Catching Rats in $H$-minor-free Graphs

We show that every $H$-minor-free graph that also excludes a $(k \times k)$-grid as a minor has treewidth/branchwidth bounded from above by a function $f(t,k)$ that is linear in $k$ and polynomial in $t := |V(H)|$. Such a result was proven originally by [Demaine & Hajiaghayi, Combinatorica, 2008], where $f$ was indeed linear in $k$. However the dependency in $t$ in this result was non-explicit (and huge). Later, [Kawarabayashi & Kobayashi, JCTB, 2020] showed that this bound can be estimated to be $f(t,k)\in 2^{\mathcal{O}(t\log t)} \cdot k$. Wood recently asked whether $f$ can be pushed further to be polynomial, while maintaining the linearity on $k$. We answer this in a particularly strong sense, by showing that the treewidth/branchwidth of $G$ is in $\mathcal{O}(gk + t^{2304}),$ where $g$ is the Euler genus of $H$. This directly yields $f(t,k)= \mathcal{O}(t^2k + t^{2304})$. Our methods build on techniques for branchwidth and on new bounds and insights for the Graph Minor Structure Theorem (GMST) due to [Gorsky, Seweryn & Wiederrecht, 2025, arXiv:2504.02532]. In particular, we prove a variant of the GMST that ensures some helpful properties for the minor relation. We further employ our methods to provide approximation algorithms for the treewidth/branchwidth of $H$-minor-free graphs. In particular, for every $\varepsilon > 0$ and every $t$-vertex graph $H$ with Euler genus $g$, we give a $(g + \varepsilon)$-approximation algorithm for the branchwidth of $H$-minor-free graphs running in $2^{\mathsf{poly}(t) / \varepsilon} \cdot \mathsf{poly}(n)$-time. Our algorithms explicitly return either an appropriate branch-decomposition or a grid-minor certifying a negative answer.