Asymptotic structure. I. Coarse tree-width

TL;DR

This paper introduces a coarse analogue of treewidth, characterizing graphs that are quasi-isometric to bounded tree-width graphs through a specialized tree-decomposition involving bounded-radius balls.

Contribution

It establishes a new equivalence between a graph's quasi-isometry to bounded tree-width graphs and a specific type of tree-decomposition with bounded-radius bags.

Findings

Characterization of graphs via coarse tree-decomposition

Equivalence between quasi-isometry and tree-decomposition properties

Generalization of previous results on bounded radius bags

Abstract

In this paper, we develop a coarse analogue of treewidth. We prove that a graph $G$ admits a tree-decomposition in which each bag is contained in the union of a bounded number of balls of bounded radius, if and only if $G$ admits a quasi-isometry to a graph with bounded tree-width. (The ``if'' half is easy, but the ``only if'' half is challenging.) This generalizes a recent result of Berger and Seymour, concerning tree-decompositions when each bag has bounded radius.