Near-optimal separators in string graphs

TL;DR

This paper improves the known separator size bounds for string graphs, showing they can be separated with fewer vertices than previously proven, moving closer to the conjectured optimal bound.

Contribution

The authors establish a new separator bound of O(\sqrt{m} \log m) vertices for string graphs, advancing the understanding of their structural properties.

Findings

Separator size improved to O(\sqrt{m} \log m) vertices

Progress towards the conjectured O(\sqrt{m}) bound

Enhanced understanding of string graph structure

Abstract

Let G be a string graph (an intersection graph of continuous arcs in the plane) with m edges. Fox and Pach proved that G has a separator consisting of O(m^{3/4}\sqrt{log m})$ vertices, and they conjectured that the bound of O(\sqrt m) actually holds. We obtain separators with O(\sqrt m \log m) vertices.