preorder digraph with 9 vertices, 27 edges>
gap> OutNeighbours(RT);
[ [ 4, 6, 5, 7, 8, 9, 1 ], [ 1, 3, 4, 5, 6, 7, 8, 9, 2 ], [ 3 ], 
  [ 5, 4 ], [ 5 ], [ 7, 8, 9, 6 ], [ 7 ], [ 8 ], [ 9 ] ]
gap> D := CycleDigraph(IsMutableDigraph, 3);
<mutable digraph with 3 vertices, 3 edges>
gap> DigraphReflexiveTransitiveClosure(D);
<mutable digraph with 3 vertices, 9 edges>
gap> D;
<mutable digraph with 3 vertices, 9 edges>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphReflexiveTransitiveReduction">
<ManSection>
  <Oper Name="DigraphTransitiveReduction" Arg="digraph"/>
  <Attr Name="DigraphTransitiveReductionAttr" Arg="digraph"/>
  <Oper Name="DigraphReflexiveTransitiveReduction" Arg="digraph"/>
  <Attr Name="DigraphReflexiveTransitiveReductionAttr" Arg="digraph"/>
  <Returns>A digraph.</Returns>
  <Description>
    If <A>digraph</A> is a topologically sortable digraph
    [see <Ref Attr="DigraphTopologicalSort"/>]
    with no multiple edges, then these  operations return the
    (reflexive) transitive reduction of <A>digraph</A>.<P/>

    The (reflexive) transitive reduction of such a digraph is the unique
    least subgraph such that the (reflexive) transitive closure of the
    subgraph is equal to the (reflexive) transitive closure of <A>digraph</A>
    [see <Ref Oper="DigraphReflexiveTransitiveClosure"/>].
    In order words, it is the least subgraph of <A>digraph</A> which
    retains the same reachability as <A>digraph</A>.<P/>

    If <A>digraph</A> is immutable, then a new immutable digraph is returned.
    Otherwise, the mutable digraph <A>digraph</A> is changed in-place into its
    (reflexive) transitive reduction.<P/>

    Let <M>n</M> be the number of vertices of an arbitrary digraph, and let
        <M>m</M> be the number of edges.
    Then these operations
    use methods with complexity <M>O(m + n + m \cdot n)</M>.
    <P/>
    <Example><![CDATA[
gap> D := Digraph([[1, 2, 3], [3], [3]]);;
gap> DigraphHasLoops(D);
true
gap> D1 := DigraphReflexiveTransitiveReduction(D);
<immutable digraph with 3 vertices, 2 edges>
gap> DigraphHasLoops(D1);
false
gap> OutNeighbours(D1);
[ [ 2 ], [ 3 ], [  ] ]
gap> D2 := DigraphTransitiveReduction(D);
<immutable digraph with 3 vertices, 4 edges>
gap> DigraphHasLoops(D2);
true
gap> OutNeighbours(D2);
[ [ 2, 1 ], [ 3 ], [ 3 ] ]
gap> DigraphReflexiveTransitiveClosure(D)
>  = DigraphReflexiveTransitiveClosure(D1);
true
gap> DigraphTransitiveClosure(D)
>  = DigraphTransitiveClosure(D2);
true
gap> D := Digraph(IsMutableDigraph, [[1], [1], [1, 2, 3]]);
<mutable digraph with 3 vertices, 5 edges>
gap> DigraphReflexiveTransitiveReduction(D);
<mutable digraph with 3 vertices, 2 edges>
gap> D;
<mutable digraph with 3 vertices, 2 edges>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="UndirectedSpanningTree">
<ManSection>
  <Oper Name="UndirectedSpanningForest" Arg="digraph"/>
  <Attr Name="UndirectedSpanningForestAttr" Arg="digraph"/>
  <Oper Name="UndirectedSpanningTree" Arg="digraph"/>
  <Attr Name="UndirectedSpanningTreeAttr" Arg="digraph"/>
  <Returns>A digraph, or <K>fail</K>.</Returns>
  <Description>
    If <A>digraph</A> is a digraph with at least one vertex, then
    <C>UndirectedSpanningForest</C> returns an undirected spanning forest of
    <A>digraph</A>, otherwise this attribute returns <K>fail</K>. See <Ref
      Oper="IsUndirectedSpanningForest" /> for the definition of an undirected
    spanning forest.<P/>

    If <A>digraph</A> is a digraph with at least one vertex and whose <Ref
      Oper="MaximalSymmetricSubdigraph"/> is connected (see <Ref
      Prop="IsConnectedDigraph" />), then <C>UndirectedSpanningTree</C> returns
    an undirected spanning tree of <A>digraph</A>, otherwise this attribute
    returns <K>fail</K>.  See <Ref Oper="IsUndirectedSpanningTree" /> for the
    definition of an undirected spanning tree.<P/>

    If <A>digraph</A> is immutable, then an immutable digraph is returned.
    Otherwise, the mutable digraph <A>digraph</A> is changed in-place into an
    undirected spanning tree of <A>digraph</A>.<P/>

    Note that for an immutable digraph that has known undirected spanning tree,
    the attribute <C>UndirectedSpanningTree</C> returns the same digraph as
    the attribute <C>UndirectedSpanningForest</C>.

    <Example><![CDATA[
gap> D := Digraph([[1, 2, 1, 3], [1], [4], [3, 4, 3]]);
<immutable multidigraph with 4 vertices, 9 edges>
gap> UndirectedSpanningTree(D);
fail
gap> forest := UndirectedSpanningForest(D);
<immutable undirected forest with 4 vertices>
gap> OutNeighbours(forest);
[ [ 2 ], [ 1 ], [ 4 ], [ 3 ] ]
gap> IsUndirectedSpanningForest(D, forest);
true
gap> DigraphConnectedComponents(forest).comps;
[ [ 1, 2 ], [ 3, 4 ] ]
gap> DigraphConnectedComponents(MaximalSymmetricSubdigraph(D)).comps;
[ [ 1, 2 ], [ 3, 4 ] ]
gap> UndirectedSpanningForest(MaximalSymmetricSubdigraph(D))
> = forest;
true
gap> D := CompleteDigraph(4);
<immutable complete digraph with 4 vertices>
gap> tree := UndirectedSpanningTree(D);
<immutable undirected tree with 4 vertices>
gap> IsUndirectedSpanningTree(D, tree);
true
gap> tree = UndirectedSpanningForest(D);
true
gap> UndirectedSpanningForest(EmptyDigraph(0));
fail
gap> D := PetersenGraph(IsMutableDigraph);
<mutable digraph with 10 vertices, 30 edges>
gap> UndirectedSpanningTree(D);
<mutable digraph with 10 vertices, 18 edges>
gap> D;
<mutable digraph with 10 vertices, 18 edges>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphMycielskian">
<ManSection>
  <Oper Name="DigraphMycielskian" Arg="digraph"/>
  <Attr Name="DigraphMycielskianAttr" Arg="digraph"/>
  <Returns>A digraph.</Returns>
  <Description>
    If <A>digraph</A> is a symmetric digraph, then
    <C>DigraphMycielskian</C> returns the Mycielskian of <A>digraph</A>.<P/>
    
    The Mycielskian of a symmetric digraph is a larger symmetric digraph
    constructed from it, which has a larger chromatic number. For further
    information, see <URL>https://en.wikipedia.org/wiki/Mycielskian</URL>.<P/>

    If <A>digraph</A> is immutable, then a new immutable digraph is returned.
    Otherwise, the mutable digraph <A>digraph</A> is changed in-place into
    its Mycielskian.<P/>

    <Example><![CDATA[
gap> D := CycleDigraph(2);
<immutable cycle digraph with 2 vertices>
gap> ChromaticNumber(D);
2
gap> D := DigraphMycielskian(D);
<immutable digraph with 5 vertices, 10 edges>
gap> ChromaticNumber(D);
3
gap> D := DigraphMycielskian(D);
<immutable digraph with 11 vertices, 40 edges>
gap> ChromaticNumber(D);
4
gap> D := CompleteBipartiteDigraph(IsMutable, 2, 3);
<mutable digraph with 5 vertices, 12 edges>
gap> DigraphMycielskian(D);
<mutable digraph with 11 vertices, 46 edges>
gap> D;
<mutable digraph with 11 vertices, 46 edges>
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphCartesianProductProjections">
<ManSection>
  <Attr Name="DigraphCartesianProductProjections" Arg="digraph"/>
  <Returns>A list of transformations.</Returns>
  <Description>
    If <A>digraph</A> is a Cartesian product digraph, 
    <C>digraph = DigraphCartesianProduct(gr_1, gr_2, ... )</C>, then
    <C>DigraphCartesianProductProjections</C> returns a list <C>proj</C>
    such that <C>proj[i]</C> is the projection onto the <C>i</C>-th
    coordinate of the product.
    <P/>

    A projection is an idempotent endomorphism of <A>digraph</A>. If
    <C>gr1, gr2, ...</C> are all loopless digraphs, then the induced 
    subdigraph of <A>digraph</A> on the image of <C>proj[i]</C> is 
    isomorphic to <C>gr_i</C>.
    <P/>

    Currently this attribute is only set upon creating an immutable 
    digraph via <C>DigraphCartesianProduct</C> and there is no way 
    of calculating it for other digraphs.
    <P/>

    For more information see
    <Ref Func="DigraphCartesianProduct" Label="for a list of digraphs"/> 

    <Example><![CDATA[
gap> D := DigraphCartesianProduct(ChainDigraph(3), CycleDigraph(4),
> Digraph([[2], [2]]));;
gap> HasDigraphCartesianProductProjections(D);
true
gap> proj := DigraphCartesianProductProjections(D);; Length(proj);
3
gap> IsIdempotent(proj[2]);
true
gap> RankOfTransformation(proj[3]);
2
gap> S := ImageSetOfTransformation(proj[2]);;
gap> IsIsomorphicDigraph(CycleDigraph(4), InducedSubdigraph(D, S));
true
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphDirectProductProjections">
<ManSection>
  <Attr Name="DigraphDirectProductProjections" Arg="digraph"/>
  <Returns>A list of transformations.</Returns>
  <Description>
    If <A>digraph</A> is a direct product digraph, 
    <C>digraph = DigraphDirectProduct(gr_1, gr_2, ... )</C>, then
    <C>DigraphDirectProductProjections</C> returns a list <C>proj</C>
    such that <C>proj[i]</C> is the projection onto the <C>i</C>-th
    coordinate of the product.
    <P/>

    A projection is an idempotent endomorphism of <A>digraph</A>. If
    <C>gr1, gr2, ...</C> are all loopless digraphs, then the image 
    of <A>digraph</A> under <C>proj[i]</C> is isomorphic to <C>gr_i</C>.
    <P/>

    Currently this attribute is only set upon creating an immutable
    digraph via <C>DigraphDirectProduct</C> and there is no way of 
    calculating it for other digraphs.
    <P/>

    For more information, see
    <Ref Func="DigraphDirectProduct" Label="for a list of digraphs"/> 

    <Example><![CDATA[
gap> D := DigraphDirectProduct(ChainDigraph(3), CycleDigraph(4),
> Digraph([[2], [2]]));;
gap> HasDigraphDirectProductProjections(D);
true
gap> proj := DigraphDirectProductProjections(D);; Length(proj);
3
gap> IsIdempotent(proj[2]);
true
gap> RankOfTransformation(proj[3]);
2
gap> P := DigraphRemoveAllMultipleEdges(
> ReducedDigraph(OnDigraphs(D, proj[2])));; 
gap> IsIsomorphicDigraph(CycleDigraph(4), P);
true
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphMaximalMatching">
<ManSection>
  <Attr Name="DigraphMaximalMatching" Arg="digraph"/>
  <Returns>A list of pairs of vertices.</Returns>
  <Description>
    This function returns a maximal matching of the digraph <A>digraph</A>.
    <P/>

    For the definition of a maximal matching, see <Ref Oper="IsMaximalMatching"/>. 

    <Example><![CDATA[
gap> D := DigraphFromDiSparse6String(".IeAoXCJU@|SHAe?d");
<immutable digraph with 10 vertices, 13 edges>
gap> M := DigraphMaximalMatching(D);; IsMaximalMatching(D, M);
true
gap> D := RandomDigraph(100);;
gap> IsMaximalMatching(D, DigraphMaximalMatching(D));
true
gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);
<mutable digraph with 18 vertices, 54 edges>
gap> IsMaximalMatching(D, DigraphMaximalMatching(D));
true
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphMaximumMatching">
<ManSection>
  <Attr Name="DigraphMaximumMatching" Arg="digraph"/>
  <Returns>A list of pairs of vertices.</Returns>
  <Description>
    This function returns a maximum matching of the digraph <A>digraph</A>.
    <P/>

    For the definition of a maximum matching, see <Ref Oper="IsMaximumMatching"/>.
    If <A>digraph</A> is bipartite (see <Ref Oper="IsBipartiteDigraph"/>), then 
    the algorithm used has complexity <C>O(m*sqrt(n))</C>. Otherwise for general 
    graphs the complexity is <C>O(m*n*log(n))</C>. Here <C>n</C> is the number of vertices 
    and <C>m</C> is the number of edges.

    <Example><![CDATA[
gap> D := DigraphFromDigraph6String("&I@EA_A?AdDp[_c??OO");
<immutable digraph with 10 vertices, 23 edges>
gap> M := DigraphMaximumMatching(D);; IsMaximalMatching(D, M);
true
gap> Length(M);
5
gap> D := Digraph([[5, 6, 7, 8], [6, 7, 8], [7, 8], [8], 
>                  [], [], [], []]);;
gap> M := DigraphMaximumMatching(D);
[ [ 1, 5 ], [ 2, 6 ], [ 3, 7 ], [ 4, 8 ] ]
gap> D := GeneralisedPetersenGraph(IsMutableDigraph, 9, 2);
<mutable digraph with 18 vertices, 54 edges>
gap> M := DigraphMaximumMatching(D);;
gap> IsMaximalMatching(D, M);
true
gap> Length(M);
9
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphVertexConnectivity">
<ManSection>
  <Attr Name="DigraphVertexConnectivity" Arg="digraph"/>
  <Returns>An non-negative integer.</Returns>
  <Description>
    This function returns the vertex connectivity of the digraph
    <A>digraph</A>. This is also sometimes called the weak vertex connectivity.
    <P/>

    The vertex connectivity of a connected digraph (in the sense of
    <Ref Prop="IsConnectedDigraph"/>) is the largest number <M>k</M> such that:
    <List>
      <Item> the digraph has at least <M>k + 1</M> vertices, and</Item>
      <Item> the digraph remains connected after the removal of any set of at
             most <M>k - 1</M> vertices.</Item>
    </List>
    If the digraph is not connected, then its vertex connectivity is 0.
    For a non-empty digraph whose symmetric closure is not complete, the vertex
    connectivity is equal to the size of the smallest set of vertices whose
    removal would disconnect the digraph.
    <P/>

    Vertex connectivity of small digraphs may be counterintuitive
    with respect to the notions of connectivity, as defined by
    <Ref Prop="IsConnectedDigraph"/>, and biconnecivity, as defined by
    <Ref Prop="IsBiconnectedDigraph"/>. Namely
    <List>
      <Item> the empty digraph is connected, but its vertex connectivity is 0;
      </Item>
      <Item> the singleton digraph is connected, but its vertex connectivity is
             0;</Item>
      <Item> the 2-cycle digraph is biconnected, but its vertex connectivity is
             only 1.</Item>
    </List>
    However, for a digraph with at least 3 vertices, having vertex connectivity
    greater than or equal to 1 is equivalent to being connected, and having
    vertex connectivity greater than or equal to 2 is equivalent to being
    biconnected.
    <P/>

    The algorithm makes <M>n - d - 1 + d \cdot (d - 1) / 2</M> calls to the
    <Ref Attr="DigraphMaximumFlow"/> maximum flow algorithm, where <M>n</M>
    is the number of vertices of <A>digraph</A>, and <M>d</M> is the
    minimum degree of the symmetric closure of <A>digraph</A>,
    see <Ref Oper="DigraphSymmetricClosure"/>.

    <Example><![CDATA[
gap> D := CompleteBipartiteDigraph(4, 5);
<immutable complete bipartite digraph with bicomponent sizes 4 and 5>
gap> DigraphVertexConnectivity(D);
4
gap> DigraphVertexConnectivity(PancakeGraph(4));
3
gap> D := Digraph([[2, 4, 5], [1, 4], [4, 7], [1, 2, 3, 5, 6, 7],
>                  [1, 4], [4, 7], [3, 4, 6]]);
<immutable digraph with 7 vertices, 20 edges>
gap> DigraphVertexConnectivity(D);
1
gap> J := JohnsonDigraph(9, 2);
<immutable symmetric digraph with 36 vertices, 504 edges>
gap> DigraphVertexConnectivity(J);
14
gap> DigraphVertexConnectivity(Digraph([]));
0
gap> DigraphVertexConnectivity(Digraph([[1]]));
0
gap> DigraphVertexConnectivity(CycleDigraph(2));
1
gap> DigraphVertexConnectivity(CompleteDigraph(5));
4
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="NonUpperSemimodularPair">
<ManSection>
  <Attr Name="NonUpperSemimodularPair" Arg="D"/>
  <Attr Name="NonLowerSemimodularPair" Arg="D"/>
  <Returns>A pair of vertices or <K>fail</K>.</Returns>
  <Description>
    <C>NonUpperSemimodularPair</C> returns a pair of vertices in the digraph
    <A>D</A> that witnesses the fact that <A>D</A> does not represent an upper
    semimodular lattice, if such a pair exists. <P/> 

    If the digraph <A>D</A> does not satisfy <Ref
    Prop="IsLatticeDigraph"/>, then an error is given. Otherwise if the
    digraph <A>D</A> does satisfy <Ref
    Prop="IsLatticeDigraph"/>, then either a non-upper semimodular pair of
    vertices is returned, or <K>fail</K> is returned if no such pair exists
    (meaning that <A>D</A> is an upper semimodular lattice. <P/>

    <C>NonLowerSemimodularPair</C> behaves in the analogous way to
    <C>NonUpperSemimodularPair</C> with respect to lower semimodularity. <P/>

    See <Ref Prop="IsUpperSemimodularDigraph"/> and <Ref
    Prop="IsLowerSemimodularDigraph"/> for the definition of upper
    semimodularity of a lattice. 

    <Example><![CDATA[
gap> D := DigraphFromDigraph6String(
> "&M~~sc`lYUZO__KIBboC_@h?U_?_GL?A_?c");
<immutable digraph with 14 vertices, 66 edges>
gap> NonLowerSemimodularPair(D);
[ 10, 9 ]
gap> NonUpperSemimodularPair(D);
fail
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphMeetTable">
<ManSection>
  <Attr Name="DigraphMeetTable" Arg="digraph"/>
  <Attr Name="DigraphJoinTable" Arg="digraph"/>
  <Returns>A matrix or <K>fail</K>.</Returns>
  <Description>
    <C>DigraphMeetTable</C> returns the <E>meet table</E> of
    <A>digraph</A> if <A>digraph</A> is a meet semilattice digraph
    <Ref Prop="IsMeetSemilatticeDigraph"/>. Similarly,
    <C>DigraphJoinTable</C> returns the <E>join table</E> of
    <A>digraph</A> if <A>digraph</A> is a join semilattice digraph.
    <Ref Prop="IsJoinSemilatticeDigraph"/>.<P/>

    When <A>digraph</A> is a meet semilattice digraph with <M>n</M> vertices,
    the <E>meet table</E> of <A>digraph</A> is the matrix <M>A</M> such that
    the <M>(i, j)</M> entry of <M>A</M> is the meet of vertices
    <M>i</M> and <M>j</M>.<P/>

    Similarly, when <A>digraph</A> is a join semilattice digraph with <M>n</M>
    vertices, the <E>join table</E> of <A>digraph</A> is the matrix <M>A</M>
    such that the <M>(i, j)</M> entry of <M>A</M> is the join of
    vertices <M>i</M> and <M>j</M>. Otherwise, each function returns <K>fail</K>.
    <Example><![CDATA[
gap> D := Digraph([[1, 2, 3, 4], [2, 4], [3, 4], [4]]);
<immutable digraph with 4 vertices, 9 edges>
gap> IsJoinSemilatticeDigraph(D);
true
gap> Display(DigraphJoinTable(D));
[ [  1,  2,  3,  4 ],
  [  2,  2,  4,  4 ],
  [  3,  4,  3,  4 ],
  [  4,  4,  4,  4 ] ]
gap> D := CycleDigraph(5);
<immutable cycle digraph with 5 vertices>
gap> DigraphJoinTable(D);
fail
]]></Example>
  </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphHash">
<ManSection>
  <Attr Name="DigraphHash" Arg="digraph"/>
  <Returns>The hash of a digraph.</Returns>
  <Description>
    This function returns a hash of <A>digraph</A>. <P/>

    Note that the underlying hashing function is system dependent, and so the
    value of this attribute is not guaranteed to be the same on different systems.
  </Description>
</ManSection>
<#/GAPDoc>