Quelle planar.xml Sprache: XML

#############################################################################
##
#W planar.xml
#Y Copyright (C) 2018 James D. Mitchell
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##

<#GAPDoc Label="IsPlanarDigraph">
<ManSection>
 <Prop Name="IsPlanarDigraph" Arg="digraph"/>
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 A <E>planar</E> digraph is a digraph that can be embedded in the plane in
 such a way that its edges do not intersect. A digraph is planar if and only
 if it does not have a subdigraph that is homeomorphic to either the
 complete graph on <C>5</C> vertices or the complete bipartite graph with
 vertex sets of sizes <C>3</C> and <C>3</C>.
 

 <C>IsPlanarDigraph</C> returns <K>true</K> if the digraph <A>digraph</A> is
 planar and <K>false</K> if it is not. The directions and multiplicities of
 any edges in <A>digraph</A> are ignored by <C>IsPlanarDigraph</C>.
 

 See also <Ref Prop="IsOuterPlanarDigraph"/>.
 

 This method uses the reference implementation in
 &EDGE_PLANARITY_SUITE; by John Boyer of the algorithms described
 in <Cite Key="BM06"/>.

 <Example><![CDATA[
gap> IsPlanarDigraph(CompleteDigraph(4));
true
gap> IsPlanarDigraph(CompleteDigraph(5));
false
gap> IsPlanarDigraph(CompleteBipartiteDigraph(2, 3));
true
gap> IsPlanarDigraph(CompleteBipartiteDigraph(3, 3));
false
gap> IsPlanarDigraph(CompleteDigraph(IsMutableDigraph, 4));
true
gap> IsPlanarDigraph(CompleteDigraph(IsMutableDigraph, 5));
false
gap> IsPlanarDigraph(CompleteBipartiteDigraph(IsMutableDigraph, 2, 3));
true
gap> IsPlanarDigraph(CompleteBipartiteDigraph(IsMutableDigraph, 3, 3));
false
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="IsOuterPlanarDigraph">
<ManSection>
 <Prop Name="IsOuterPlanarDigraph" Arg="digraph"/>
 <Returns><K>true</K> or <K>false</K>.</Returns>
 <Description>
 An <E>outer planar</E> digraph is a digraph that can be embedded in the
 plane in such a way that its edges do not intersect, and all vertices
 belong to the unbounded face of the embedding. A digraph is outer planar
 if and only if it does not have a subdigraph that is homeomorphic to either
 the complete graph on <C>4</C> vertices or the complete bipartite graph
 with vertex sets of sizes <C>2</C> and <C>3</C>.
 

 <C>IsOuterPlanarDigraph</C> returns <K>true</K> if the digraph
 <A>digraph</A> is outer planar and <K>false</K> if it is not. The
 directions and multiplicities of any edges in <A>digraph</A> are ignored by
 <C>IsPlanarDigraph</C>. 

 See also <Ref Prop="IsPlanarDigraph"/>.

 This method uses the reference implementation in
 &EDGE_PLANARITY_SUITE; by John Boyer of the algorithms described
 in <Cite Key="BM06"/>.

<Example><![CDATA[
gap> IsOuterPlanarDigraph(CompleteDigraph(4));
false
gap> IsOuterPlanarDigraph(CompleteDigraph(5));
false
gap> IsOuterPlanarDigraph(CompleteBipartiteDigraph(2, 3));
false
gap> IsOuterPlanarDigraph(CompleteBipartiteDigraph(3, 3));
false
gap> IsOuterPlanarDigraph(CycleDigraph(10));
true
gap> IsOuterPlanarDigraph(CompleteDigraph(IsMutableDigraph, 4));
false
gap> IsOuterPlanarDigraph(CompleteDigraph(IsMutableDigraph, 5));
false
gap> IsOuterPlanarDigraph(CompleteBipartiteDigraph(IsMutableDigraph,
> 2, 3));
false
gap> IsOuterPlanarDigraph(CompleteBipartiteDigraph(IsMutableDigraph,
> 3, 3));
false
gap> IsOuterPlanarDigraph(CycleDigraph(IsMutableDigraph, 10));
true

]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="KuratowskiPlanarSubdigraph">
<ManSection>
 <Attr Name="KuratowskiPlanarSubdigraph" Arg="digraph"/>
 <Returns>A list or <K>fail</K>.</Returns>
 <Description>
 <C>KuratowskiPlanarSubdigraph</C> returns the immutable list of lists of
 out-neighbours of an induced subdigraph (excluding multiple edges and loops)
 of the digraph <A>digraph</A> that witnesses the fact that <A>digraph</A> is
 not planar, or <K>fail</K> if <A>digraph</A> is planar. In other words,
 <C>KuratowskiPlanarSubdigraph</C> returns the out-neighbours of a subdigraph
 of <A>digraph</A> that is homeomorphic to the complete graph with <C>5</C>
 vertices, or to the complete bipartite graph with vertex sets of sizes
 <C>3</C> and <C>3</C>.
 

 The directions and multiplicities of any edges in <A>digraph</A> are
 ignored when considering whether or not <A>digraph</A> is planar. 

 See also <Ref Prop="IsPlanarDigraph"/>
 and <Ref Attr="SubdigraphHomeomorphicToK33"/>.
 

 This method uses the reference implementation in
 &EDGE_PLANARITY_SUITE; by John Boyer of the algorithms described
 in <Cite Key="BM06"/>.

<Example><![CDATA[
gap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6],
> [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);
<immutable digraph with 11 vertices, 25 edges>
gap> KuratowskiPlanarSubdigraph(D);
fail
gap> D := Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10],
> [2, 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],
> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);
<immutable digraph with 10 vertices, 50 edges>
gap> IsPlanarDigraph(D);
false
gap> KuratowskiPlanarSubdigraph(D);
[ [ 2, 9, 7 ], [ 1, 3 ], [ 6 ], [ 5, 9 ], [ 6, 4 ], [ 3, 5 ], [ 4 ],
 [ 7, 9, 3 ], [ 8 ], [ ] ]
gap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],
> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);
<mutable digraph with 11 vertices, 25 edges>
gap> KuratowskiPlanarSubdigraph(D);
fail
gap> D := Digraph(IsMutableDigraph, [[2, 4, 7, 9, 10],
> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],
> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],
> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);
<mutable digraph with 10 vertices, 50 edges>
gap> IsPlanarDigraph(D);
false
gap> KuratowskiPlanarSubdigraph(D);
[ [ 2, 9, 7 ], [ 1, 3 ], [ 6 ], [ 5, 9 ], [ 6, 4 ], [ 3, 5 ], [ 4 ],
 [ 7, 9, 3 ], [ 8 ], [ ] ]
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="KuratowskiOuterPlanarSubdigraph">
<ManSection>
 <Attr Name="KuratowskiOuterPlanarSubdigraph" Arg="digraph"/>
 <Returns>A list or <K>fail</K>.</Returns>
 <Description>
 <C>KuratowskiOuterPlanarSubdigraph</C> returns the immutable list of
 immutable lists of out-neighbours of an induced subdigraph (excluding
 multiple edges and loops) of the digraph <A>digraph</A> that witnesses the
 fact that <A>digraph</A> is not outer planar, or <K>fail</K> if
 <A>digraph</A> is outer planar. In other words,
 <C>KuratowskiOuterPlanarSubdigraph</C> returns the out-neighbours of a
 subdigraph of <A>digraph</A> that is homeomorphic to the complete graph with
 <C>4</C> vertices, or to the complete bipartite graph with vertex sets of
 sizes <C>2</C> and <C>3</C>.
 

 The directions and multiplicities of any edges in <A>digraph</A> are ignored
 when considering whether or not <A>digraph</A> is outer planar.
 

 See also
 <Ref Prop="IsOuterPlanarDigraph"/>,
 <Ref Attr="SubdigraphHomeomorphicToK4"/>, and
 <Ref Attr="SubdigraphHomeomorphicToK23"/>.
 

 This method uses the reference implementation in
 &EDGE_PLANARITY_SUITE; by John Boyer of the algorithms described
 in <Cite Key="BM06"/>.

<Example><![CDATA[
gap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6],
> [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);
<immutable digraph with 11 vertices, 25 edges>
gap> KuratowskiOuterPlanarSubdigraph(D);
[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4, 1 ], [ 6, 3 ], [ 1, 11 ],
 [ 7, 4 ], [ 8, 6 ], [ 2, 7 ], [ 11, 2 ], [ 2, 1 ], [ 5, 9 ] ]
gap> D := Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10],
> [2, 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],
> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);
<immutable digraph with 10 vertices, 50 edges>
gap> IsOuterPlanarDigraph(D);
false
gap> KuratowskiOuterPlanarSubdigraph(D);
[ [ ], [ ], [ ], [ 8, 9 ], [ ], [ ], [ 9, 4 ], [ 7, 9, 4 ],
 [ 8, 7 ], [ ] ]
gap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],
> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);
<mutable digraph with 11 vertices, 25 edges>
gap> KuratowskiOuterPlanarSubdigraph(D);
[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4, 1 ], [ 6, 3 ], [ 1, 11 ],
 [ 7, 4 ], [ 8, 6 ], [ 2, 7 ], [ 11, 2 ], [ 2, 1 ], [ 5, 9 ] ]
gap> D := Digraph(IsMutableDigraph, [[2, 4, 7, 9, 10],
> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],
> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],
> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);
<mutable digraph with 10 vertices, 50 edges>
gap> IsOuterPlanarDigraph(D);
false
gap> KuratowskiOuterPlanarSubdigraph(D);
[ [ ], [ ], [ ], [ 8, 9 ], [ ], [ ], [ 9, 4 ], [ 7, 9, 4 ],
 [ 8, 7 ], [ ] ]]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="PlanarEmbedding">
<ManSection>
 <Attr Name="PlanarEmbedding" Arg="digraph"/>
 <Returns>A list or <K>fail</K>.</Returns>
 <Description>
 If <A>digraph</A> is a planar digraph, then <C>PlanarEmbedding</C> returns
 the immutable list of lists of out-neighbours of <A>digraph</A> (excluding
 multiple edges and loops) such that each vertex's neighbours are given in
 clockwise order. If <A>digraph</A> is not planar, then <K>fail</K> is
 returned. 

 The directions and multiplicities of any edges in <A>digraph</A> are ignored
 by <C>PlanarEmbedding</C>.
 

 See also
 <Ref Prop="IsPlanarDigraph"/>.
 

 This method uses the reference implementation in
 &EDGE_PLANARITY_SUITE; by John Boyer of the algorithms described
 in <Cite Key="BM06"/>.

<Example><![CDATA[
gap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6],
> [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);
<immutable digraph with 11 vertices, 25 edges>
gap> PlanarEmbedding(D);
[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4, 1 ], [ 6, 3 ], [ 1, 11, 7 ],
 [ 7, 4 ], [ 8, 6 ], [ 7, 2 ], [ 2, 11 ], [ 1, 2 ], [ 9, 5 ] ]
gap> D := Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10],
> [2, 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],
> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);
<immutable digraph with 10 vertices, 50 edges>
gap> PlanarEmbedding(D);
fail
gap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],
> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);
<mutable digraph with 11 vertices, 25 edges>
gap> PlanarEmbedding(D);
[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4, 1 ], [ 6, 3 ], [ 1, 11, 7 ],
 [ 7, 4 ], [ 8, 6 ], [ 7, 2 ], [ 2, 11 ], [ 1, 2 ], [ 9, 5 ] ]
gap> D := Digraph(IsMutableDigraph, [[2, 4, 7, 9, 10],
> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],
> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],
> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);
<mutable digraph with 10 vertices, 50 edges>
gap> PlanarEmbedding(D);
fail
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="OuterPlanarEmbedding">
<ManSection>
 <Attr Name="OuterPlanarEmbedding" Arg="digraph"/>
 <Returns>A list or <K>fail</K>.</Returns>
 <Description>
 If <A>digraph</A> is an outer planar digraph, then
 <C>OuterPlanarEmbedding</C> returns the immutable list of lists of
 out-neighbours of <A>digraph</A> (excluding multiple edges and loops) such
 that each vertex's neighbours are given in clockwise order. If
 <A>digraph</A> is not outer planar, then <K>fail</K> is returned. 

 The directions and multiplicities of any edges in <A>digraph</A> are
 ignored by <C>OuterPlanarEmbedding</C>.
 

 See also <Ref Prop="IsOuterPlanarDigraph"/>.
 

 This method uses the reference implementation in
 &EDGE_PLANARITY_SUITE; by John Boyer of the algorithms described
 in <Cite Key="BM06"/>.

<Example><![CDATA[
gap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6],
> [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);
<immutable digraph with 11 vertices, 25 edges>
gap> OuterPlanarEmbedding(D);
fail
gap> D := Digraph([[2, 4, 7, 9, 10], [1, 3, 4, 6, 9, 10], [6, 10],
> [2, 5, 8, 9], [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],
> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);
<immutable digraph with 10 vertices, 50 edges>
gap> OuterPlanarEmbedding(D);
fail
gap> OuterPlanarEmbedding(CompleteBipartiteDigraph(2, 2));
[ [ 3, 4 ], [ 4, 3 ], [ 2, 1 ], [ 1, 2 ] ]
gap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],
> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);
<mutable digraph with 11 vertices, 25 edges>
gap> OuterPlanarEmbedding(D);
fail
gap> D := Digraph(IsMutableDigraph, [[2, 4, 7, 9, 10],
> [1, 3, 4, 6, 9, 10], [6, 10], [2, 5, 8, 9],
> [1, 2, 3, 4, 6, 7, 9, 10], [3, 4, 5, 7, 9, 10],
> [3, 4, 5, 6, 9, 10], [3, 4, 5, 7, 9], [2, 3, 5, 6, 7, 8], [3, 5]]);
<mutable digraph with 10 vertices, 50 edges>
gap> OuterPlanarEmbedding(D);
fail
gap> OuterPlanarEmbedding(CompleteBipartiteDigraph(2, 2));
[ [ 3, 4 ], [ 4, 3 ], [ 2, 1 ], [ 1, 2 ] ]
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="SubdigraphHomeomorphicToK">
<ManSection>
 <Attr Name="SubdigraphHomeomorphicToK23" Arg="digraph"/>
 <Attr Name="SubdigraphHomeomorphicToK33" Arg="digraph"/>
 <Attr Name="SubdigraphHomeomorphicToK4" Arg="digraph"/>
 <Returns>A list or <K>fail</K>.</Returns>
 <Description>
 These attributes return the immutable list of lists of
 out-neighbours of a subdigraph of the digraph <A>digraph</A> which is
 homeomorphic to one of the following:
 the complete bipartite graph with vertex sets of sizes <C>2</C> and
 <C>3</C>; the complete bipartite graph with vertex sets of sizes <C>3</C>
 and <C>3</C>; or the complete graph with <C>4</C> vertices. If
 <A>digraph</A> has no such subdigraphs, then <K>fail</K> is returned.
 

 See also <Ref Prop="IsPlanarDigraph"/> and <Ref
 Prop="IsOuterPlanarDigraph"/> for more details.

 This method uses the reference implementation in
 &EDGE_PLANARITY_SUITE; by John Boyer of the algorithms described
 in <Cite Key="BM06"/>.

<Example><![CDATA[
gap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], [1, 7, 11],
> [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);
<immutable digraph with 11 vertices, 25 edges>
gap> SubdigraphHomeomorphicToK4(D);
[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4, 1 ], [ 6, 3 ], [ 1, 7, 11 ],
 [ 7, 4 ], [ 8, 6 ], [ 2, 7 ], [ 11, 2 ], [ 2, 1 ], [ 5, 9 ] ]
gap> SubdigraphHomeomorphicToK23(D);
[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4, 1 ], [ 6, 3 ], [ 1, 11 ],
 [ 7, 4 ], [ 8, 6 ], [ 2, 7 ], [ 11, 2 ], [ 2, 1 ], [ 5, 9 ] ]
gap> D := Digraph([[3, 5, 10], [8, 9, 10], [1, 4], [3, 6], [1, 11],
> [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);
<immutable digraph with 11 vertices, 24 edges>
gap> SubdigraphHomeomorphicToK4(D);
fail
gap> SubdigraphHomeomorphicToK23(D);
[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4, 1 ], [ 6, 3 ], [ 11, 1 ],
 [ 7, 4 ], [ 8, 6 ], [ 2, 7 ], [ 2, 11 ], [ 1, 2 ], [ 9, 5 ] ]
gap> SubdigraphHomeomorphicToK33(D);
fail
gap> SubdigraphHomeomorphicToK23(NullDigraph(0));
fail
gap> SubdigraphHomeomorphicToK33(CompleteDigraph(5));
fail
gap> SubdigraphHomeomorphicToK33(CompleteBipartiteDigraph(3, 3));
[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ 1, 2, 3 ], [ 3, 2, 1 ],
 [ 2, 3, 1 ] ]
gap> SubdigraphHomeomorphicToK4(CompleteDigraph(3));
fail
gap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],
> [3, 6], [1, 7, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);
<mutable digraph with 11 vertices, 25 edges>
gap> SubdigraphHomeomorphicToK4(D);
[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4, 1 ], [ 6, 3 ], [ 1, 7, 11 ],
 [ 7, 4 ], [ 8, 6 ], [ 2, 7 ], [ 11, 2 ], [ 2, 1 ], [ 5, 9 ] ]
gap> SubdigraphHomeomorphicToK23(D);
[ [ 3, 5, 10 ], [ 9, 8, 10 ], [ 4, 1 ], [ 6, 3 ], [ 1, 11 ],
 [ 7, 4 ], [ 8, 6 ], [ 2, 7 ], [ 11, 2 ], [ 2, 1 ], [ 5, 9 ] ]
gap> D := Digraph(IsMutableDigraph, [[3, 5, 10], [8, 9, 10], [1, 4],
> [3, 6], [1, 11], [4, 7], [6, 8], [2, 7], [2, 11], [1, 2], [5, 9]]);
<mutable digraph with 11 vertices, 24 edges>
gap> SubdigraphHomeomorphicToK4(D);
fail
gap> SubdigraphHomeomorphicToK23(D);
[ [ 3, 10, 5 ], [ 10, 8, 9 ], [ 4, 1 ], [ 6, 3 ], [ 11, 1 ],
 [ 7, 4 ], [ 8, 6 ], [ 2, 7 ], [ 2, 11 ], [ 1, 2 ], [ 9, 5 ] ]
gap> SubdigraphHomeomorphicToK33(D);
fail
gap> SubdigraphHomeomorphicToK23(NullDigraph(0));
fail
gap> SubdigraphHomeomorphicToK33(CompleteDigraph(5));
fail
gap> SubdigraphHomeomorphicToK33(CompleteBipartiteDigraph(3, 3));
[ [ 4, 6, 5 ], [ 4, 5, 6 ], [ 6, 5, 4 ], [ 1, 2, 3 ], [ 3, 2, 1 ],
 [ 2, 3, 1 ] ]
gap> SubdigraphHomeomorphicToK4(CompleteDigraph(3));
fail
]]></Example>
 </Description>
</ManSection>
<#/GAPDoc>

<#GAPDoc Label="DualPlanarGraph">
<ManSection>
 <Attr Name="DualPlanarGraph" Arg="digraph"/>
 <Returns>A digraph or <K>fail</K>.</Returns>
 <Description>
 If <A>digraph</A> is a planar digraph, then <Ref Attr="DualPlanarGraph"/> returns the the dual graph of <A>digraph</A>.
 If <A>digraph</A> is not planar, then <K>fail</K> is returned.

 The dual graph of a planar digraph <A>digraph</A> has a vertex for each face of <A>digraph</A> and an edge for
 each pair of faces that are separated by an edge from each other.
 Vertex <A>i</A> of the dual graph corresponds to the facial walk at the <A>i</A>-th position calling
 <Ref Oper="FacialWalks"/> of <A>digraph</A> with the rotation system returned by <Ref Attr="PlanarEmbedding"/>.
 

 Note that <Ref Attr="PlanarEmbedding"/>, and therefore
 <Ref Attr="DualPlanarGraph"/>, uses the reference implementation in
 &EDGE_PLANARITY_SUITE; by John Boyer of the algorithms described in
 <Cite Key="BM06"/>.

<Example><![CDATA[
gap> D := CompleteDigraph(4);;
gap> dualD := DualPlanarGraph(D);
<immutable digraph with 4 vertices, 12 edges>
gap> IsIsomorphicDigraph(D, dualD);
true
gap> cube := Digraph([[2, 4, 5], [1, 3, 6], [2, 4, 7], [1, 3, 8],
> [1, 6, 8], [2, 5, 7], [3, 6, 8], [4, 5, 7]]);
<immutable digraph with 8 vertices, 24 edges>
gap> oct := DualPlanarGraph(cube);;
gap> IsIsomorphicDigraph(oct, CompleteMultipartiteDigraph([2, 2, 2]));
true
]]></Example>
 </Description>
</ManSection>

<#/GAPDoc>

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