A coarse block-cut tree theorem

2026-07-08

AI Summary

This work investigates the large-scale connectivity structure of graphs within the framework of coarse geometry and proposes a coarse-grained decomposition analogous to the classical block–cutpoint tree. The method constructs a tree decomposition in which any two vertices within the same bag cannot be separated by a set of weak diameter at most \(d\) that lies far from both, while controlling the size of adhesion sets via weak diameter constraints. The main contribution is the first coarse-geometric analogue of the block–cutpoint tree theorem, accompanied by a dual characterization based on the existence of two disjoint paths. Leveraging tools from coarse geometry, tree decomposition theory, and the coarse Menger theorem, the authors prove that for any graph \(G\) and positive integer \(d\), there exists such a tree decomposition whose adhesion sets have weak diameter at most \(3d + 2\).

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 $v$.

Problem

• coarse geometry
• block-cut tree
• graph decomposition
• weak diameter
• coarse connectivity

Innovation

• coarse block-cut tree
• tree decomposition
• weak diameter
• Coarse Menger's Theorem
• adhesion sets