Product structure extension of the Alon--Seymour--Thomas theorem

TL;DR

This paper improves bounds on the treewidth of graphs excluding certain minors, showing it can be reduced to 4 generally, to 2 for planar graphs, and extended to $K_{3,t}$-minor-free graphs, with controlled blowup sizes.

Contribution

It proves that the treewidth bounds for $K_t$-minor-free graphs can be significantly lowered, solving an open problem and extending results to broader graph classes.

Findings

Treewidth can be reduced to 4 for $K_t$-minor-free graphs.

Treewidth can be reduced to 2 for planar graphs.

Extended results to $K_{3,t}$-minor-free graphs with controlled blowups.

Abstract

Alon, Seymour and Thomas [1990] proved that every $n$-vertex graph excluding $K_t$ as a minor has treewidth less than $t^{3/2}\sqrt{n}$. Illingworth, Scott and Wood [2022] recently refined this result by showing that every such graph is a subgraph of some graph with treewidth $t-2$, where each vertex is blown up by a complete graph of order $O(\sqrt{tn})$. Solving an open problem of Illingworth, Scott and Wood [2022], we prove that the treewidth bound can be reduced to $4$ while keeping blowups of order $O_t(\sqrt{n})$. As an extension of the Lipton--Tarjan theorem, in the case of planar graphs, we show that the treewidth can be further reduced to $2$, which is best possible. We generalise this result for $K_{3,t}$-minor-free graphs, with blowups of order $O(t\sqrt{n})$. This setting includes graphs embeddable on any fixed surface.