Graphs that are quasi-isometric to graphs with bounded treewidth

TL;DR

This paper characterizes graphs that are quasi-isometric to graphs with bounded treewidth, providing a new understanding of their structure through tree-decompositions involving bounded balls and diameters.

Contribution

It extends existing characterizations to include graphs quasi-isometric to those with bounded treewidth, pathwidth, and linewidth, and links these to graphs with bounded rank-width, tree independence number, and sim-width.

Findings

Graphs quasi-isometric to bounded treewidth graphs have specific tree-decompositions.

Characterization extends to graphs with bounded pathwidth and linewidth.

Graphs with bounded rank-width, tree independence number, and sim-width are quasi-isometric to bounded treewidth graphs.

Abstract

In this paper, we characterise graphs that are quasi-isometric to graphs with bounded treewidth. Specifically, we prove that a graph is quasi-isometric to a graph with bounded treewidth if and only if it has a tree-decomposition where each bag consists of a bounded number of balls of bounded diameter. This result extends a characterisation by Berger and Seymour (2024) of graphs that are quasi-isometric to trees. Additionally, we characterise graphs that are quasi-isometric to graphs with bounded pathwidth and graphs that are quasi-isometric to graphs with bounded linewidth. As an application of these results, we show that graphs with bounded rank-width, graphs with bounded tree independence number, and graphs with bounded sim-width are quasi-isometric to graphs with bounded treewidth.