In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph $G$ and a positive integer $k$, and the objective is to decide whether $G$ contains a minimal vertex cover of size at least $k$. This problem has been considered in several articles in the last years. We focus on its kernelization, which had been almost unexplored so far. We prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover, even on bipartite graphs, unless ${\sf NP} \subseteq {\sf coNP} / {\sf poly}$. Motivated by a question of Boria et al. [Discret. Appl. Math. 2015] about the existence of subquadratic kernels for MMVC parameterized by $k$, we rule out their existence unless $P=NP$, if we restrict the kernelization algorithms to apply only a type of natural reduction rules that we call "large optimal preserving rules". In particular, these rules contain the typical reduction rules to obtain linear kernels for Vertex Cover. On the positive side, we provide subquadratic kernels on $H$-free graphs for several graphs $H$, such as the bull, the paw, or the complete graphs, by making use of the Erd\H{o}s-Hajnal property in order to find an appropriate decomposition.

翻译：在最大最小螺旋覆盖( MMVC) 问题中, 我们得到一个图形 $G$ 和正整数 $k$, 目标是决定$G$是否包含最小的顶层覆盖至少为$k$。 这个问题在过去几年中已经在几个文章中得到了考虑。 我们关注它的内核化问题, 这个问题迄今为止几乎还没有被探索。 我们证明 MMVC 不接受以最小的纸浆覆盖大小为多孔内核参数的多孔内核, 甚至在双面图形上也是如此, 除非 $@subseteq $\ subseteq $@sf coNP} / ssubseteqeq suff suff policate suppliates suff $H polical rules。 具体地说, 在“ 最优化的纸浆纸浆规则” 中, 提供相应的纸浆规则。