Product structure of graphs with an excluded minor

TL;DR

This paper demonstrates that graphs excluding certain minors can be represented as subgraphs of products of a bounded treewidth graph and a complete graph, providing optimal bounds and generalizing key separator and product structure theorems.

Contribution

It establishes a new product structure theorem for minor-free graphs with optimal bounds, extending previous separator and planar graph results.

Findings

Graphs with excluded minors are subgraphs of products of bounded treewidth graphs and complete graphs.

Provides optimal bounds on the treewidth of the factor graph.

Achieves near-optimal bounds on the size of the complete graph factor.

Abstract

This paper shows that $K_t$-minor-free (and $K_{s, t}$-minor-free) graphs $G$ are subgraphs of products of a tree-like graph $H$ (of bounded treewidth) and a complete graph $K_m$. Our results include optimal bounds on the treewidth of $H$ and optimal bounds (to within a constant factor) on $m$ in terms of the number of vertices of $G$ and the treewidth of $G$. These results follow from a more general theorem whose corollaries include a strengthening of the celebrated separator theorem of Alon, Seymour, and Thomas [J. Amer. Math. Soc. 1990] and the Planar Graph Product Structure Theorem of Dujmovi\'c et al. [J. ACM 2020].