Polynomial Kernels for Generalized Domination Problems

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