Graphs Excluding a Minor in Blowups of Treewidth 3 Graphs

Alon, Seymour, and Thomas [J. Amer. Math. Soc. 1990] famously showed that every $n$-vertex $K_h$-minor-free graph has treewidth $O_h(\sqrt{n})$. Recently, Distel, Dujmović, Eppstein, Hickingbotham, Joret, Micek, Morin, Seweryn, and Wood [SIAM J. Discrete Math. 2024] refined this by showing that these graphs are $O_h(\sqrt{n})$-blowups of treewidth $4$ graphs. We improve this by showing that these graphs are $O_h(\sqrt{n})$-blowups of treewidth $3$ graphs.