On the parameterized complexity of vertex cover and edge cover with connectivity constraints

We investigate the effect of certain natural connectivity constraints on the parameterized complexity of two fundamental graph covering problems, namely Vertex Cover and 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) for a fixed positive integer t, and call the problem t-Total Vertex Cover (resp. t-Total Edge Cover). In both cases the parameter k is the size of the solution. We show that
• both problems remain fixed-parameter tractable with these restrictions, with running times of the form O⋆(ck)O⋆(ck) for some constant c>0c>0 in each case, where the O⋆O⋆ notation hides polynomial factors;
• for each fixed t≥2t≥2, t-Total Vertex Cover has no polynomial kernel unless CoNP⊆NP/polyCoNP⊆NP/poly;
• for each fixed t≥2t≥2, t-Total Edge Cover has a linear vertex kernel of size t+1tk. These results significantly improve earlier work on these problems. We illustrate the utility of the technique used to solve t-Total Vertex Cover, by applying it to derive an O⋆(ck)O⋆(ck)-time FPT algorithm for the t-Total Edge Dominating Set problem.Our no-poly-kernel result for t-Total Vertex Cover, and the known NP-hardness result for t-Total Edge Cover, are in stark contrast to the fact that Vertex Cover has a 2k vertex kernel, and that Edge Cover is solvable in polynomial time. This illustrates how even the slightest connectivity requirement results in a drastic change in the tractability of problems—the curse of connectivity!