Parameterized complexity of generalized domination problems

Abstract

Given two sets σ,ρ of non-negative 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 the 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 parameterization by n−k, and results on certain infinite sets (in particular for σ,ρ being the sets of even and odd integers).