Separators in region intersection graphs

TL;DR

This paper proves that region intersection graphs over graphs excluding a complete minor have small balanced separators proportional to the square root of their edges, confirming a conjecture for string graphs and improving previous bounds.

Contribution

It establishes a universal separator size bound for region intersection graphs over minor-excluding graphs, confirming a conjecture for string graphs and extending separator theory.

Findings

String graphs with m edges have O(√m) balanced separators.

Separator size bound generalizes the planar separator theorem.

Confirms a conjecture of Fox and Pach (2010).

Abstract

For undirected graphs $G=(V,E)$ and $G_0=(V_0,E_0)$, say that $G$ is a region intersection graph over $G_0$ if there is a family of connected subsets $\{ R_u \subseteq V_0 : u \in V \}$ of $G_0$ such that $\{u,v\} \in E \iff R_u \cap R_v \neq \emptyset$. We show if $G_0$ excludes the complete graph $K_h$ as a minor for some $h \geq 1$, then every region intersection graph $G$ over $G_0$ with $m$ edges has a balanced separator with at most $c_h \sqrt{m}$ nodes, where $c_h$ is a constant depending only on $h$. If $G$ additionally has uniformly bounded vertex degrees, then such a separator is found by spectral partitioning. A string graph is the intersection graph of continuous arcs in the plane. The preceding result implies that every string graph with $m$ edges has a balanced separator of size $O(\sqrt{m})$. This bound is optimal, as it generalizes the planar separator theorem. It confirms a conjecture of Fox and Pach (2010), and improves over the $O(\sqrt{m} \log m)$ bound of Matousek (2013).