Induced Minors and Coarse Tree Decompositions

TL;DR

This paper investigates a conjecture about the structure of graphs excluding certain minors, proving a weaker version involving bounds on the independence number within tree decompositions.

Contribution

It proves a weaker form of a conjecture relating excluded minors to bounded independence numbers in tree decompositions.

Findings

Proved a version with bags having distance 16(log n + 1)-independence number at most c(log n)^d.

Established a version with bags having distance 8-independence number at most 2^{c(log n)^{1-(1/d)}}.

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 $\boxplus_t$ 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 $2^{c (\log n)^{1-(1/d)}}$.