Parameterized complexity of generalized domination problems

Abstract

Given two sets σ, ρ of nonnegative integers, a set S of vertices of a graph G is (σ,ρ)-dominating if |S∩N(v)| ∈ σ for every vertex v ∈ S, and |S ∩ N(v)| ∈ ρ for every v ∉ S. This concept, introduced by Telle in 1990's, generalizes and unifies several variants of graph domination studied separately before. We study the parameterized complexity of (σ,ρ)-domination in this general setting. Among other results we show that existence of a (σ,ρ)-dominating set of size k (and at most k) are W[1]-complete problems (when parameterized by k) for any pair of finite sets σ and ρ. We further present results on dual parametrization by n∈-∈k, and results on certain infinite sets (in particular for σ, ρ being the sets of even and odd integers). © 2010 Springer-Verlag.