We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: $p$-Edge-Connected $p$-Vertex-Connected VC problem (where $p \geq 2$ is a fixed integer). Like VC, both new problems are FPT, but do not admit polynomial kernel unless $NP \subseteq coNP/poly$, which highly unlikely. prove however that time efficient sized approximate kernelization schemes. obtain an $...