Representation of Graphs on a Torus

Representation of Graphs on a Torus
F.J. Cobos, J.C. Dana, A. Márquez and F. Mateos
Dpto. de Matemática Aplicada I. Facultad de Informática y Estadı́stica.
Universidad de Sevilla. Avda. Reina Mercedes s/n. 41012-Sevilla (Spain)
A complete characterization of the class of graphs that admit a
toroidal visibility representation is given. In that representation, vertices map to arcs of parallels in the torus and two vertices are joined
by an edge if their corresponded intervals are joined by a strip parallel
to the meridians in the torus. Moreover, linear time algorithms are
given for testing the existence and constructing such a representation.
We also study the relation of these graphs with those admitting a
planar visibility representation and those with a cylindrical visibility