Product Structure Extension of the Alon–Seymour–Thomas Theorem

Pub Date : 2024-07-09 DOI:10.1137/23m1591773

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

引用次数: 0

Abstract

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.

查看原文

微信好友朋友圈 QQ好友复制链接

阿伦-塞缪尔-托马斯定理的乘积结构扩展

SIAM 离散数学杂志》，第 38 卷第 3 期，第 2095-2107 页，2024 年 9 月。摘要。Alon、Seymour 和 Thomas [J. Amer. Math. Soc., 3 (1990), pp.最近，Illingworth、Scott 和 Wood [Product Structure of Graphs with an Excluded Minor, preprint, arXiv:2104.06627, 2022]完善了这一结果，证明了每个这样的图都是某个树宽为 [math] 的图的子图，其中每个顶点都被一个阶为 [math] 的完整图炸开。通过解决伊林沃斯、斯科特和伍德[2022]的一个未决问题，我们证明了树宽约束可以减小到 4，同时保持阶数[数学]的炸开。作为 Lipton-Tarjan 定理的扩展，在平面图的情况下，我们证明树宽可以进一步减小到 2，这是最好的可能。我们将这一结果推广到[math]-minor-free 图中，其炸裂阶数为[math]。这种情况包括可嵌入任何固定表面的图。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助