Grid induced minor theorem for graphs of small degree

Abstract

A graph H is an induced minor of a graph G if H can be obtained from G by vertex deletions and edge contractions. We show that there is a function f (k, d) = O(k10 +2d5 ) so that if a graph has treewidth at least f (k, d) and maximum degree at most d, then it contains a k × k-grid as an induced minor. This proves the conjecture of Aboulker, Adler, Kim, Sintiari, and Trotignon (2021) [1] that any graph with large treewidth and bounded maximum degree contains a large wall or the line graph of a large wall as an induced subgraph. It also implies that for any fixed planar graph H, there is a subexponential time algorithm for maximum weight independent set on H- induced-minor-free graphs.