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<div class="chlinktop">Goto Chapter: <a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chapA.html">A</a> <a href="chapB.html">B</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>

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<a href="chap7_mj.html">[MathJax on]</a>
<a id="X84975388859F203D" name="X84975388859F203D"></a>
<div class="ChapSects"><a href="chap7.html#X84975388859F203D">7 Homomorphisms</a>
<div class="ContSect"> <a href="chap7.html#X8200192B80AD2071">7.1 Acting on digraphs</a>

<div class="ContSSBlock">
 <a href="chap7.html#X808972017C486F1F">7.1-1 OnDigraphs</a>
 <a href="chap7.html#X7AFBA7608498F9CE">7.1-2 OnMultiDigraphs</a>
 <a href="chap7.html#X7F6A97E6870CDDA3">7.1-3 OnTuplesDigraphs</a>
</div></div>
<div class="ContSect"> <a href="chap7.html#X7E48B9F87A0F22D4">7.2 Isomorphisms and canonical labellings</a>

<div class="ContSSBlock">
 <a href="chap7.html#X83E593F3855B122E">7.2-1 DigraphsUseNauty</a>
 <a href="chap7.html#X858C32127A190175">7.2-2 AutomorphismGroup</a>
 <a href="chap7.html#X7E7B0D88865A89F6">7.2-3 BlissAutomorphismGroup</a>
 <a href="chap7.html#X857758B18144C0CD">7.2-4 NautyAutomorphismGroup</a>
 <a href="chap7.html#X877732B1783C391B">7.2-5 AutomorphismGroup</a>
 <a href="chap7.html#X7DA2C4FE837FFE01">7.2-6 AutomorphismGroup</a>
 <a href="chap7.html#X7D676FE67A6684FF">7.2-7 BlissCanonicalLabelling</a>
 <a href="chap7.html#X87DA265D803DB337">7.2-8 BlissCanonicalLabelling</a>
 <a href="chap7.html#X877B1D377EC197D7">7.2-9 BlissCanonicalDigraph</a>
 <a href="chap7.html#X803ACEDA7BBAC5B3">7.2-10 DigraphGroup</a>
 <a href="chap7.html#X8096C5287E459279">7.2-11 DigraphOrbits</a>
 <a href="chap7.html#X8386028782F2D3FF">7.2-12 DigraphOrbitReps</a>
 <a href="chap7.html#X8657604E87A25E5F">7.2-13 DigraphSchreierVector</a>
 <a href="chap7.html#X78913684795FB256">7.2-14 DigraphStabilizer</a>
 <a href="chap7.html#X7B4F24B283C9EE28">7.2-15 IsIsomorphicDigraph</a>
 <a href="chap7.html#X7D4A1FA8868DA930">7.2-16 IsIsomorphicDigraph</a>
 <a href="chap7.html#X8219FE6C839D9457">7.2-17 IsomorphismDigraphs</a>
 <a href="chap7.html#X7ED93C0F86D9D34F">7.2-18 IsomorphismDigraphs</a>
 <a href="chap7.html#X7E1C806D81DFE15E">7.2-19 RepresentativeOutNeighbours</a>
 <a href="chap7.html#X80183D4A7C51365A">7.2-20 IsDigraphIsomorphism</a>
 <a href="chap7.html#X7F16B8B3825A627A">7.2-21 IsDigraphColouring</a>
 <a href="chap7.html#X7D4378F27B49C9AC">7.2-22 MaximalCommonSubdigraph</a>
 <a href="chap7.html#X7B99C75B8021FCDA">7.2-23 MinimalCommonSuperdigraph</a>
</div></div>
<div class="ContSect"> <a href="chap7.html#X7E1A02FE8384C03C">7.3 Homomorphisms of digraphs</a>

<div class="ContSSBlock">
 <a href="chap7.html#X79ABF0E783FD67C7">7.3-1 HomomorphismDigraphsFinder</a>
 <a href="chap7.html#X85E9B019877AD7FE">7.3-2 DigraphHomomorphism</a>
 <a href="chap7.html#X8653EDA183E06D05">7.3-3 HomomorphismsDigraphs</a>
 <a href="chap7.html#X82D0FCD87D47ACA8">7.3-4 DigraphMonomorphism</a>
 <a href="chap7.html#X84B4BDB984C2A9A8">7.3-5 MonomorphismsDigraphs</a>
 <a href="chap7.html#X7CB5AD9F861684FD">7.3-6 DigraphEpimorphism</a>
 <a href="chap7.html#X7DFB1F5D873937B3">7.3-7 EpimorphismsDigraphs</a>
 <a href="chap7.html#X7AD04CC186E86CCA">7.3-8 DigraphEmbedding</a>
 <a href="chap7.html#X82EBDF137EEDC5FD">7.3-9 EmbeddingsDigraphs</a>
 <a href="chap7.html#X80FE9FCB79718A61">7.3-10 IsDigraphHomomorphism</a>
 <a href="chap7.html#X7D6B5A0887903810">7.3-11 IsDigraphEmbedding</a>
 <a href="chap7.html#X87E60FB17BC444EF">7.3-12 SubdigraphsMonomorphisms</a>
 <a href="chap7.html#X7F80709A80CBFA98">7.3-13 DigraphsRespectsColouring</a>
 <a href="chap7.html#X7E93B268823F6478">7.3-14 GeneratorsOfEndomorphismMonoid</a>
 <a href="chap7.html#X86D0A74B83D6B9A7">7.3-15 DigraphColouring</a>
 <a href="chap7.html#X7AB7200D831013C1">7.3-16 DigraphGreedyColouring</a>
 <a href="chap7.html#X7F9CB3B27B9590DB">7.3-17 DigraphWelshPowellOrder</a>
 <a href="chap7.html#X807D49358529A37C">7.3-18 ChromaticNumber</a>
 <a href="chap7.html#X863487EF7D64EF56">7.3-19 DigraphCore</a>
 <a href="chap7.html#X7EEA340C81FCA52D">7.3-20 LatticeDigraphEmbedding</a>
 <a href="chap7.html#X7E183D4979D324EF">7.3-21 IsLatticeHomomorphism</a>
</div></div>
</div>

<h3>7 Homomorphisms</h3>

<a id="X8200192B80AD2071" name="X8200192B80AD2071"></a>

<h4>7.1 Acting on digraphs</h4>

<a id="X808972017C486F1F" name="X808972017C486F1F"></a>

<h5>7.1-1 OnDigraphs</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnDigraphs</code>( <var class="Arg">digraph</var>, <var class="Arg">perm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnDigraphs</code>( <var class="Arg">digraph</var>, <var class="Arg">trans</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A digraph.

If <var class="Arg">digraph</var> is a digraph, and the second argument <var class="Arg">perm</var> is a permutation of the vertices of <var class="Arg">digraph</var>, then this operation returns a digraph constructed by relabelling the vertices of <var class="Arg">digraph</var> according to <var class="Arg">perm</var>. Note that for an automorphism <code class="code">f</code> of a digraph, we have <code class="code">OnDigraphs(<var class="Arg">digraph</var>, f) = </code><var class="Arg">digraph</var>.

If the second argument is a transformation <var class="Arg">trans</var> of the vertices of <var class="Arg">digraph</var>, then this operation returns a digraph constructed by transforming the source and range of each edge according to <var class="Arg">trans</var>. Thus a vertex which does not appear in the image of <var class="Arg">trans</var> will be isolated in the returned digraph, and the returned digraph may contain multiple edges, even if <var class="Arg">digraph</var> does not. If <var class="Arg">trans</var> is mathematically a permutation, then the result coincides with <code class="code">OnDigraphs(<var class="Arg">digraph</var>, AsPermutation(<var class="Arg">trans</var>))</code>.

The <code class="func">DigraphVertexLabels</code> (<a href="chap5.html#X7E51F2FE87140B32">5.1-12</a>) of <var class="Arg">digraph</var> will not be retained in the returned digraph.

If <var class="Arg">digraph</var> belongs to <code class="func">IsMutableDigraph</code> (<a href="chap3.html#X7D7EDF83820ED6F5">3.1-2</a>), then relabelling of the vertices is performed directly on <var class="Arg">digraph</var>. If <var class="Arg">digraph</var> belongs to <code class="func">IsImmutableDigraph</code> (<a href="chap3.html#X7CAFAA89804F80BD">3.1-3</a>), an immutable copy of <var class="Arg">digraph</var> with the vertices relabelled is returned.

<div class="example"><pre>
gap> D := Digraph([[3], [1, 3, 5], [1], [1, 2, 4], [2, 3, 5]]);
<immutable digraph with 5 vertices, 11 edges>
gap> new := OnDigraphs(D, (1, 2));
<immutable digraph with 5 vertices, 11 edges>
gap> OutNeighbours(new);
[ [ 2, 3, 5 ], [ 3 ], [ 2 ], [ 2, 1, 4 ], [ 1, 3, 5 ] ]
gap> D := Digraph([[2], [], [2]]);
<immutable digraph with 3 vertices, 2 edges>
gap> t := Transformation([1, 2, 1]);;
gap> new := OnDigraphs(D, t);
<immutable multidigraph with 3 vertices, 2 edges>
gap> OutNeighbours(new);
[ [ 2, 2 ], [ ], [ ] ]
gap> ForAll(DigraphEdges(D),
> e -> IsDigraphEdge(new, [e[1] ^ t, e[2] ^ t]));
true
</pre></div>

<a id="X7AFBA7608498F9CE" name="X7AFBA7608498F9CE"></a>

<h5>7.1-2 OnMultiDigraphs</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnMultiDigraphs</code>( <var class="Arg">digraph</var>, <var class="Arg">pair</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnMultiDigraphs</code>( <var class="Arg">digraph</var>, <var class="Arg">perm1</var>, <var class="Arg">perm2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A digraph.

If <var class="Arg">digraph</var> is a digraph, and <var class="Arg">pair</var> is a pair consisting of a permutation of the vertices and a permutation of the edges of <var class="Arg">digraph</var>, then this operation returns a digraph constructed by relabelling the vertices and edges of <var class="Arg">digraph</var> according to <var class="Arg">perm[1]</var> and <var class="Arg">perm[2]</var>, respectively.

In its second form, <code class="code">OnMultiDigraphs</code> returns a digraph with vertices and edges permuted by <var class="Arg">perm1</var> and <var class="Arg">perm2</var>, respectively.

Note that <code class="code">OnDigraphs(<var class="Arg">digraph</var>, perm)=OnMultiDigraphs(<var class="Arg">digraph</var>, [perm, ()])</code> where <code class="code">perm</code> is a permutation of the vertices of <var class="Arg">digraph</var>. If you are only interested in the action of a permutation on the vertices of a digraph, then you can use <code class="code">OnDigraphs</code> instead of <code class="code">OnMultiDigraphs</code>. If <var class="Arg">digraph</var> belongs to <code class="func">IsMutableDigraph</code> (<a href="chap3.html#X7D7EDF83820ED6F5">3.1-2</a>), then relabelling of the vertices is performed directly on <var class="Arg">digraph</var>. If <var class="Arg">digraph</var> belongs to <code class="func">IsImmutableDigraph</code> (<a href="chap3.html#X7CAFAA89804F80BD">3.1-3</a>), an immutable copy of <var class="Arg">digraph</var> with the vertices relabelled is returned.

<div class="example"><pre>
gap> D1 := Digraph([
> [3, 6, 3], [], [3], [9, 10], [9], [], [], [10, 4, 10], [], []]);
<immutable multidigraph with 10 vertices, 10 edges>
gap> p := BlissCanonicalLabelling(D1);
[ (1,7)(3,6)(4,10)(5,9), () ]
gap> D2 := OnMultiDigraphs(D1, p);
<immutable multidigraph with 10 vertices, 10 edges>
gap> OutNeighbours(D2);
[ [ ], [ ], [ ], [ ], [ ], [ 6 ], [ 6, 3, 6 ], [ 4, 10, 4 ],
 [ 5 ], [ 5, 4 ] ]</pre></div>

<a id="X7F6A97E6870CDDA3" name="X7F6A97E6870CDDA3"></a>

<h5>7.1-3 OnTuplesDigraphs</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnTuplesDigraphs</code>( <var class="Arg">list</var>, <var class="Arg">perm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnSetsDigraphs</code>( <var class="Arg">list</var>, <var class="Arg">perm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A list or set of digraphs.

If <var class="Arg">list</var> is a list of digraphs, and <var class="Arg">perm</var> is a permutation of the vertices of the digraphs in <var class="Arg">list</var>, then <code class="func">OnTuplesDigraphs</code> returns a new list constructed by applying <var class="Arg">perm</var> via <code class="func">OnDigraphs</code> (<a href="chap7.html#X808972017C486F1F">7.1-1</a>) to a copy (with the same mutability) of each entry of <var class="Arg">list</var> in turn.

More precisely, <code class="code">OnTuplesDigraphs(<var class="Arg">list</var>,<var class="Arg">perm</var>)</code> is a list of length <code class="code">Length(<var class="Arg">list</var>)</code>, whose <code class="code">i</code>-th entry is <code class="code">OnDigraphs(DigraphCopy(<var class="Arg">list</var>[i]), <var class="Arg">perm</var>)</code>.

If <var class="Arg">list</var> is moreover a GAP set (i.e. a duplicate-free sorted list), then <code class="func">OnSetsDigraphs</code> returns the sorted output of <code class="func">OnTuplesDigraphs</code>, which is therefore again a set.

<div class="example"><pre>
gap> list := [CycleDigraph(IsMutableDigraph, 6),
> DigraphReverse(CycleDigraph(6))];
[ <mutable digraph with 6 vertices, 6 edges>,
 <immutable digraph with 6 vertices, 6 edges> ]
gap> p := (1, 6)(2, 5)(3, 4);;
gap> result_tuples := OnTuplesDigraphs(list, p);
[ <mutable digraph with 6 vertices, 6 edges>,
 <immutable digraph with 6 vertices, 6 edges> ]
gap> result_tuples[2] = OnDigraphs(list[2], p);
true
gap> result_tuples = list;
false
gap> result_tuples = Reversed(list);
true
gap> result_sets := OnSetsDigraphs(list, p);
[ <immutable digraph with 6 vertices, 6 edges>,
 <mutable digraph with 6 vertices, 6 edges> ]
gap> result_sets = list;
true</pre></div>

<a id="X7E48B9F87A0F22D4" name="X7E48B9F87A0F22D4"></a>

<h4>7.2 Isomorphisms and canonical labellings</h4>

From version 0.11.0 of Digraphs it is possible to use either <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> or <a href="https://pallini.di.uniroma1.it">nauty</a> (via <a href="https://github.com/gap-packages/NautyTracesInterface">NautyTracesInterface</a> ) to calculate canonical labellings and automorphism groups of digraphs; see <a href="chapBib.html#biBJK07">[JK07]</a> and <a href="chapBib.html#biBMP14">[MP14]</a> for more details about <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> and <a href="https://pallini.di.uniroma1.it">nauty</a> , respectively.

<a id="X83E593F3855B122E" name="X83E593F3855B122E"></a>

<h5>7.2-1 DigraphsUseNauty</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DigraphsUseNauty</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DigraphsUseBliss</code>( )</td><td class="tdright">( function )</td></tr></table></div>
Returns: Nothing.

These functions can be used to specify whether <a href="https://pallini.di.uniroma1.it">nauty</a> or <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> should be used by default by Digraphs. If <a href="https://github.com/gap-packages/NautyTracesInterface">NautyTracesInterface</a> is not available, then these functions do nothing. Otherwise, by calling <code class="code">DigraphsUseNauty</code> subsequent computations will default to using <a href="https://pallini.di.uniroma1.it">nauty</a> rather than <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> , where possible.

You can call these functions at any point in a GAP session, as many times as you like, it is guaranteed that existing digraphs remain valid, and that comparison of existing digraphs and newly created digraphs via <code class="func">IsIsomorphicDigraph</code> (<a href="chap7.html#X7B4F24B283C9EE28">7.2-15</a>), <code class="func">IsIsomorphicDigraph</code> (<a href="chap7.html#X7D4A1FA8868DA930">7.2-16</a>), <code class="func">IsomorphismDigraphs</code> (<a href="chap7.html#X8219FE6C839D9457">7.2-17</a>), and <code class="func">IsomorphismDigraphs</code> (<a href="chap7.html#X7ED93C0F86D9D34F">7.2-18</a>) are also valid.

It is also possible to compute the automorphism group of a specific digraph using both <a href="https://pallini.di.uniroma1.it">nauty</a> and <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> using <code class="func">NautyAutomorphismGroup</code> (<a href="chap7.html#X857758B18144C0CD">7.2-4</a>) and <code class="func">BlissAutomorphismGroup</code> (<a href="chap7.html#X7E7B0D88865A89F6">7.2-3</a>), respectively.

<a id="X858C32127A190175" name="X858C32127A190175"></a>

<h5>7.2-2 AutomorphismGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismGroup</code>( <var class="Arg">digraph</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A permutation group.

If <var class="Arg">digraph</var> is a digraph, then this attribute contains the group of automorphisms of <var class="Arg">digraph</var>. An automorphism of <var class="Arg">digraph</var> is an isomorphism from <var class="Arg">digraph</var> to itself. See <code class="func">IsomorphismDigraphs</code> (<a href="chap7.html#X8219FE6C839D9457">7.2-17</a>) for more information about isomorphisms of digraphs.

If <var class="Arg">digraph</var> is not a multidigraph then the automorphism group is returned as a group of permutations on the set of vertices of <var class="Arg">digraph</var>.

If <var class="Arg">digraph</var> is a multidigraph then the automorphism group is returned as the direct product of a group of permutations on the set of vertices of <var class="Arg">digraph</var> with a group of permutations on the set of edges of <var class="Arg">digraph</var>. These groups can be accessed using <code class="func">Projection</code> (<a href="/home/runner/gap/doc/ref/chap32.html#X8769E8DA80BC96C1">Reference: Projection for a domain and a positive integer</a>) on the returned group.

By default, the automorphism group is found using <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> by Tommi Junttila and Petteri Kaski. If <a href="https://github.com/gap-packages/NautyTracesInterface">NautyTracesInterface</a> is available, then <a href="https://pallini.di.uniroma1.it">nauty</a> by Brendan Mckay and Adolfo Piperno can be used instead; see <code class="func">BlissAutomorphismGroup</code> (<a href="chap7.html#X7E7B0D88865A89F6">7.2-3</a>), <code class="func">NautyAutomorphismGroup</code> (<a href="chap7.html#X857758B18144C0CD">7.2-4</a>), <code class="func">DigraphsUseBliss</code> (<a href="chap7.html#X83E593F3855B122E">7.2-1</a>), and <code class="func">DigraphsUseNauty</code> (<a href="chap7.html#X83E593F3855B122E">7.2-1</a>).

If the argument <var class="Arg">digraph</var> is mutable, then the return value of this attribute is recomputed every time it is called.

<div class="example"><pre>
gap> johnson := DigraphFromGraph6String("E}lw");
<immutable symmetric digraph with 6 vertices, 24 edges>
gap> G := AutomorphismGroup(johnson);
Group([ (3,4), (2,3)(4,5), (1,2)(5,6) ])
gap> cycle := CycleDigraph(9);
<immutable cycle digraph with 9 vertices>
gap> G := AutomorphismGroup(cycle);
Group([ (1,2,3,4,5,6,7,8,9) ])
gap> IsCyclic(G) and Size(G) = 9;
true</pre></div>

<a id="X7E7B0D88865A89F6" name="X7E7B0D88865A89F6"></a>

<h5>7.2-3 BlissAutomorphismGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlissAutomorphismGroup</code>( <var class="Arg">digraph</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlissAutomorphismGroup</code>( <var class="Arg">digraph</var>, <var class="Arg">vertex_colours</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlissAutomorphismGroup</code>( <var class="Arg">digraph</var>, <var class="Arg">vertex_colours</var>, <var class="Arg">edge_colours</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A permutation group.

If <var class="Arg">digraph</var> is a digraph, then this attribute contains the group of automorphisms of <var class="Arg">digraph</var> as calculated using <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> by Tommi Junttila and Petteri Kaski.

The attribute <code class="func">AutomorphismGroup</code> (<a href="chap7.html#X858C32127A190175">7.2-2</a>) and operation <code class="func">AutomorphismGroup</code> (<a href="chap7.html#X877732B1783C391B">7.2-5</a>) returns the value of either <code class="code">BlissAutomorphismGroup</code> or <code class="func">NautyAutomorphismGroup</code> (<a href="chap7.html#X857758B18144C0CD">7.2-4</a>). These groups are, of course, equal but their generating sets may differ.

The attribute <code class="func">AutomorphismGroup</code> (<a href="chap7.html#X7DA2C4FE837FFE01">7.2-6</a>) returns the value of <code class="code">BlissAutomorphismGroup</code> as it is not implemented for <a href="https://pallini.di.uniroma1.it">nauty</a> The requirements for the optional arguments <var class="Arg">vertex_colours</var> and <var class="Arg">edge_colours</var> are documented in <code class="func">AutomorphismGroup</code> (<a href="chap7.html#X7DA2C4FE837FFE01">7.2-6</a>). See also <code class="func">DigraphsUseBliss</code> (<a href="chap7.html#X83E593F3855B122E">7.2-1</a>), and <code class="func">DigraphsUseNauty</code> (<a href="chap7.html#X83E593F3855B122E">7.2-1</a>).

If the argument <var class="Arg">digraph</var> is mutable, then the return value of this attribute is recomputed every time it is called.

<div class="example"><pre>
gap> G := BlissAutomorphismGroup(JohnsonDigraph(5, 2));;
gap> IsSymmetricGroup(G);
true
gap> Size(G);
120</pre></div>

<a id="X857758B18144C0CD" name="X857758B18144C0CD"></a>

<h5>7.2-4 NautyAutomorphismGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NautyAutomorphismGroup</code>( <var class="Arg">digraph</var>[, <var class="Arg">vert_colours</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A permutation group.

If <var class="Arg">digraph</var> is a digraph, then this attribute contains the group of automorphisms of <var class="Arg">digraph</var> as calculated using <a href="https://pallini.di.uniroma1.it">nauty</a> by Brendan Mckay and Adolfo Piperno via <a href="https://github.com/gap-packages/NautyTracesInterface">NautyTracesInterface</a> . The attribute <code class="func">AutomorphismGroup</code> (<a href="chap7.html#X858C32127A190175">7.2-2</a>) and operation <code class="func">AutomorphismGroup</code> (<a href="chap7.html#X877732B1783C391B">7.2-5</a>) returns the value of either <code class="code">NautyAutomorphismGroup</code> or <code class="func">BlissAutomorphismGroup</code> (<a href="chap7.html#X7E7B0D88865A89F6">7.2-3</a>). These groups are, of course, equal but their generating sets may differ.

See also <code class="func">DigraphsUseBliss</code> (<a href="chap7.html#X83E593F3855B122E">7.2-1</a>), and <code class="func">DigraphsUseNauty</code> (<a href="chap7.html#X83E593F3855B122E">7.2-1</a>).

If the argument <var class="Arg">digraph</var> is mutable, then the return value of this attribute is recomputed every time it is called.

<div class="example"><pre>
gap> NautyAutomorphismGroup(JohnsonDigraph(5, 2));
Group([ (3,4)(6,7)(8,9), (2,3)(5,6)(9,10), (2,5)(3,6)(4,7),
(1,2)(6,8)(7,9) ])</pre></div>

<a id="X877732B1783C391B" name="X877732B1783C391B"></a>

<h5>7.2-5 AutomorphismGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismGroup</code>( <var class="Arg">digraph</var>, <var class="Arg">vert_colours</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A permutation group.

This operation computes the automorphism group of a vertex-coloured digraph. A vertex-coloured digraph can be specified by its underlying digraph <var class="Arg">digraph</var> and its colouring <var class="Arg">vert_colours</var>. Let <code class="code">n</code> be the number of vertices of <var class="Arg">digraph</var>. The colouring <var class="Arg">vert_colours</var> may have one of the following two forms:

<ul>
<li>a list of <code class="code">n</code> integers, where <var class="Arg">vert_colours</var><code class="code">[i]</code> is the colour of vertex <code class="code">i</code>, using the colours <code class="code">[1 .. m]</code> for some <code class="code">m <= n</code>; or

</li>
<li>a list of non-empty disjoint lists whose union is <code class="code">DigraphVertices(<var class="Arg">digraph</var>)</code>, such that <var class="Arg">vert_colours</var><code class="code">[i]</code> is the list of all vertices with colour <code class="code">i</code>.

</li>
</ul>
The automorphism group of a coloured digraph <var class="Arg">digraph</var> with colouring <var class="Arg">vert_colours</var> is the group consisting of its automorphisms; an automorphism of <var class="Arg">digraph</var> is an isomorphism of coloured digraphs from <var class="Arg">digraph</var> to itself. This group is equal to the subgroup of <code class="code">AutomorphismGroup(<var class="Arg">digraph</var>)</code> consisting of those automorphisms that preserve the colouring specified by <var class="Arg">vert_colours</var>. See <code class="func">AutomorphismGroup</code> (<a href="chap7.html#X858C32127A190175">7.2-2</a>), and see <code class="func">IsomorphismDigraphs</code> (<a href="chap7.html#X7ED93C0F86D9D34F">7.2-18</a>) for more information about isomorphisms of coloured digraphs.

If <var class="Arg">digraph</var> is not a multidigraph then the automorphism group is returned as a group of permutations on the set of vertices of <var class="Arg">digraph</var>.

If <var class="Arg">digraph</var> is a multidigraph then the automorphism group is returned as the direct product of a group of permutations on the set of vertices of <var class="Arg">digraph</var> with a group of permutations on the set of edges of <var class="Arg">digraph</var>. These groups can be accessed using <code class="func">Projection</code> (<a href="/home/runner/gap/doc/ref/chap32.html#X8769E8DA80BC96C1">Reference: Projection for a domain and a positive integer</a>) on the returned group.

By default, the automorphism group is found using <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> by Tommi Junttila and Petteri Kaski. If <a href="https://github.com/gap-packages/NautyTracesInterface">NautyTracesInterface</a> is available, then <a href="https://pallini.di.uniroma1.it">nauty</a> by Brendan Mckay and Adolfo Piperno can be used instead; see <code class="func">BlissAutomorphismGroup</code> (<a href="chap7.html#X7E7B0D88865A89F6">7.2-3</a>), <code class="func">NautyAutomorphismGroup</code> (<a href="chap7.html#X857758B18144C0CD">7.2-4</a>), <code class="func">DigraphsUseBliss</code> (<a href="chap7.html#X83E593F3855B122E">7.2-1</a>), and <code class="func">DigraphsUseNauty</code> (<a href="chap7.html#X83E593F3855B122E">7.2-1</a>).

<div class="example"><pre>
gap> cycle := CycleDigraph(9);
<immutable cycle digraph with 9 vertices>
gap> G := AutomorphismGroup(cycle);;
gap> IsCyclic(G) and Size(G) = 9;
true
gap> colours := [[1, 4, 7], [2, 5, 8], [3, 6, 9]];;
gap> H := AutomorphismGroup(cycle, colours);;
gap> Size(H);
3
gap> H = AutomorphismGroup(cycle, [1, 2, 3, 1, 2, 3, 1, 2, 3]);
true
gap> H = SubgroupByProperty(G, p -> OnTuplesSets(colours, p) = colours);
true
gap> IsTrivial(AutomorphismGroup(cycle, [1, 1, 2, 2, 2, 2, 2, 2, 2]));
true</pre></div>

<a id="X7DA2C4FE837FFE01" name="X7DA2C4FE837FFE01"></a>

<h5>7.2-6 AutomorphismGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismGroup</code>( <var class="Arg">digraph</var>, <var class="Arg">vert_colours</var>, <var class="Arg">edge_colours</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A permutation group.

This operation computes the automorphism group of a vertex- and/or edge-coloured digraph. A coloured digraph can be specified by its underlying digraph <var class="Arg">digraph</var> and colourings <var class="Arg">vert_colours</var>, <var class="Arg">edge_colours</var>. Let <code class="code">n</code> be the number of vertices of <var class="Arg">digraph</var>. The colourings must have the following forms:

<ul>
<li><var class="Arg">vert_colours</var> must be <code class="keyw">fail</code> or a list of <code class="code">n</code> integers, where <var class="Arg">vert_colours</var><code class="code">[i]</code> is the colour of vertex <code class="code">i</code>, using the colours <code class="code">[1 .. m]</code> for some <code class="code">m <= n</code>;

</li>
<li><var class="Arg">edge_colours</var> must be <code class="keyw">fail</code> or a list of <code class="code">n</code> lists of integers of the same shape as <code class="code">OutNeighbours(digraph)</code>, where <var class="Arg">edge_colours</var><code class="code">[i][j]</code> is the colour of the edge <code class="code">OutNeighbours(digraph)[i][j]</code>, using the colours <code class="code">[1 .. k]</code> for some <code class="code">k <= n</code>;

</li>
</ul>
Giving <var class="Arg">vert_colours</var> [<var class="Arg">edge_colours</var>] as <code class="code">fail</code> is equivalent to setting all vertices [edges] to be the same colour.

Unlike <code class="func">AutomorphismGroup</code> (<a href="chap7.html#X858C32127A190175">7.2-2</a>), it is possible to obtain the automorphism group of an edge-coloured multidigraph (see <code class="func">IsMultiDigraph</code> (<a href="chap6.html#X7BB84CFC7E8B2B26">6.2-11</a>)) when no two edges share the same source, range, and colour. The automorphism group of a vertex/edge-coloured digraph <var class="Arg">digraph</var> with colouring <var class="Arg">c</var> is the group consisting of its vertex/edge-colour preserving automorphisms; an automorphism of <var class="Arg">digraph</var> is an isomorphism of vertex/edge-coloured digraphs from <var class="Arg">digraph</var> to itself. This group is equal to the subgroup of <code class="code">AutomorphismGroup(<var class="Arg">digraph</var>)</code> consisting of those automorphisms that preserve the colouring specified by <var class="Arg">colours</var>. See <code class="func">AutomorphismGroup</code> (<a href="chap7.html#X858C32127A190175">7.2-2</a>), and see <code class="func">IsomorphismDigraphs</code> (<a href="chap7.html#X7ED93C0F86D9D34F">7.2-18</a>) for more information about isomorphisms of coloured digraphs.

If <var class="Arg">digraph</var> is not a multidigraph then the automorphism group is returned as a group of permutations on the set of vertices of <var class="Arg">digraph</var>.

If <var class="Arg">digraph</var> is a multidigraph then the automorphism group is returned as the direct product of a group of permutations on the set of vertices of <var class="Arg">digraph</var> with a group of permutations on the set of edges of <var class="Arg">digraph</var>. These groups can be accessed using <code class="func">Projection</code> (<a href="/home/runner/gap/doc/ref/chap32.html#X8769E8DA80BC96C1">Reference: Projection for a domain and a positive integer</a>) on the returned group.

By default, the automorphism group is found using <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> by Tommi Junttila and Petteri Kaski. If <a href="https://github.com/gap-packages/NautyTracesInterface">NautyTracesInterface</a> is available, then <a href="https://pallini.di.uniroma1.it">nauty</a> by Brendan Mckay and Adolfo Piperno can be used instead; see <code class="func">BlissAutomorphismGroup</code> (<a href="chap7.html#X7E7B0D88865A89F6">7.2-3</a>), <code class="func">NautyAutomorphismGroup</code> (<a href="chap7.html#X857758B18144C0CD">7.2-4</a>), <code class="func">DigraphsUseBliss</code> (<a href="chap7.html#X83E593F3855B122E">7.2-1</a>), and <code class="func">DigraphsUseNauty</code> (<a href="chap7.html#X83E593F3855B122E">7.2-1</a>).

<div class="example"><pre>
gap> cycle := CycleDigraph(12);
<immutable cycle digraph with 12 vertices>
gap> vert_colours := List([1 .. 12], x -> x mod 3 + 1);;
gap> edge_colours := List([1 .. 12], x -> [x mod 2 + 1]);;
gap> Size(AutomorphismGroup(cycle));
12
gap> Size(AutomorphismGroup(cycle, vert_colours));
4
gap> Size(AutomorphismGroup(cycle, fail, edge_colours));
6
gap> Size(AutomorphismGroup(cycle, vert_colours, edge_colours));
2
gap> IsTrivial(AutomorphismGroup(cycle,
> vert_colours, List([1 .. 12], x -> [x mod 4 + 1])));
true
</pre></div>

<a id="X7D676FE67A6684FF" name="X7D676FE67A6684FF"></a>

<h5>7.2-7 BlissCanonicalLabelling</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlissCanonicalLabelling</code>( <var class="Arg">digraph</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NautyCanonicalLabelling</code>( <var class="Arg">digraph</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A permutation, or a list of two permutations.

A function ρ that maps a digraph to a digraph is a canonical representative map if the following two conditions hold for all digraphs G and H:

<ul>
<li>ρ(G) and G are isomorphic as digraphs; and

</li>
<li>ρ(G)=ρ(H) if and only if G and H are isomorphic as digraphs.

</li>
</ul>
A canonical labelling of a digraph G (under ρ) is an isomorphism of G onto its canonical representative, ρ(G). See <code class="func">IsomorphismDigraphs</code> (<a href="chap7.html#X8219FE6C839D9457">7.2-17</a>) for more information about isomorphisms of digraphs.

<code class="code">BlissCanonicalLabelling</code> returns a canonical labelling of the digraph <var class="Arg">digraph</var> found using <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> by Tommi Junttila and Petteri Kaski. <code class="code">NautyCanonicalLabelling</code> returns a canonical labelling of the digraph <var class="Arg">digraph</var> found using <a href="https://pallini.di.uniroma1.it">nauty</a> by Brendan McKay and Adolfo Piperno. Note that the canonical labellings returned by <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> and <a href="https://pallini.di.uniroma1.it">nauty</a> are not usually the same (and may depend of the version used).

<code class="code">BlissCanonicalLabelling</code> can only be computed if <var class="Arg">digraph</var> has no multiple edges; see <code class="func">IsMultiDigraph</code> (<a href="chap6.html#X7BB84CFC7E8B2B26">6.2-11</a>).

<div class="example"><pre>
gap> digraph1 := DigraphFromDiSparse6String(".ImNS_AiB?qRN");
<immutable digraph with 10 vertices, 8 edges>
gap> BlissCanonicalLabelling(digraph1);
(1,9,5,7)(3,6,4,10)
gap> p := (1, 2, 7, 5)(3, 9)(6, 10, 8);;
gap> digraph2 := OnDigraphs(digraph1, p);
<immutable digraph with 10 vertices, 8 edges>
gap> digraph1 = digraph2;
false
gap> OnDigraphs(digraph1, BlissCanonicalLabelling(digraph1)) =
> OnDigraphs(digraph2, BlissCanonicalLabelling(digraph2));
true</pre></div>

<a id="X87DA265D803DB337" name="X87DA265D803DB337"></a>

<h5>7.2-8 BlissCanonicalLabelling</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlissCanonicalLabelling</code>( <var class="Arg">digraph</var>, <var class="Arg">colours</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NautyCanonicalLabelling</code>( <var class="Arg">digraph</var>, <var class="Arg">colours</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A permutation.

A function ρ that maps a coloured digraph to a coloured digraph is a canonical representative map if the following two conditions hold for all coloured digraphs G and H:

<ul>
<li>ρ(G) and G are isomorphic as coloured digraphs; and

</li>
<li>ρ(G)=ρ(H) if and only if G and H are isomorphic as coloured digraphs.

</li>
</ul>
A canonical labelling of a coloured digraph G (under ρ) is an isomorphism of G onto its canonical representative, ρ(G). See <code class="func">IsomorphismDigraphs</code> (<a href="chap7.html#X7ED93C0F86D9D34F">7.2-18</a>) for more information about isomorphisms of coloured digraphs.

A coloured digraph can be specified by its underlying digraph <var class="Arg">digraph</var> and its colouring <var class="Arg">colours</var>. Let <code class="code">n</code> be the number of vertices of <var class="Arg">digraph</var>. The colouring <var class="Arg">colours</var> may have one of the following two forms:

<ul>
<li>a list of <code class="code">n</code> integers, where <var class="Arg">colours</var><code class="code">[i]</code> is the colour of vertex <code class="code">i</code>, using the colours <code class="code">[1 .. m]</code> for some <code class="code">m <= n</code>; or

</li>
<li>a list of non-empty disjoint lists whose union is <code class="code">DigraphVertices(<var class="Arg">digraph</var>)</code>, such that <var class="Arg">colours</var><code class="code">[i]</code> is the list of all vertices with colour <code class="code">i</code>.

</li>
</ul>
If <var class="Arg">digraph</var> and <var class="Arg">colours</var> together form a coloured digraph, <code class="code">BlissCanonicalLabelling</code> returns a canonical labelling of the digraph <var class="Arg">digraph</var> found using <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> by Tommi Junttila and Petteri Kaski. Similarly, <code class="code">NautyCanonicalLabelling</code> returns a canonical labelling of the digraph <var class="Arg">digraph</var> found using <a href="https://pallini.di.uniroma1.it">nauty</a> by Brendan McKay and Adolfo Piperno. Note that the canonical labellings returned by <a href="http://www.tcs.tkk.fi/Software/bliss/">bliss</a> and <a href="https://pallini.di.uniroma1.it">nauty</a> are not usually the same (and may depend of the version used).

<code class="code">BlissCanonicalLabelling</code> can only be computed if <var class="Arg">digraph</var> has no multiple edges; see <code class="func">IsMultiDigraph</code> (<a href="chap6.html#X7BB84CFC7E8B2B26">6.2-11</a>). The canonical labelling of <var class="Arg">digraph</var> is given as a permutation of its vertices. The canonical representative of <var class="Arg">digraph</var> can be created from <var class="Arg">digraph</var> and its canonical labelling <code class="code">p</code> by using the operation <code class="func">OnDigraphs</code> (<a href="chap7.html#X808972017C486F1F">7.1-1</a>):

<div class="example"><pre>gap> OnDigraphs(digraph, p);</pre></div>

The colouring of the canonical representative can easily be constructed. A vertex <code class="code">v</code> (in <var class="Arg">digraph</var>) has colour <code class="code">i</code> if and only if the vertex <code class="code">v ^ p</code> (in the canonical representative) has colour <code class="code">i</code>, where <code class="code">p</code> is the permutation of the canonical labelling that acts on the vertices of <var class="Arg">digraph</var>. In particular, if <var class="Arg">colours</var> has the first form that is described above, then the colouring of the canonical representative is given by:

<div class="example"><pre>gap> List(DigraphVertices(digraph), i -> colours[i / p]);</pre></div>

On the other hand, if <var class="Arg">colours</var> has the second form above, then the canonical representative has colouring:

<div class="example"><pre>gap> OnTuplesSets(colours, p);</pre></div>

<div class="example"><pre>
gap> digraph := DigraphFromDiSparse6String(".ImNS_AiB?qRN");
<immutable digraph with 10 vertices, 8 edges>
gap> colours := [[1, 2, 8, 9, 10], [3, 4, 5, 6, 7]];;
gap> p := BlissCanonicalLabelling(digraph, colours);
(1,5,8,4,10,3,9)(6,7)
gap> OnDigraphs(digraph, p);
<immutable digraph with 10 vertices, 8 edges>
gap> OnTuplesSets(colours, p);
[ [ 1, 2, 3, 4, 5 ], [ 6, 7, 8, 9, 10 ] ]
gap> colours := [1, 1, 1, 1, 2, 3, 1, 3, 2, 1];;
gap> p := BlissCanonicalLabelling(digraph, colours);
(1,6,9,7)(3,4,5,8,10)
gap> OnDigraphs(digraph, p);
<immutable digraph with 10 vertices, 8 edges>
gap> List(DigraphVertices(digraph), i -> colours[i / p]);
[ 1, 1, 1, 1, 1, 1, 2, 2, 3, 3 ]</pre></div>

<a id="X877B1D377EC197D7" name="X877B1D377EC197D7"></a>

<h5>7.2-9 BlissCanonicalDigraph</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlissCanonicalDigraph</code>( <var class="Arg">digraph</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NautyCanonicalDigraph</code>( <var class="Arg">digraph</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A digraph.

The attribute <code class="code">BlissCanonicalLabelling</code> returns the canonical representative found by applying <code class="func">BlissCanonicalLabelling</code> (<a href="chap7.html#X7D676FE67A6684FF">7.2-7</a>). The digraph returned is canonical in the sense that

<ul>
<li><code class="code">BlissCanonicalDigraph(<var class="Arg">digraph</var>)</code> and <var class="Arg">digraph</var> are isomorphic as digraphs; and

</li>
<li>If <code class="code">gr</code> is any digraph then <code class="code">BlissCanonicalDigraph(gr)</code> and <code class="code">BlissCanonicalDigraph(<var class="Arg">digraph</var>)</code> are equal if and only if <code class="code">gr</code> and <var class="Arg">digraph</var> are isomorphic as digraphs.

</li>
</ul>
Analogously, the attribute <code class="code">NautyCanonicalLabelling</code> returns the canonical representative found by applying <code class="func">NautyCanonicalLabelling</code> (<a href="chap7.html#X7D676FE67A6684FF">7.2-7</a>).

If the argument <var class="Arg">digraph</var> is mutable, then the return value of this attribute is recomputed every time it is called.

<div class="example"><pre>
gap> digraph := Digraph([[1], [2, 3], [3], [1, 2, 3]]);
<immutable digraph with 4 vertices, 7 edges>
gap> canon := BlissCanonicalDigraph(digraph);
<immutable digraph with 4 vertices, 7 edges>
gap> OutNeighbours(canon);
[ [ 1 ], [ 2 ], [ 3, 2 ], [ 1, 3, 2 ] ]
</pre></div>

<a id="X803ACEDA7BBAC5B3" name="X803ACEDA7BBAC5B3"></a>

<h5>7.2-10 DigraphGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DigraphGroup</code>( <var class="Arg">digraph</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: A permutation group.

If <var class="Arg">digraph</var> is immutable and was created knowing a subgroup of its automorphism group, then this group is stored in the attribute <code class="code">DigraphGroup</code>. If <var class="Arg">digraph</var> is mutable, or was not created knowing a subgroup of its automorphism group, then <code class="code">DigraphGroup</code> returns the entire automorphism group of <var class="Arg">digraph</var>. Note that if <var class="Arg">digraph</var> is mutable, then the automorphism group is recomputed every time this function is called.

Note that certain other constructor operations such as <code class="func">CayleyDigraph</code> (<a href="chap3.html#X7FCADADC7EC28478">3.1-12</a>), <code class="func">BipartiteDoubleDigraph</code> (<a href="chap3.html#X7C6E6CB284982C7A">3.3-44</a>), and <code class="func">DoubleDigraph</code> (<a href="chap3.html#X7FB8B48C87C0ED16">3.3-43</a>), may not require a group as one of the arguments, but use the standard constructor method using a group, and hence set the <code class="code">DigraphGroup</code> attribute for the resulting digraph.

<div class="example"><pre>
gap> n := 4;;
gap> adj := function(x, y)
> return (((x - y) mod n) = 1) or (((x - y) mod n) = n - 1);
> end;;
gap> group := CyclicGroup(IsPermGroup, n);
Group([ (1,2,3,4) ])
gap> D := Digraph(IsMutableDigraph, group, [1 .. n], \^, adj);
<mutable digraph with 4 vertices, 8 edges>
gap> HasDigraphGroup(D);
false
gap> DigraphGroup(D);
Group([ (2,4), (1,2)(3,4) ])
gap> AutomorphismGroup(D);
Group([ (2,4), (1,2)(3,4) ])
gap> D := Digraph(group, [1 .. n], \^, adj);
<immutable digraph with 4 vertices, 8 edges>
gap> HasDigraphGroup(D);
true
gap> DigraphGroup(D);
Group([ (1,2,3,4) ])
gap> D := DoubleDigraph(D);
<immutable digraph with 8 vertices, 32 edges>
gap> HasDigraphGroup(D);
true
gap> DigraphGroup(D);
Group([ (1,2,3,4)(5,6,7,8), (1,5)(2,6)(3,7)(4,8) ])
gap> AutomorphismGroup(D) =
> Group([(6, 8), (5, 7), (4, 6), (3, 5), (2, 4),
> (1, 2)(3, 4)(5, 6)(7, 8)]);
true
gap> D := Digraph([[2, 3], [], []]);
<immutable digraph with 3 vertices, 2 edges>
gap> HasDigraphGroup(D);
false
gap> HasAutomorphismGroup(D);
false
gap> DigraphGroup(D);
Group([ (2,3) ])
gap> HasAutomorphismGroup(D);
true
gap> group := DihedralGroup(8);
<pc group of size 8 with 3 generators>
gap> D := CayleyDigraph(group);
<immutable digraph with 8 vertices, 24 edges>
gap> HasDigraphGroup(D);
true
gap> GeneratorsOfGroup(DigraphGroup(D));
[ (1,2)(3,5)(4,6)(7,8), (1,7,4,3)(2,5,6,8), (1,4)(2,6)(3,7)(5,8) ]</pre></div>

<a id="X8096C5287E459279" name="X8096C5287E459279"></a>

<h5>7.2-11 DigraphOrbits</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DigraphOrbits</code>( <var class="Arg">digraph</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: An immutable list of lists of integers.

<code class="code">DigraphOrbits</code> returns the orbits of the action of the <code class="func">DigraphGroup</code> (<a href="chap7.html#X803ACEDA7BBAC5B3">7.2-10</a>) on the set of vertices of <var class="Arg">digraph</var>.

<div class="example"><pre>
gap> G := Group([(2, 3)(7, 8, 9), (1, 2, 3)(4, 5, 6)(8, 9)]);;
gap> D := EdgeOrbitsDigraph(G, [1, 2]);
<immutable digraph with 9 vertices, 6 edges>
gap> DigraphOrbits(D);
[ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]
gap> D := DigraphMutableCopy(D);
<mutable digraph with 9 vertices, 6 edges>
gap> DigraphOrbits(D);
[ [ 1, 2, 3 ], [ 4, 5, 6, 7, 8, 9 ] ]
</pre></div>

<a id="X8386028782F2D3FF" name="X8386028782F2D3FF"></a>

<h5>7.2-12 DigraphOrbitReps</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DigraphOrbitReps</code>( <var class="Arg">digraph</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: An immutable list of integers.

<code class="code">DigraphOrbitReps</code> returns a list of orbit representatives of the action of the <code class="func">DigraphGroup</code> (<a href="chap7.html#X803ACEDA7BBAC5B3">7.2-10</a>) on the set of vertices of <var class="Arg">digraph</var>.

<div class="example"><pre>
gap> D := CayleyDigraph(AlternatingGroup(4));
<immutable digraph with 12 vertices, 24 edges>
gap> DigraphOrbitReps(D);
[ 1 ]
gap> D := DigraphMutableCopy(D);
<mutable digraph with 12 vertices, 24 edges>
gap> DigraphOrbitReps(D);
[ 1 ]
gap> D := DigraphFromDigraph6String("&IGO??S?`?_@?a?CK?O");
<immutable digraph with 10 vertices, 14 edges>
gap> DigraphOrbitReps(D);
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
gap> DigraphOrbitReps(DigraphMutableCopy(D));
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
</pre></div>

<a id="X8657604E87A25E5F" name="X8657604E87A25E5F"></a>

<h5>7.2-13 DigraphSchreierVector</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DigraphSchreierVector</code>( <var class="Arg">digraph</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: An immutable list of integers.

<code class="code">DigraphSchreierVector</code> returns the so-called Schreier vector of the action of the <code class="func">DigraphGroup</code> (<a href="chap7.html#X803ACEDA7BBAC5B3">7.2-10</a>) on the set of vertices of <var class="Arg">digraph</var>. The Schreier vector is a list <code class="code">sch</code> of integers with length <code class="code">DigraphNrVertices(<var class="Arg">digraph</var>)</code> where:

<dl>
<dt><code class="code">sch[i] < 0:</code></dt>
<dd>implies that <code class="code">i</code> is an orbit representative and <code class="code">DigraphOrbitReps(<var class="Arg">digraph</var>)[-sch[i]] = i</code>.

</dd>
<dt><code class="code">sch[i] > 0:</code></dt>
<dd>implies that <code class="code">i / gens[sch[i]]</code> is one step closer to the root (or representative) of the tree, where <code class="code">gens</code> is the generators of <code class="code">DigraphGroup(<var class="Arg">digraph</var>)</code>.

</dd>
</dl>

<div class="example"><pre>
gap> n := 4;;
gap> adj := function(x, y)
> return (((x - y) mod n) = 1) or (((x - y) mod n) = n - 1);
> end;;
gap> group := CyclicGroup(IsPermGroup, n);
Group([ (1,2,3,4) ])
gap> D := Digraph(IsMutableDigraph, group, [1 .. n], \^, adj);
<mutable digraph with 4 vertices, 8 edges>
gap> sch := DigraphSchreierVector(D);
[ -1, 2, 2, 1 ]
gap> D := CayleyDigraph(AlternatingGroup(4));
<immutable digraph with 12 vertices, 24 edges>
--> --------------------

--> maximum size reached

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