Shortest Path Separators in Unit Disk Graphs

TL;DR

This paper presents a new balanced separator theorem for unit-disk graphs using shortest paths and neighborhood structures, improving previous results and employing novel geometric and combinatorial techniques.

Contribution

It introduces a new separator theorem involving two shortest paths and neighborhoods, solving an open problem and extending classical separator results.

Findings

Provides a balanced separator theorem for unit-disk graphs

Improves upon previous separator results requiring larger neighborhoods

Uses novel geometric methods including Delaunay triangulations

Abstract

We introduce a new balanced separator theorem for unit-disk graphs involving two shortest paths combined with the 1-hop neighbours of those paths and two other vertices. This answers an open problem of Yan, Xiang and Dragan [CGTA '12] and improves their result that requires removing the 3-hop neighborhood of two shortest paths. Our proof uses very different ideas, including Delaunay triangulations and a generalization of the celebrated balanced separator theorem of Lipton and Tarjan [J. Appl. Math. '79] to systems of non-intersecting paths.