Mathematics > Combinatorics
[Submitted on 16 Dec 2022 (v1), last revised 6 Apr 2024 (this version, v4)]
Title:Product structure extension of the Alon--Seymour--Thomas theorem
View PDF HTML (experimental)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.
Submission history
From: David Wood [view email][v1] Fri, 16 Dec 2022 22:26:31 UTC (12 KB)
[v2] Tue, 20 Dec 2022 08:06:08 UTC (12 KB)
[v3] Thu, 3 Aug 2023 20:37:08 UTC (27 KB)
[v4] Sat, 6 Apr 2024 06:40:22 UTC (28 KB)
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