Polynomial Kernels for Generalized Domination Problems

TL;DR

This paper investigates the parameterized complexity of a generalized domination problem, identifying conditions for polynomial kernels and extending results to broader parameters.

Contribution

It characterizes when the [σ, ρ] Dominating Set problem admits polynomial kernels based on properties of σ and ρ, especially for finite sets.

Findings

Identifies exact conditions for polynomial kernels with finite σ and ρ.

Extends kernelization bounds to more general parameters.

Provides lower and upper bounds for kernel sizes.

Abstract

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.