Separators for intersection graphs of spheres

TL;DR

This paper establishes optimal separator bounds for intersection graphs of spheres and convex bodies in any dimension, providing tight bounds on separator size relative to the number of edges and vertices.

Contribution

It proves the existence of size-optimal separators for intersection graphs of spheres and convex bodies in arbitrary dimensions, extending previous results.

Findings

Separator size is O_d(m^{1/d}n^{1-2/d}) for intersection graphs of spheres.

The bound is proven to be tight and optimal.

Results apply to both spheres and fat convex bodies.

Abstract

We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension $d$. One of our results is that if an intersection graph of $n$ spheres in $\mathbb{R}^d$ has $m$ edges, then it contains a balanced separator of size $O_d(m^{1/d}n^{1-2/d})$. This bound is best possible in terms of the parameters involved. The same result holds if the balls and spheres are replaced by fat convex bodies and their boundaries.