‣ ReembeddingsOfDigraph( digraph, g, oriented ) | ( operation ) |
‣ ReembeddingsOfSimplicialSphere( surf, g, oriented ) | ( operation ) |
Returns: a list
The method ReembeddingsOfSimplicialSphere computes all edge-face equivalent simplicial surfaces of the given vertex-faithful simplicial sphere surf with the given genus g if these simplicial surfaces are orientable or not is given by oriented. Note that two simplicial surfaces are edge-face equivalent if the corresponding face graphs are isomorphic (see 15.4-2 for a definition of the face graph). The method ReembeddingsOfDigraph computes for a 3-connected cubic planar graph all simplicial surfaces with the given genus g and if these simplicial surfaces are orientable or not is given by oriented that have digraph as their face graph. We call this a re-embedding of a digraph or a simplicial sphere. If surf is not a vertex-faithful simplicial sphere or digraph is not planar and cubic, an error is printed. It is not checked whether digraph is a 3-connected graph.
Note that, non-orientable surfaces of genus one are projective planes, orientable surfaces of genus one are tori and non-orientable surfaces of genus two are Klein bottles.
For example, consider the complete graph on four vertices:
gap> digraph:=CompleteDigraph(4);; gap> ReembeddingsOfDigraph(digraph,1,false); [ simplicial surface (3 vertices, 6 edges, and 4 faces) ] gap> ReembeddingsOfDigraph(digraph,1,true); [ ] gap> ReembeddingsOfDigraph(digraph,2,false); [ ]
So the complete graph on four vertices has exactly one re-embedding on a projective plane but no re-embedding on the torus or the Klein bottle. Note that the complete graph on four vertices is the face graph of the tetrahedron. The octahedron has for example no edge-face equivalent projective plane but three edge-face equivalent tori and two edge-face equivalent Klein bottles.
gap> oct:=Octahedron();; gap> ReembeddingsOfSimplicialSphere(Octahedron(),1,false); [ ] gap> ReembeddingsOfSimplicialSphere(Octahedron(),1,true); [ simplicial surface (4 vertices, 12 edges, and 8 faces), simplicial surface (4 vertices, 12 edges, and 8 faces), simplicial surface (4 vertices, 12 edges, and 8 faces) ] gap> ReembeddingsOfSimplicialSphere(Octahedron(),2,false); [ simplicial surface (4 vertices, 12 edges, and 8 faces), simplicial surface (4 vertices, 12 edges, and 8 faces) ]
generated by GAPDoc2HTML