Excluding a rectangular grid

For every positive integer $k$, we define the $k$-treedepth as the largest graph parameter $\mathrm{td}_k$ satisfying (i) $\mathrm{td}_k(\emptyset)=0$; (ii) $\mathrm{td}_k(G) \leq 1+ \mathrm{td}_k(G-u)$ for every graph $G$ and every vertex $u \in V(G)$; and (iii) if $G$ is a $(<k)$-clique-sum of $G_1$ and $G_2$, then $\mathrm{td}_k(G) \leq \max \{\mathrm{td}_k(G_1),\mathrm{td}_k(G_2)\}$, for all graphs $G_1,G_2$. This parameter coincides with treedepth if $k=1$, and with treewidth plus $1$ if $k \geq |V(G)|$. We prove that for every positive integer $k$, a class of graphs $\mathcal{C}$ has bounded $k$-treedepth if and only if there is a positive integer $\ell$ such that for every tree $T$ on $k$ vertices, no graph in $\mathcal{C}$ contains $T \square P_\ell$ as a minor. This implies for $k=1$ that a minor-closed class of graphs has bounded treedepth if and only if it excludes a path, for $k=2$ that a minor-closed class of graphs has bounded $2$-treedepth if and only if it excludes as a minor a ladder (Huynh, Joret, Micek, Seweryn, and Wollan; Combinatorica, 2021), and for large values of $k$ that a minor-closed class of graphs has bounded treewidth if and only if it excludes a grid (Grid-Minor Theorem, Robertson and Seymour; JCTB, 1986). As a corollary, we obtain the following qualitative strengthening of the Grid-Minor Theorem in the case of bounded-height grids. For all positive integers $k, \ell$, every graph that does not contain the $k \times \ell$ grid as a minor has $(2k-1)$-treedepth at most a function of $(k, \ell)$.