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Product structure extension of the Alon-Seymour-Thomas theorem

par Distel, Marc;Dujmović, Vida V.;Eppstein, David;Hickingbotham, Robert;Joret, Gwenaël ;Micek, Piotr;Morin, Pat P.;Seweryn, Micha̷l M.T.;Wood, David
Référence SIAM journal on discrete mathematics, 38, 3, page (2095-2107)
Publication Publié, 2024-07-01

Article révisé par les pairs

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Résumé :

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 surdn. 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 scrO(surdtn). 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 scrOt(surdn). 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 scrO(tsurdn). This setting includes graphs embeddable on any fixed surface.