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.

翻译：我们证明：对于每个树深为$h$的平面图$X$，存在正整数$c$，使得每个不含$X$为挖子的图$G$，都存在一个树宽至多为$f(h)$的图$H$，使得$G$同构于$H\boxtimes K_c$的一个子图。这是Robertson与Seymour（JCTB 1986）的网格挖子定理在定性意义上的强化，且树深是该结果中的最优参数。作为示例应用，我们利用该结果改进了不含固定图为挖子的图的弱着色数的上界。