For a fixed graph $H$, the $H$-SUBGRAPH HITTING problem consists in deleting the minimum number of vertices from an input graph to obtain a graph without any occurrence of $H$ as a subgraph. This problem can be seen as a generalization of VERTEX COVER, which corresponds to the case $H = K_2$. We initiate a study of $H$-SUBGRAPH HITTING from the point of view of characterizing structural parameterizations that allow for polynomial kernels, within the recently active framework of taking as the parameter the number of vertex deletions to obtain a graph in a "simple" class $C$. Our main contribution is to identify graph parameters that, when $H$-SUBGRAPH HITTING is parameterized by the vertex-deletion distance to a class $C$ where any of these parameters is bounded, and assuming standard complexity assumptions and that $H$ is biconnected, allow us to prove the following sharp dichotomy: the problem admits a polynomial kernel if and only if $H$ is a clique. These new graph parameters are inspired by the notion of $C$-elimination distance introduced by Bulian and Dawar [Algorithmica 2016], and generalize it in two directions. Our results also apply to the version of the problem where one wants to hit $H$ as an induced subgraph, and imply in particular, that the problems of hitting minors and hitting (induced) subgraphs have a substantially different behavior with respect to the existence of polynomial kernels under structural parameterizations.

翻译：对于固定图$H$，$H$-子图击中问题要求从输入图中删除最少顶点数，使得得到的图不包含任何$H$作为子图。该问题可视为顶点覆盖问题的推广（后者对应于$H = K_2$的情形）。我们在近期活跃的研究框架下（以删除顶点至某"简单"类$C$中的图所需的顶点数作为参数），从刻画允许存在多项式核的结构化参数化特征的角度，首次对$H$-子图击中问题展开研究。我们的主要贡献在于识别出一类图参数，当$H$-子图击中问题以删除顶点至某类$C$（其中这些参数有界）的距离为参数时，在标准复杂度假设且$H$为双连通图的条件下，可证明以下尖锐二分性：该问题存在多项式核当且仅当$H$为团。这些新图参数受Bulian与Dawar [Algorithmica 2016]提出的$C$-消去距离概念启发，并在两个方向上对其进行了推广。我们的结果同样适用于需要将$H$作为导出子图击中的问题变体，并特别表明：在结构化参数化下，击中小图与击中（导出）子图问题在多项式核的存在性方面具有显著不同的行为特征。