Tesselation and visibility representations of maps on the torus
A well-known model of the torus is given by a parallelogram whose opposite sides are pairwise identified. An important concept related to drawing of graphs in the plane is the notion of upward drawings where each edge of a graph is oriented and drawn in \(R^2\) such that the ordinate monotonically increases when we traverse the edge according to its orientation. In view of the parallelogram mentioned above, the authors define two families of parallel lines and tessellate the torus, and then they associate to every tesselation a toroidal map with an upward drawing. On this basis they prove that a toroidal map admits a tessellation representation if and only if it is essentially 2-connected. Further on, several characterizations of essentially 2-connected maps on the torus are given, and it is shown that they can be obtained from two minimal maps by vertex splitting and creating digons. And also those graphs are characterized that can be embedded in the torus as an essentially 2-connected map.
scientific article; zbMATH DE number 1127609