Improved Enumeration of Simple Topological Graphs

A simple topological graph is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs and are isomorphic if can be obtained from by a homeomorphism of the sphere, and weakly isomorphic if and have the same set of pairs of crossing edges. We generalize results of Pach and Tth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph with vertices, edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize is at most , and at most if . As a consequence we obtain a new upper bound on the number of intersection graphs of pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with vertices to , using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize is at most and at least for graphs with .