Improved enumeration of simple topological graphs

A simple topological graph T = (V (T), E(T)) 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 G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We generalize results of Pach and Tóth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2 O(n 2 log(m/n)) , and at most 2 O(mn 1/2 log n) if m ≤ n 3/2. As a consequence we obtain a new upper bound 2 O(n 3/2 log n) on the number of intersection graphs of n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with n vertices to 2 n 2 ·α(n) O(1) , 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 G is at most 2 m 2 +O(mn) and at least 2 Ω(m 2) for graphs with m > (6 + ε)n.