We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: the $p$-Edge-Connected and $p$-Vertex-Connected VC problem (where $p \geq 2$ is a fixed integer). Like Connected VC, both new VC problems are FPT, but do not admit a polynomial kernel unless $NP \subseteq coNP/poly$, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. We obtain an $O(2^{O(pk)}n^{O(1)})$-time algorithm for the $p$-Edge-Connected VC and an $O(2^{O(k^2)}n^{O(1)})$-time algorithm for the $p$-Vertex-Connected VC. Finally, we describe a $2(p+1)$-approximation algorithm for the $p$-Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning $p$-vertex/edge-connected subgraph of a $p$-vertex/edge-connected graph obtained independently by Nishizeki and Poljak (1994) and Nagamochi and Ibaraki (1992).

中文翻译：

---

p-Edge/Vertex-Connected Vertex Cover：参数化和近似算法

我们介绍并研究了 Connected VertexCover (VC) 问题的两个自然概括：$p$-Edge-Connected 和 $p$-Vertex-Connected VC 问题（其中 $p\geq 2$ 是固定整数）。与 Connected VC 一样，两个新的 VC 问题都是 FPT，但不承认多项式核，除非 $NP \subseteq coNP/poly$，这是极不可能的。然而，我们证明这两个问题都承认时间高效的多项式大小的近似核化方案。我们为 $p$-Edge-Connected VC 获得 $O(2^{O(pk)}n^{O(1)})$-time 算法和 $O(2^{O(k^2) )}n^{O(1)})$-时间算法用于 $p$-Vertex-Connected VC。最后，我们描述了 $p$-Edge-Connected VC 的 $2(p+1)$-近似算法。新 VC 问题的证明需要比 Connected VC 更复杂的论证。特别是，