  In this paper, we study the parameterized complexity of a generalized
domination problem called the [${\sigma}, {\rho}$] 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
[${\sigma}, {\rho}$] Dominating Set problem parameterized by treewidth is well
studied. Here the properties of the sets ${\sigma}$ and ${\rho}$ that make the
problem tractable are identified [1]. We consider a larger parameter and
investigate the existence of polynomial sized kernels. When ${\sigma}$ and
${\rho}$ are finite, we identify the exact condition when the [${\sigma},
{\rho}$] 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.
