Enumeration of intersection graphs of $x$-monotone curves

TL;DR

This paper investigates the combinatorial complexity of intersection graphs of x-monotone pseudo-segments, establishing exponential bounds on their number and introducing new upper bounds based on VC-dimension and graph properties.

Contribution

It provides the first exponential lower bound and near-matching upper bound on the number of intersection graphs of x-monotone pseudo-segments, utilizing new VC-dimension bounds.

Findings

Constructed exponentially many distinct intersection graphs

Established upper bounds on the number of such graphs

Improved bounds for bipartite and bounded chromatic number cases

Abstract

A curve in the plane is $x$-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct $2^{\Omega(n^{4/3})}$ families, each consisting of $n$ labelled $x$-monotone pseudo-segments such that their intersection graphs are different. On the other hand, we show that the number of such intersection graphs is at most $2^{O(n^{4/3}\log^2n)}$. Our proof uses a new upper bound on the number of set systems of size $m$ on a ground set of size $n$, with VC-dimension at most $d$. Much better upper bounds are obtained if we only count bipartite intersection graphs, or, in general, intersection graphs with bounded chromatic number.