The Curse of Connectivity: t-Total Vertex (Edge) Cover

Abstract

We investigate the effect of certain natural connectivity constraints on the parameterized complexity of two fundamental graph covering problems, namely k-Vertex Cover and k-Edge Cover. Specifically, we impose the additional requirement that each connected component of a solution have at least t vertices (resp. edges from the solution), and call the problem t-total vertex cover (resp. t-total edge cover). We show that both problems remain fixed-parameter tractable with these restrictions, with running times of the form \({\mathcal O}^{*}\left(c^{k}\right)\) for some constant c > 0 in each case; for every t ≥ 2, t-total vertex cover has no polynomial kernel unless the Polynomial Hierarchy collapses to the third level; for every t ≥ 2, t-total edge cover has a linear vertex kernel of size \(\frac{t+1}{t}k\). KeywordsCover ProblemVertex CoverPolynomial KernelVertex Cover ProblemMinimal Vertex CoverThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.