An improved bound on the treewidth of planar graphs excluding a grid minor

We show that every planar graph with no $t\times t$ grid minor has treewidth at most $4t+4$. This improves on the previously best known bound of $\frac{9}{2}t-\frac{11}{2}$, due to Gu and Tamaki (2012), and is within a factor $2$ of optimal. A key step in the proof is showing the following result, which might be of independent interest: Every $2$-connected plane graph $G$ with radius $d$ and faces of size at most $k$ has a tree-decomposition of width at most $\max \{3d+k+5,2d+2k+1\}$ such that the vertex set of every face of $G$ is contained in some bag.