Induced Minors and Coarse Tree Decompositions

Abstract

Let $G$ be a graph, $S\subseteq V(G)$ be a vertex set in $G$ and $r$ be a positive integer. The distance $r$-independence number of $S$ is the size of the largest subset $I\subseteq S$ such that no pair $u$, $v$ of vertices in $I$ have a path on at most $r$ edges between them in $G$. It has been conjectured [Chudnovsky et al., arXiv, 2025] that for every positive integer $t$ there exist positive integers $c$, $d$ such that every graph $G$ that excludes both the complete bipartite graph $K_{t,t}$ and the grid ${⊞}_t$ as an induced minor has a tree decomposition in which every bag has (distance $1$) independence number at most $c(\log n{)}^d$. We prove a weaker version of this conjecture where every bag of the tree decomposition has distance $16(\log n+1)$-independence number at most $c(\log n{)}^d$. On the way we also prove a version of the conjecture where every bag of the decomposition has distance $8$-independence number at most $2^{c(\log n{)}^{1-(1/d)}}$.