An alternative characterisation of graphs quasi-isometric to graphs of bounded treewidth

TL;DR

This paper introduces a new structural characterization of graphs quasi-isometric to graphs of bounded treewidth, focusing on partitions with bounded weak diameter and quotient treewidth, complementing existing characterizations.

Contribution

It provides a novel structural description based on partitions with bounded weak diameter and quotient treewidth, differing from previous quasi-isometry characterizations.

Findings

Graphs are characterized by partitions with bounded weak diameter and quotient treewidth.

The new characterization is structurally different and complementary to existing ones.

It offers insights into the large-scale geometry of graphs related to bounded treewidth.

Abstract

Quasi-isometry is a measure of how similar two graphs are at `large-scale'. Nguyen, Scott, and Seymour [arXiv:2501.09839] and Hickingbotham [arXiv:2501.10840] independently gave a characterisation of graphs quasi-isometric to graphs of treewidth $k$. In this paper, we give a new characterisation of such graphs. Specifically, we show that such graphs $G$ are characterised by the existence of a partition whose quotient has treewidth at most $k$ and such that each part has bounded weak diameter in $G$. The primary contribution of our characterisation is a structural description of graphs that admit such a quasi-isometry. This differs from the characterisation mentioned above, which primarily shows the existence of such a quasi-isometry. The characterisations are complementary, and neither immediately implies the other.