Induced minors and subpolynomial treewidth

TL;DR

This paper proves that graphs excluding certain induced minors, specifically complete bipartite graphs and hexagonal grids, have subpolynomial treewidth if they also have bounded clique number, advancing understanding of graph structure.

Contribution

It establishes that classes of graphs excluding specific induced minors have subpolynomial treewidth, a significant structural property with implications for graph algorithms.

Findings

Graphs excluding $K_{t,t}$ and $W_{t\times t}$ as induced minors have subpolynomial treewidth.

The treewidth bound is $2^{c\log^{1-\epsilon} n}$ for some $\epsilon \in (0,1]$ and constant $c$.

The result applies to graphs with no large cliques, broadening the scope of structural graph theory.

Abstract

Given a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-induced-minor-free if no induced minor of $G$ is isomorphic to a member of $\mathcal{H}$, We denote by $W_{t\times t}$ the $t$-by-$t$ hexagonal grid, and by $K_{t,t}$ the complete bipartite graph with both sides of the bipartition of size $t$. We show that the class of $\{K_{t,t},W_{t\times t}\}$-induced minor-free graphs with bounded clique number has subpolynomial treewidth. Specifically, we prove that for every integer $t$ there exist $\epsilon \in (0,1]$ and $c \in \mathbb{N}$ such that every $n$-vertex $\{K_{t,t},W_{t\times t}\}$-induced minor-free graph with no clique of size $t$ has treewidth at most $2^{c\log^{1-\epsilon}n}$.