An improved quasi-isometry between graphs of bounded cliquewidth and graphs of bounded treewidth

TL;DR

This paper improves the understanding of the relationship between graphs of bounded cliquewidth and bounded treewidth by establishing a tighter quasi-isometry with a smaller treewidth bound and a better quasi-isometry parameter.

Contribution

It introduces a new partitioning method that yields a 3-quasi-isometry between graphs of cliquewidth k and graphs of treewidth k-1, refining previous results.

Findings

Constructed a 3-quasi-isometry between graphs of cliquewidth k and treewidth k-1

Provided a partition with dense parts whose quotient has treewidth k-1

Showed the treewidth bound is tight up to an additive constant

Abstract

Cliquewidth is a dense analogue of treewidth. It can be deduced from recent results by Hickingbotham [arXiv:2501.10840] and Nguyen, Scott, and Seymour [arXiv:2501.09839] that graphs of bounded cliquewidth are quasi-isometric to graphs of bounded treewidth. We improve on this by showing that graphs of cliquewidth $k$ admit a partition with `local, but dense' parts whose quotient has treewidth $k-1$. Specifically, each part is contained within the closed neighbourhood of some vertex. We use this to construct a $3$-quasi-isometry between graphs of cliquewidth $k$ and graphs of treewidth $k-1$. This is an improvement in both the quasi-isometry parameter and the treewidth. We also show that the bound on the treewidth is tight up to an additive constant.