p-Edge/Vertex-Connected Vertex Cover: Parameterized and Approximation Algorithms

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 $O(2^{O(pk)}n^{O(1)})$-time algorithm for $O(2^{O(k^2)}n^{O(1)})$-time VC. Finally, we describe $2(p+1)$-approximation The proofs require more sophisticated arguments than In particular, approximation use Gomory-Hu trees kernels result on small-size spanning $p$-vertex/edge-connected subgraph graph obtained independently by Nishizeki Poljak (1994) Nagamochi Ibaraki (1992).