Coarse Balanced Separators in Fat-Minor-Free Graphs

TL;DR

This paper extends the concept of balanced separators to fat-minor-free graphs, providing bounds on separator size and an efficient algorithm for their computation.

Contribution

It introduces a coarse analogue of classic separator theorems for fat-minor-free graphs, including a randomized polynomial-time algorithm.

Findings

Balanced separators of size O(√n) exist in fat-minor-free graphs.

Such separators can be covered by a small number of balls of fixed radius.

The algorithm can find either a separator or a fat minor model efficiently.

Abstract

Fat minors are a coarse analogue of graph minors where the subgraphs modeling vertices and edges of the embedded graph are required to be distant from each other, instead of just being disjoint. In this paper, we give a coarse analogue of the classic theorem that an $n$-vertex graph excluding a fixed minor admits a balanced separator of size $O(\sqrt{n})$. Specifically, we prove that for every integer $d$, real $\varepsilon>0$, and graph $H$, there exist constants $c$ and $r$ such that every $n$-vertex graph $G$ excluding $H$ as a $d$-fat minor admits a set $S \subseteq V(G)$ that is a balanced separator of $G$ and can be covered by $c n^{\frac{1}{2}+\varepsilon}$ balls of radius $r$ in $G$. Our proof also works in the weighted setting where the balance of the separator is measured with respect to any weight function on the vertices, and is effective: we obtain a randomized polynomial-time algorithm to compute either such a balanced separator, or a $d$-fat model of $H$ in $G$.