Induced Minors, Asymptotic Dimension, and Baker's Technique

Abstract

Asymptotic dimension is a large-scale invariant of metric spaces that was introduced by Gromov (1993). We prove that every hereditary class of bounded-degree graphs that excludes some graph as a fat minor has asymptotic dimension at most $2$, which is optimal. This makes substantial progress on a question of Bonamy, Bousquet, Esperet, Groenland, Liu, Pirot, and Scott (J. Eur. Math. Soc. 2023). The key to our proof is a notion inspired by Baker's technique (J. ACM 1994). We say that a graph class $\mathcal{G}$ has bounded Baker-treewidth if there exists a function $f \colon \mathbb{N} \to \mathbb{N}$ such that, for every graph $G\in \mathcal{G}$, there is a layering of $G$ such that the subgraph induced by the union of any $\ell$ consecutive layers has treewidth at most $f(\ell)$. We show that every class of bounded-degree graphs that excludes some graph as an induced minor has bounded Baker-treewidth. We discuss further applications of this result to clustered colouring and the design of linear-time approximate schemes.