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 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 . © 2010 Springer-Verlag Berlin Heidelberg.