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arXiv · 2212.08739

Product structure extension of the Alon--Seymour--Thomas theorem

Marc Distel·Vida Dujmović·David Eppstein·Robert Hickingbotham·Gwenaël Joret·Piotr Micek·Pat Morin·Michał T. Seweryn·David R. Wood
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Abstract

Alon, Seymour and Thomas [1990] proved that every $n$-vertex graph excluding $K_t$ as a minor has treewidth less than $t^{3/2}\sqrt{n}$. Illingworth, Scott and Wood [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 $O(\sqrt{tn})$. 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 $O_t(\sqrt{n})$. 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 generalise this result for $K_{3,t}$-minor-free graphs, with blowups of order $O(t\sqrt{n})$. This setting includes graphs embeddable on any fixed surface.

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Marc Distel, Vida Dujmović, David Eppstein, Robert Hickingbotham, Gwenaël Joret, Piotr Micek, Pat Morin, Michał T. Seweryn, David R. Wood. 2022-12-16. Product structure extension of the Alon--Seymour--Thomas theorem. https://doi.org/10.1137/23m1591773

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