Treewidth, Hadwiger Number, and Induced Minors

TL;DR

This paper characterizes graph classes where large treewidth guarantees large clique minors, showing they exclude certain planar induced minors and establishing bounds on their treewidth in terms of Hadwiger number.

Contribution

It provides a characterization of (tw,had)-bounded graph classes, establishes bounds on their treewidth, and supports a conjecture that these bounds are linear for all such classes.

Findings

Characterization of (tw,had)-bounded classes via excluded planar induced minors.

Bound on treewidth in terms of Hadwiger number with polynomial factors.

Verification of the linear bound conjecture for specific graph classes.

Abstract

Treewidth and Hadwiger number are two of the most important parameters in structural graph theory. This paper studies graph classes in which large treewidth implies the existence of a large complete graph minor. To formalise this, we say that a graph class $\mathcal{G}$ is (tw,had)-bounded if there is a function $f$ (called the (tw,had)-bounding function) such that tw$(G)$ $\leq$ $f$(had$(G)$) for every graph $G \in \mathcal{G}$. We characterise (tw,had)-bounded graph classes as those that exclude some planar graph as an induced minor, and use this characterisation to show that every proper vertex-minor-closed class is (tw,had)-bounded. Furthermore, we demonstrate that any (tw,had)-bounded graph class has a (tw,had)-bounding function in O(had$(G)^9$polylog(had$(G)$)). Our bound comes from the bound for the Grid Minor Theorem given by Chuzhoy and Tan, and any quantitative improvement to their result will lead directly to an improvement to our result. More strongly, we conjecture that every (tw,had)-bounded graph class has a linear (tw,had)-bounding function. In support of this conjecture, we show that it holds for the class of outer-string graphs, and for a natural generalisation of outer-string graphs: intersection graphs of strings rooted at the boundary of a fixed surface. We also verify our conjecture for low-rank perturbations of circle graphs, which is an important step towards verifying it for all proper vertex-minor-closed classes.