Non-existence of annular separators in geometric graphs

TL;DR

This paper demonstrates that in geometric graphs of dimension three or higher with polynomial volume growth, there are no small annular separators, contrasting with known results in planar graphs.

Contribution

It proves the non-existence of low-dimensional annular separators in higher-dimensional geometric graphs with polynomial growth.

Findings

Existence of geometric graphs with polynomial growth and large annular separators

Annular separators in these graphs have size at least R^s for any s ≥ 1

Contrasts with planar graph separator properties

Abstract

Benjamini and Papasoglou (2011) showed that planar graphs with uniform polynomial volume growth admit $1$-dimensional annular separators: The vertices at graph distance $R$ from any vertex can be separated from those at distance $2R$ by removing at most $O(R)$ vertices. They asked whether geometric $d$-dimensional graphs with uniform polynomial volume growth similarly admit $(d-1)$-dimensional annular separators when $d > 2$. We show that this fails in a strong sense: For any $d \geq 3$ and every $s \geq 1$, there is a collection of interior-disjoint spheres in $\mathbb{R}^d$ whose tangency graph $G$ has uniform polynomial growth, but such that all annular separators in $G$ have cardinality at least $R^s$.