Enumeration of intersection graphs of $x$-monotone curves

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 $2^{Ω(n^{4/3})}$ 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.

Figures (4)

• Figure 1: Modifying lines through $p$.
• Figure 2: For $\alpha_1,\alpha_2 \in \mathcal{A}$ and $\beta \in \mathcal{B}'$, $\beta$ crosses $\alpha_1$ but does not cross $\alpha_2$.
• Figure 3: Vertical decomposition of $\mathcal{A}$.
• Figure 4: Cell $\Delta_i$ bounded above by $\alpha$ and contains the left endpoint of $\beta$.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 2.1
• proof
• Lemma 3.1: H
• Theorem 3.2