Product Structure Extension of the Alon–Seymour–Thomas Theorem

Alon, Seymour, and Thomas [J. Amer. Math. Soc., 3 (1990), pp. 801-808] proved that every n-vertex graph excluding Kt as a minor has treewidth less than t3/2 \. Illingworth, Scott, and Wood [Product Structure of Graphs with an Excluded Minor, preprint, arXiv:2104.06627, 2022] recently refined this result by showing that every such graph is a subgraph of some graph with treewidth t - 2, where each vertex is blown up by a complete graph of order \(\). Solving an open problem of Illingworth, Scott, and Wood [2022], we prove that the treewidth bound can be reduced to 4 while keeping blowups of order \(\). As an extension of the Lipton-Tarjan theorem, in the case of planar graphs, we show that the treewidth can be further reduced to 2, which is best possible. We generalize this result for K3,t-minor-free graphs, with blowups of order \(t\). This setting includes graphs embeddable on any fixed surface.