Coarse Balanced Separators and Tree-Decompositions

TL;DR

This paper proves a conjecture linking coarse balanced separators made of small-radius balls to bounded treewidth in certain graphs, advancing understanding of graph structure excluding planar minors.

Contribution

It confirms a conjecture that graphs with small-radius balanced separators are quasi-isometric to bounded treewidth graphs, specifically for $K_{t,t}$-induced-subgraph-free graphs.

Findings

Confirmed the conjecture for graphs excluding $K_{t,t}$ as an induced subgraph.

Bridged two major conjectures about graphs excluding planar minors.

Established a structural link between coarse separators and treewidth.

Abstract

A classical result of Robertson and Seymour (1986) states that the treewidth of a graph is linearly tied to its separation number: the smallest integer $k$ such that, for every weighting of the vertices, the graph admits a balanced separator of size at most $k$. Motivated by recent progress on coarse treewidth, Abrishami, Czy\.{z}ewska, Kluk, Pilipczuk, Pilipczuk, and Rz\k{a}\.{z}ewski (2025) conjectured a coarse analogue to this result: every graph that has a balanced separator consisting of a bounded number of balls of bounded radius is quasi-isometric to a graph with bounded treewidth. In this paper, we confirm their conjecture for $K_{t,t}$-induced-subgraph-free graphs when the separator consists of a bounded number of balls of radius $1$. In doing so, we bridge two important conjectures concerning the structure of graphs that exclude a planar graph as an induced minor.