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arXiv · 2505.09834

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

Marc Distel
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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.

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Marc Distel. 2025-05-23. An improved quasi-isometry between graphs of bounded cliquewidth and graphs of bounded treewidth. https://arxiv.org/abs/2505.09834

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