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.

翻译：我们研究顶点覆盖问题的结构参数化的核化复杂性。这里，目标是找到一个多项式时间预处理算法，能将顶点覆盖问题的任意实例$(G,k)$约简为一个等价实例，其规模是$G$的预定复杂度参数的多项式。先前一系列研究涉及基于将$G$减少到简单图类$\mathcal{F}$（如森林、有界树深图、最大度为2的图）所需顶点删除次数的参数化。我们着手寻找最一般的图类$\mathcal{F}$，使得以输入图到$\mathcal{F}$的顶点删除距离为参数的顶点覆盖问题允许多项式核化。我们给出了允许此类核化的子式封闭图族$\mathcal{F}$的完整刻画。我们引入了一个称为桥深度的新图参数，并证明多项式核化存在当且仅当$\mathcal{F}$具有有界桥深度。该证明基于桥深度与图中最小阻塞集（其移除会降低独立数的顶点集）大小之间的有趣联系。