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arXiv · 2405.20547

Enumeration of intersection graphs of $x$-monotone curves

Jacob Fox·Janos Pach·Andrew Suk
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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.

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Jacob Fox, Janos Pach, Andrew Suk. 2024-05-31. Enumeration of intersection graphs of $x$-monotone curves. https://arxiv.org/abs/2405.20547

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