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Allow specifying vertex colours in IsDigraphHomomorphism etc. #283

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Jan 24, 2020
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Allow specifying vertex colours in IsDigraphHomomorphism etc. #283

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a20be56
allow specifying colours in IsDigraphHomomorphism etc.
MT-resource-bot Jan 20, 2020
6b94e2e
linting and coverage
MT-resource-bot Jan 21, 2020
3b52ce2
refactor
MT-resource-bot Jan 23, 2020
7b7ec18
Add K tags to true.
james-d-mitchell Jan 23, 2020
19c02e0
Merge branch 'master' into coloured-homomorphisms
james-d-mitchell Jan 24, 2020
0db5724
Update isomorph.xml
james-d-mitchell Jan 24, 2020
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64 changes: 63 additions & 1 deletion doc/grahom.xml
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Original file line number Diff line number Diff line change
Expand Up @@ -565,9 +565,13 @@ gap> EmbeddingsDigraphs(D1, D2);
<#GAPDoc Label="IsDigraphHomomorphism">
<ManSection>
<Oper Name="IsDigraphHomomorphism" Arg="src, ran, x"/>
<Oper Name="IsDigraphHomomorphism" Arg="src, ran, x, col1, col2"/>
<Oper Name="IsDigraphEpimorphism" Arg="src, ran, x"/>
<Oper Name="IsDigraphEpimorphism" Arg="src, ran, x, col1, col2"/>
<Oper Name="IsDigraphMonomorphism" Arg="src, ran, x"/>
<Oper Name="IsDigraphMonomorphism" Arg="src, ran, x, col1, col2"/>
<Oper Name="IsDigraphEndomorphism" Arg="digraph, x"/>
<Oper Name="IsDigraphEndomorphism" Arg="digraph, x, col"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
<C>IsDigraphHomomorphism</C> returns <K>true</K> if the permutation
Expand Down Expand Up @@ -603,6 +607,22 @@ gap> EmbeddingsDigraphs(D1, D2);
</List>
See also <Ref Func="GeneratorsOfEndomorphismMonoid"/>.<P/>

If <A>col1</A> and <A>col2</A>, or <A>col</A>, are given then they must
represent vertex colourings; see <Ref Oper="AutomorphismGroup" Label="for a
digraph and a homogeneous list"/> for details of the permissible values for
these argument. The homomorphism must then also have the property:

<List>
<Item>
<C>col[i] = col[i ^ x]</C> for all vertices <C>i</C> of <A>digraph</A>,
in the case of <C>IsDigraphEndomorphism</C>.</Item>
<Item>
<C>col1[i] = col2[i ^ x]</C> for all vertices <C>i</C> of <A>src</A>,
in the cases of the other operations.</Item>
</List>

See also <Ref Oper="DigraphsRespectsColouring"/>.

<Example><![CDATA[
gap> src := Digraph([[1], [1, 2], [1, 3]]);
<immutable digraph with 3 vertices, 5 edges>
Expand All @@ -616,8 +636,18 @@ gap> IsDigraphHomomorphism(src, ran, Transformation([3, 3, 3]));
false
gap> IsDigraphHomomorphism(src, src, Transformation([3, 3, 3]));
true
gap> IsDigraphHomomorphism(src, ran, Transformation([1, 2, 2]),
> [1, 2, 2], [1, 2]);
true
gap> IsDigraphHomomorphism(src, ran, Transformation([1, 2, 2]),
> [2, 1, 1], [1, 2]);
false
gap> IsDigraphEndomorphism(src, Transformation([3, 3, 3]));
true
gap> IsDigraphEndomorphism(src, Transformation([3, 3, 3]), [1, 1, 1]);
true
gap> IsDigraphEndomorphism(src, Transformation([3, 3, 3]), [1, 1, 2]);
false
gap> IsDigraphEpimorphism(src, ran, Transformation([3, 3, 3]));
false
gap> IsDigraphMonomorphism(src, ran, Transformation([1, 2, 2]));
Expand All @@ -633,11 +663,13 @@ true]]></Example>
<#GAPDoc Label="IsDigraphEmbedding">
<ManSection>
<Oper Name="IsDigraphEmbedding" Arg="src, ran, x"/>
<Oper Name="IsDigraphEmbedding" Arg="src, ran, x, col1, col2"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
<C>IsDigraphEmbedding</C> returns <K>true</K> if the permutation
or transformation <A>x</A> is a embedding of the digraph
<A>src</A> into the digraph <A>ran</A>.
<A>src</A> into the digraph <A>ran</A>, while respecting the colourings
<A>col1</A> and <A>col2</A> if given.
<P/>

A permutation or transformation <A>x</A> is a <E>embedding</E> of a digraph
Expand All @@ -657,6 +689,10 @@ gap> IsDigraphMonomorphism(src, ran, ());
true
gap> IsDigraphEmbedding(src, ran, ());
true
gap> IsDigraphEmbedding(src, ran, (), [2, 1], [2, 1, 1]);
true
gap> IsDigraphEmbedding(src, ran, (), [2, 1], [1, 2, 1]);
false
gap> ran := Digraph([[1, 2], [1, 2], [1, 3]]);
<immutable digraph with 3 vertices, 6 edges>
gap> IsDigraphMonomorphism(src, ran, IdentityTransformation);
Expand Down Expand Up @@ -710,3 +746,29 @@ true
</Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DigraphsRespectsColouring">
<ManSection>
<Oper Name="DigraphsRespectsColouring" Arg="src, ran, x, col1, col2"/>
<Returns> <K>true</K> or <K>false</K>. </Returns>
<Description>
The operation <C>DigraphsRespectsColouring</C> verifies whether or not
the permutation or transformation <A>x</A> respects the vertex colourings
<A>col1</A> and <A>col2</A> of the digraphs <A>src</A> and <A>range</A>.
That is, <C>DigraphsRespectsColouring</C> returns true if and only if for
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@james-d-mitchell james-d-mitchell Jan 23, 2020 •
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"true" -> "<K>true</K>"

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@flsmith

all vertices <C>i</C> of <A>src</A>, <C>col1[i] = col2[i ^ x]</C>.
<P/>

<Example><![CDATA[
gap> src := Digraph([[1], [1, 2]]);
<immutable digraph with 2 vertices, 3 edges>
gap> ran := Digraph([[1], [1, 2], [1, 3]]);
<immutable digraph with 3 vertices, 5 edges>
gap> DigraphsRespectsColouring(src, ran, (1, 2), [2, 1], [1, 2, 1]);
true
gap> DigraphsRespectsColouring(src, ran, (1, 2), [2, 1], [2, 1, 1]);
false
]]></Example>
</Description>
</ManSection>
<#/GAPDoc>
25 changes: 25 additions & 0 deletions doc/isomorph.xml
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Expand Up @@ -753,7 +753,9 @@ gap> OutNeighbours(canon);
<#GAPDoc Label="IsDigraphAutomorphism">
<ManSection>
<Oper Name="IsDigraphIsomorphism" Arg="src, ran, x"/>
<Oper Name="IsDigraphIsomorphism" Arg="src, ran, x, col1, col2"/>
<Oper Name="IsDigraphAutomorphism" Arg="digraph, x"/>
<Oper Name="IsDigraphAutomorphism" Arg="digraph, x, col"/>
<Returns><K>true</K> or <K>false</K>.</Returns>
<Description>
<C>IsDigraphIsomorphism</C> returns <K>true</K> if the permutation or
Expand Down Expand Up @@ -783,6 +785,21 @@ gap> OutNeighbours(canon);
</List>
See also <Ref Attr="AutomorphismGroup" Label="for a digraph"
/>.<P/>


If <A>col1</A> and <A>col2</A>, or <A>col</A>, are given, then they must
represent vertex colourings; see <Ref Oper="AutomorphismGroup" Label="for a
digraph and a homogeneous list"/> for details of the permissible values for
these arguments. The homomorphism must then also have the property:

<List>
<Item>
<C>col1[i] = col2[i ^ x]</C> for all vertices <C>i</C> of <A>src</A>,
for <C>IsDigraphIsomorphism</C>. </Item>
<Item>
<C>col[i] = col[i ^ x]</C> for all vertices <C>i</C> of <A>digraph</A>,
for <C>IsDigraphAutomorphism</C>. </Item>
</List>

For some digraphs, it can be faster to use <C>IsDigraphAutomorphism</C>
than to test membership in the automorphism group of <A>digraph</A>.
Expand All @@ -794,6 +811,10 @@ gap> IsDigraphAutomorphism(src, (1, 2, 3));
false
gap> IsDigraphAutomorphism(src, (2, 3));
true
gap> IsDigraphAutomorphism(src, (2, 3), [2, 1, 1]);
true
gap> IsDigraphAutomorphism(src, (2, 3), [2, 2, 1]);
false
gap> IsDigraphAutomorphism(src, (2, 3)(4, 5));
false
gap> IsDigraphAutomorphism(src, (1, 4));
Expand All @@ -810,6 +831,10 @@ gap> IsDigraphIsomorphism(ran, src, (1, 2));
true
gap> IsDigraphIsomorphism(src, Digraph([[3], [1, 3], [2]]), (1, 2, 3));
false
gap> IsDigraphIsomorphism(src, ran, (1, 2), [1, 2, 3], [2, 1, 3]);
true
gap> IsDigraphIsomorphism(src, ran, (1, 2), [1, 2, 2], [2, 1, 3]);
false
]]></Example>
</Description>
</ManSection>
Expand Down
1 change: 1 addition & 0 deletions doc/z-chap6.xml
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Expand Up @@ -62,6 +62,7 @@ from} $E_a$ \emph{to} $E_b$. In this case we say that $E_a$ and $E_b$ are
<#Include Label="EmbeddingsDigraphs">
<#Include Label="IsDigraphHomomorphism">
<#Include Label="IsDigraphEmbedding">
<#Include Label="DigraphsRespectsColouring">
<#Include Label="GeneratorsOfEndomorphismMonoid">
<#Include Label="DigraphColouring">
<#Include Label="DigraphGreedyColouring">
Expand Down
29 changes: 29 additions & 0 deletions gap/grahom.gd
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Expand Up @@ -43,23 +43,52 @@ DeclareOperation("IsDigraphEndomorphism", [IsDigraph, IsTransformation]);
DeclareOperation("IsDigraphHomomorphism",
[IsDigraph, IsDigraph, IsTransformation]);

DeclareOperation("IsDigraphEndomorphism",
[IsDigraph, IsTransformation, IsList]);
DeclareOperation("IsDigraphHomomorphism",
[IsDigraph, IsDigraph, IsTransformation, IsList, IsList]);

DeclareOperation("IsDigraphEndomorphism", [IsDigraph, IsPerm]);
DeclareOperation("IsDigraphHomomorphism",
[IsDigraph, IsDigraph, IsPerm]);

DeclareOperation("IsDigraphEndomorphism",
[IsDigraph, IsPerm, IsList]);
DeclareOperation("IsDigraphHomomorphism",
[IsDigraph, IsDigraph, IsPerm, IsList, IsList]);

DeclareOperation("IsDigraphEpimorphism",
[IsDigraph, IsDigraph, IsTransformation]);
DeclareOperation("IsDigraphMonomorphism",
[IsDigraph, IsDigraph, IsTransformation]);
DeclareOperation("IsDigraphEmbedding",
[IsDigraph, IsDigraph, IsTransformation]);

DeclareOperation("IsDigraphEpimorphism",
[IsDigraph, IsDigraph, IsTransformation, IsList, IsList]);
DeclareOperation("IsDigraphMonomorphism",
[IsDigraph, IsDigraph, IsTransformation, IsList, IsList]);
DeclareOperation("IsDigraphEmbedding",
[IsDigraph, IsDigraph, IsTransformation, IsList, IsList]);

DeclareOperation("IsDigraphEpimorphism",
[IsDigraph, IsDigraph, IsPerm]);
DeclareOperation("IsDigraphMonomorphism",
[IsDigraph, IsDigraph, IsPerm]);
DeclareOperation("IsDigraphEmbedding",
[IsDigraph, IsDigraph, IsPerm]);

DeclareOperation("IsDigraphEpimorphism",
[IsDigraph, IsDigraph, IsPerm, IsList, IsList]);
DeclareOperation("IsDigraphMonomorphism",
[IsDigraph, IsDigraph, IsPerm, IsList, IsList]);
DeclareOperation("IsDigraphEmbedding",
[IsDigraph, IsDigraph, IsPerm, IsList, IsList]);

DeclareOperation("IsDigraphColouring", [IsDigraph, IsList]);
DeclareOperation("IsDigraphColouring", [IsDigraph, IsTransformation]);

DeclareOperation("DigraphsRespectsColouring",
[IsDigraph, IsDigraph, IsTransformation, IsList, IsList]);
DeclareOperation("DigraphsRespectsColouring",
[IsDigraph, IsDigraph, IsPerm, IsList, IsList]);
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