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arXiv · 2505.06550

Coarse Balanced Separators and Tree-Decompositions

Maria Chudnovsky·Robert Hickingbotham
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Abstract

A classical result of Robertson and Seymour (1986) states that the treewidth of a graph is linearly tied to its separation number: the smallest integer $k$ such that, for every weighting of the vertices, the graph admits a balanced separator of size at most $k$. Motivated by recent progress on coarse treewidth, Abrishami, Czyżewska, Kluk, Pilipczuk, Pilipczuk, and Rzążewski (2025) conjectured a coarse analogue to this result: every graph that has a balanced separator consisting of a bounded number of balls of bounded radius is quasi-isometric to a graph with bounded treewidth. In this paper, we confirm their conjecture for $K_{t,t}$-induced-subgraph-free graphs when the separator consists of a bounded number of balls of radius $1$. In doing so, we bridge two important conjectures concerning the structure of graphs that exclude a planar graph as an induced minor.

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Maria Chudnovsky, Robert Hickingbotham. 2025-07-22. Coarse Balanced Separators and Tree-Decompositions. https://arxiv.org/abs/2505.06550

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