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. We give three applications of this result: (1) improved upper bounds for the weak coloring numbers of graphs excluding a given minor, (2) an improved product structure theorem for apex-minor-free graphs, and (3) improved upper bounds for the p-centered chromatic number of graphs excluding a given minor.
