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arXiv · 2607.07111

A coarse block-cut tree theorem

Júlia Baligács·Václav Blažej·Jadwiga Czyżewska·Michał Pilipczuk·Evangelos Protopapas
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Abstract

We prove a coarse analogue of the classic fact that every graph can be decomposed along its cut-vertices into $2$-connected components. Precisely, we prove that for every graph $G$ and a positive integer $d$, $G$ admits a tree decomposition whose adhesion sets have weak diameter at most $3d+2$ so that no two vertices $u,v$ lying in the same bag can be separated by a set of weak diameter at most $d$ whose distance from $u$ and $v$ is more than $d$. By the Coarse Menger's Theorem for two paths, this condition admits also a dual formulation, phrased in terms of the existence of two paths that are far from each other and connect the vicinity of $u$ with the vicinity of $v$.

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Júlia Baligács, Václav Blažej, Jadwiga Czyżewska, Michał Pilipczuk, Evangelos Protopapas. 2026-07-08. A coarse block-cut tree theorem. https://arxiv.org/abs/2607.07111

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