Coarse Balanced Separators in Biclique-Induced-Minor-Free Graphs

Abstract

It is a classical theorem of Robertson and Seymour (1986) that the treewidth of a graph is linearly related to its separation number: the smallest integer $k$ such that, for every weight function on 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 Rzażewski (2025) conjectured the following coarse analogue: for every $r\in \mathbb{N}$ there exists an $r'\in \mathbb{N}$ such that every graph that admits balanced separators that can be covered by a bounded number of balls of bounded radius $r$ admits a tree decomposition where every bag can be covered by a bounded number of balls of radius $r'$. We verify a stronger variant of this conjecture for all $r \in \mathbb{N}$ for the hereditary class of $K_{t,t}$-induced-minor-free graphs of bounded clique number. A key step in the proof is the following result, which we expect to be of independent interest. In $K_{t,t}$-induced-minor-free graphs with clique number bounded by $s$, given a large subset of vertices $Y \subseteq V(G)$, there is a set $Z$ whose size is bounded by a function polynomial in $s$, such that no ball of radius $r$ in $G- Z$ covers a large proportion of $Y$.