In this paper, we study the parameterized complexity of a generalized
domination problem called the [σ,ρ] Dominating Set problem. This
problem generalizes a large number of problems including the Minimum Dominating
Set problem and its many variants. The parameterized complexity of the
[σ,ρ] Dominating Set problem parameterized by treewidth is well
studied. Here the properties of the sets σ and ρ that make the
problem tractable are identified [1]. We consider a larger parameter and
investigate the existence of polynomial sized kernels. When σ and
ρ are finite, we identify the exact condition when the [σ,ρ] Dominating Set problem parameterized by vertex cover admits polynomial
kernels. Our lower and upper bound results can also be extended to more general
conditions and provably smaller parameters as well.Comment: 19 pages, 6 figure