Improved enumeration of simple topological graphs

TL;DR

This paper provides new upper bounds on the number of weak isomorphism classes of simple topological graphs, improving understanding of their combinatorial complexity and related intersection graph classes.

Contribution

It generalizes previous results by establishing tighter bounds on the enumeration of simple topological graphs under isomorphism and weak isomorphism.

Findings

Upper bound of 2^O(n^2 log(m/n)) for weak isomorphism classes

New bound of 2^O(mn^{1/2} log n) when m < n^{3/2}

Improved upper bound of 2^{n^2 alpha(n)^O(1)} for complete graphs

Abstract

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 Toth 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 alpha(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^Omega(m^2) for graphs with m > (6+epsilon)n.