Bridge-Depth刻画了顶点覆盖的哪些结构参数化存在多项式核
Bridge-Depth Characterizes which Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel
AI总结：
该研究引入新图参数bridge-depth，完整刻画了存在多项式核的、以输入图到 minor-闭图类的顶点删除距离为参数的顶点覆盖问题的minor-闭图族特征。
摘要：
我们研究顶点覆盖（Vertex Cover）问题结构参数化的核化复杂度。该研究的目标是找到一种多项式时间预处理算法，能将顶点覆盖问题的任意实例$(G,k)$归约为等价实例，且等价实例的规模以图$G$的预设复杂度参数的规模为多项式上界。过往大量研究围绕基于顶点删除数的参数化展开，这类参数衡量将图$G$归约为简单图类$\mathcal{F}$（如森林、有界树深度图、最大度为2的图）的成员所需删除的顶点数量。我们致力于找出最通用的图类$\
英文摘要：
We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance $(G,k)$ of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a pre-determined complexity parameter of $G$. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce $G$ to a member of a simple graph class $\mathcal{F}$, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes $\mathcal{F}$ for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to $\mathcal{F}$, admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families $\mathcal{F}$ for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if $\mathcal{F}$ has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number.
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