Improved Bounds for the Flat Wall Theorem
Improved Bounds for the Flat Wall Theorem
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平壁定理的改进界限
DOI：
10.1137/1.9781611973730.20
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Julia Chuzhoy
中科院分区：
文献类型：
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作者：
Julia Chuzhoy
罗伯逊（Robertson）和西摩（Seymour）的扁平壁定理指出，对于所有整数W，t> 1，每个图形g都包含尺寸f（w，t）的壁，必须包含（i）一个kt-nor，或者（ii）一个小子集a v（g）a v（g）的v（g）a refter a in thos a g y a.ka.ka.kawarabay thom具有以下两组参数：（1）F（w，t） θ（T24（T2 + W））|我们表明，如果g包含大小ω（dt（t + w））的壁，则是g包含一个kt少量，或者在G中有一个尺寸W的平坦墙。在边缘 - 偶口路径问题的算法中，出现了d≤4的证明，我们的证明是独立的，我们的证明是自我限制的，除非使用众所周知的distiales a a a a ang a ang a ang a a a a a a a a a a a a a a a a and a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a的cotity a and a a and a a a的使用量有高度效率。和g \ a中的尺寸W的平坦墙低度场景的结果是通过提供几乎匹配的下限：即，对于所有整数W，t> 1，都有一个图G，其中包含大小ω（wt）的壁，因此G中的最大顶点度为5，而G中没有大小W的平坦壁，也不包含KT-Minor。
The Flat Wall Theorem of Robertson and Seymour states that there is some function f, such that for all integers w, t > 1, every graph G containing a wall of size f(w, t), must contain either (i) a Kt-minor; or (ii) a small subset A ⊂ V(G) of vertices, and a flat wall of size w in G \ A. Kawarabayashi, Thomas and Wollan recently showed a self-contained proof of this theorem with the following two sets of parameters: (1) f(w, t) = Θ(t24(t2 + w)) with |A| = O(t24), and (2) f(w, t) = [EQUATION] with |A| ≤ t − 5. The latter result gives the best possible bound on |A|. In this paper we improve their bounds to f(w, t) = Θ(t(t + w)) with |A| ≤ t − 5. For the special case where the maximum vertex degree in G is bounded by D, we show that, if G contains a wall of size Ω(Dt(t + w)), then either G contains a Kt-minor, or there is a flat wall of size w in G. This setting naturally arises in algorithms for the Edge-Disjoint Paths problem, with D ≤ 4. Like the proof of Kawarabayashi et al., our proof is self-contained, except for using a well-known theorem on routing pairs of disjoint paths. We also provide efficient algorithms that return either a model of the Kt-minor, or a vertex set A and a flat wall of size w in G\A. We complement our result for the low-degree scenario by proving an almost matching lower bound: namely, for all integers w, t > 1, there is a graph G, containing a wall of size Ω(wt), such that the maximum vertex degree in G is 5, and G contains no flat wall of size w, and no Kt-minor.