Enumeration of intersection graphs of $x$-monotone curves
Jacob Fox, Janos Pach, Andrew Suk

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 -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 families, each consisting of labelled -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 . Our proof uses a new upper bound on the number of set systems of size on a ground set of size , with VC-dimension at most . Much better upper bounds are obtained if we only count bipartite intersection graphs, or, in general, intersection graphs with bounded chromatic number.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Numerical Analysis Techniques · Topological and Geometric Data Analysis
