Product Structure Extension of the Alon–Seymour–Thomas Theorem

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics 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.

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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。

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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.