[论文解读] Graphs without large $K_{2,n}$-minors

Lochet, William, Sharma, Roohani|arXiv (Cornell University)|Feb 5, 2017

Advanced Graph Theory Research参考文献 2被引用 3

一句话总结

本文通过证明图无大$K_{2,n}$-子式当且仅当其由两类图通过有界2-和构造而成：第一类图（外平面图的推广，允许有限的弦交叉）和有界大小图的增强形式，对无大$K_{2,n}$-子式的图进行了刻画。关键结果是：每个最小度数≥6的足够大的3-连通图、每个含高阶顶点的4-连通图，以及每个足够大的5-连通图，必然包含一个$K_{2,n}$-子式（$n$较大），从而确立了子式包含的结构阈值。

ABSTRACT

In the F-Deletion problem, where F is a fixed finite family of graphs, the input is a graph G and an integer k, and the goal is to determine if there exists a set of at most k vertices whose deletion results in a graph that does not contain any graph of F as a minor. The F-Deletion problem encapsulates a large class of natural and interesting graph problems like Vertex Cover, Feedback Vertex Set, Treewidth-η Deletion, Treedepth-η Deletion, Pathwidth-η Deletion, Outerplanar Deletion, Vertex Planarization and many more. We study the F-Deletion problem from the kernelization perspective. In a seminal work, Fomin et al. [FOCS 2012] gave a polynomial kernel for this problem when the family F contains at least one planar graph. The asymptotic growth of the size of the kernel is not uniform with respect to the family F: that is, the size of the kernel is k^{f(F)}, for some function f that depends only on F. Later Giannopoulou et al. [TALG 2017] showed that the non-uniformity in the kernel size bound is unavoidable as Treewidth-η Deletion cannot admit a kernel of size 𝒪(k^{(η+1)/2 - ε}), for any ε > 0, unless NP ⊆ coNP/poly. On the other hand it was also shown that Treedepth-η Deletion admits a uniform kernel of size f(F) ⋅ k⁶ depicting that there are subclasses of F where the asymptotic kernel sizes do not grow as a function of the family F. This work led to the question of determining classes of F where the problem admits uniform polynomial kernels. In this paper, we show that if all the graphs in F are connected and ℱ contains K_{2,p} (a bipartite graph with 2 vertices on one side and p vertices on the other), then the problem admits a uniform kernel of size f(F) ⋅ k^10. The graph K_{2,p} is one natural extension of the graph θ_p, where θ_p is a graph on two vertices and p parallel edges. The case when F contains θ_p has been studied earlier and serves as (the only) other example where the problem admits a uniform polynomial kernel.

研究动机与目标

• 刻画对任意给定$n$，不包含大$K_{2,n}$-子式的图的结构。
• 将已知的$K_{1,n}$-子式自由图的结构刻画推广至更复杂的$K_{2,n}$-子式情形。
• 确立$K_{2,n}$-子式在大图中必然存在的充分连通性条件。
• 通过2-和与有界增强形式，形式化一个分解定理，以描述$K_{2,n}$-子式自由图。

提出的方法

• 引入第一类图为外平面图的推广，其定义基于哈密顿圈上受限制的弦交叉。
• 定义两类增强形式：扇形与狭条，用于在有界大小图上添加，构成第二类构造块。
• 使用2-和运算组合图，保持结构约束的同时构建复杂的$K_{2,n}$-子式自由图。
• 应用归纳法与子式收缩/删除论证，证明第一类图不包含$K_{2,5}$-子式。
• 在2-和中采用递归子式回避论证，证明若各分量图避免$K_{2,m}$与$K_{2,n}$-子式，则其2-和也避免$K_{2,mn}$-子式。
• 利用狭条与扇形中的边分离集合，限制不相交连接的数量，从而对子式大小建立指数上界。

实验结果

研究问题

• RQ1 对给定的$n$，不包含大$K_{2,n}$-子式的图具有何种结构特征？
• RQ2 $K_{2,n}$-子式自由图的类是否可使用有限组构造块与运算进行分解？
• RQ3 何种连通性阈值会强制大图中出现$K_{2,n}$-子式？
• RQ4 2-和运算如何影响具有有界大小增强形式的图的子式闭包性质？

主要发现

• 每个$K_{2,n}$-子式自由图均可通过有限次2-和操作，由第一类图与有界大小图的增强形式构造而成。
• 第一类图不包含$K_{2,5}$-子式，确立了子式包含的结构障碍。
• 两个$K_{2,m}$-与$K_{2,n}$-子式自由图的2-和是$K_{2,mn}$-子式自由的，从而实现递归子式回避边界的构建。
• 对最多$n$个顶点的图的任意增强形式，均避免$K_{2,f_{6.3}(n)}$-子式，其中$f_{6.3}(n) = 10n \cdot 2^n$，提供指数上界。
• 对任意固定$n$，每个足够大的3-连通图，若其最小度数至少为六，则必含$K_{2,n}$-子式。
• 每个含足够高阶顶点的4-连通图，以及每个足够大的5-连通图，对大$n$必然包含$K_{2,n}$-子式。