Product Structure Extension of the Alon–Seymour–Thomas Theorem

SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2095-2107, September 2024.
Abstract. Alon, Seymour, and Thomas [J. Amer. Math. Soc., 3 (1990), pp. 801–808] proved that every [math]-vertex graph excluding [math] as a minor has treewidth less than [math]. 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 [math], where each vertex is blown up by a complete graph of order [math]. 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 [math]. 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 [math]-minor-free graphs, with blowups of order [math]. This setting includes graphs embeddable on any fixed surface.