The grid-minor theorem revisited

TL;DR

This paper strengthens the Grid-Minor Theorem by showing that graphs excluding a fixed minor can be embedded into products of graphs with bounded treewidth and a complete graph, with implications for graph coloring.

Contribution

It provides a qualitative strengthening of the Grid-Minor Theorem using treedepth, which is shown to be optimal, and applies this to improve bounds on weak coloring numbers.

Findings

Graphs excluding a fixed minor can be embedded into products of bounded treewidth graphs and complete graphs.

Treedepth is the optimal parameter for such embeddings.

Improved bounds for weak coloring numbers of minor-excluded graphs.

Abstract

We prove that for every planar graph $X$ of treedepth $h$, there exists a positive integer $c$ such that for every $X$-minor-free graph $G$, there exists a graph $H$ of treewidth at most $f(h)$ such that $G$ is isomorphic to a subgraph of $H\boxtimes K_c$. This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB 1986), and treedepth is the optimal parameter in such a result. As an example application, we use this result to improve the upper bound for weak coloring numbers of graphs excluding a fixed graph as a minor.