Computational Complexity of Generalized Domination: A Complete Dichotomy for Chordal Graphs

Abstract

The so called (σ,ρ)-domination, introduced by J.A. Telle, is a concept which provides a unifying generalization for many variants of domination in graphs. (A set S of vertices of a graph G is called (σ,ρ)-dominating if for every vertex v ∈ S, |S ∩ N(v)| ∈ σ, and for every v ∉ S, |S ∩ N(v)| ∈ ρ, where σ and ρ are sets of nonnegative integers and N(v) denotes the open neighborhood of the vertex v in G.) It was known that for any two nonempty finite sets σ and ρ (such that 0 ∉ ρ), the decision problem whether an input graph contains a (σ,ρ)-dominating set is NP-complete, but that when restricted to chordal graphs, some polynomial time solvable instances occur. We show that for chordal graphs, the problem performs a complete dichotomy: it is polynomial time solvable if σ, ρ are such that every chordal graph contains at most one (σ,ρ)-dominating set, and NP-complete otherwise. The proof involves certain flavor of existentionality - we are not able to characterize such pairs (σ,ρ) by a structural description, but at least we can provide a recursive algorithm for their recognition. If ρ contains the 0 element, every graph contains a (σ,ρ)-dominating set (the empty one), and so the nontrivial question here is to ask for a maximum such set. We show that MAX-(σ,ρ)-domination problem is NP-complete for chordal graphs whenever ρ contains, besides 0, at least one more integer.