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

TL;DR

The paper delivers a direct, algorithmic link between bounded cliquewidth and bounded treewidth by constructing, for any graph with cliquewidth at most $k$, a dominated partition whose quotient has treewidth at most $k-1$ and showing the original graph is $3$-quasi-isometric to this quotient. This improves prior quasi-isometry bounds and tightens the treewidth bound, while proving the bound is near-tight via a subdivision-based lower bound. The results yield a 3-quasi-isometry between $k$-cliquewidth graphs and $(k-1)$-treewidth graphs and imply that graphs of cliquewidth at most $k$ have Assouad--Nagata dimension $1$ for $k\ge 3$. The approach combines structural decompositions with quotient-based distance control, offering a constructive pathway from dense-width parameters to large-scale geometric properties.

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.

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4: Distel2023properLiu2025
• Theorem 5
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 8