The grid-minor theorem revisited
This provides a qualitative strengthening of a foundational result in graph minor theory, with implications for graph structure and coloring problems.
The authors prove that for any planar graph X of treedepth h, every X-minor-free graph G is a subgraph of the strong product of a graph H (with treewidth bounded by a function of h) and a complete graph K_c. This strengthens the Grid-Minor Theorem and yields improved upper bounds for weak coloring numbers of minor-free graphs.
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.