排除网格定理:改进与简化

Julia Chuzhoy
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引用次数: 48

摘要

研究了Robertson和Seymour的排除网格定理。这是图论中的一个基本结果,它表明存在某个函数f:Z+→Z+,使得对于任何整数g> 0,任何树宽至少为f(g)的图,都包含(g x g)-网格作为次要项。直到最近,最著名的f的上界是g的超指数上界。Chekuri和Chuzhoy最近的一项工作提供了第一个多项式边界,通过证明树宽度f(g)=O(g98 poly log g)足以确保在任何图中存在(g x g)-网格。在本文中,我们提供了一个更简单的排除网格定理的证明,得到$f(g)=O(g^{36} poly log g)$的界。我们的证明是自包含的,除了使用先前的工作将输入图的最大顶点度降低到一个常数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Excluded Grid Theorem: Improved and Simplified
We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function f:Z+→ Z+, such that for any integer g> 0, any graph of treewidth at least f(g), contains the (g x g)-grid as a minor. Until recently, the best known upper bounds on f were super-exponential in g. A recent work of Chekuri and Chuzhoy provided the first polynomial bound, by showing that treewidth f(g)=O(g98 poly log g) is sufficient to ensure the existence of the (g x g)-grid minor in any graph. In this paper we provide a much simpler proof of the Excluded Grid Theorem, achieving a bound of $f(g)=O(g^{36} poly log g)$. Our proof is self-contained, except for using prior work to reduce the maximum vertex degree of the input graph to a constant.
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