B-disjunctive total domination in graphs: Algorithm and hardness results

Abstract

Let G = (V,E) be a connected graph with at least two vertices. For a fixed positive integer b > 1, a set D ⊆ V is called a b-disjunctive total dominating set of G if for every vertex v ∈ V, v is either adjacent to a vertex of D or has at least b vertices in D at distance 2 from it. The minimum cardinality of a b-disjunctive total dominating set of G is called the b-disjunctive total domination number of G, and is denoted by (formula presented) (G). The Minimum b-Disj Total Domination problem is to find a b-disjunctive total dominating set of cardinality (formula presented) (G). Given a positive integer k and a graph G, the b-Disj Total Dom Decision problem is to decide whether G has a b-disjunctive total dominating set of cardinality at most k. In this paper, we initiate the algorithmic study of the Minimum b-Disj Total Domination problem. We prove that the b-Disj Total Dom Decision problem is NP-complete even for bipartite graphs and chordal graphs, two important graph classes. On the positive side, we propose a ln(Δ2 + (b − 1)Δ) + 1-approximation algorithm for the Minimum b-Disj Total Domination problem. We prove that the Minimum b-Disj Total Domination problem cannot be approximated within (Formula presented) unless NP ⊆ DTIME(|V |O(log log|V|)). Finally, we show that the Minimum b-Disj Total Domination problem is APX-complete for bipartite graphs with maximum degree b + 3.