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 $D\subseteq V$ is called a b-disjunctive total dominating set of G if for every vertex $v\in V$, 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 $\gamma _{b}^{td}(G)$. The Minimum b-Disj Total Domination problem is to find a b-disjunctive total dominating set of cardinality $\gamma _{b}^{td}(G)$. 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 $\ln (\varDelta ^{2}+(b-1)\varDelta )+1$-approximation algorithm for the Minimum b-Disj Total Domination problem. We prove that the Minimum b-Disj Total Domination problem cannot be approximated within $\frac{1}{2}(1-\epsilon ) \ln |V|$ for any $\epsilon > 0$ unless NP $\subseteq$ 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 $b+3$.