Algorithmic aspects of b-disjunctive domination in graphs

Abstract

For a graph \(G=(V,E)\), a set \(D\subseteq V\) is called a disjunctive dominating set of G if for every vertex \(v\in V\setminus D\), v is either adjacent to a vertex of D or has at least two vertices in D at distance 2 from it. The cardinality of a minimum disjunctive dominating set of G is called the disjunctive domination number of graph G, and is denoted by \(\gamma _{2}^{d}(G)\). The Minimum Disjunctive Domination Problem (MDDP) is to find a disjunctive dominating set of cardinality \(\gamma _{2}^{d}(G)\). Given a positive integer k and a graph G, the Disjunctive Domination Decision Problem (DDDP) is to decide whether G has a disjunctive dominating set of cardinality at most k. In this article, we first propose a polynomial time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a \((\ln (\Delta ^{2}+\Delta +2)+1)\)-approximation algorithm for MDDP, where \(\Delta \) is the maximum degree of input graph \(G=(V,E)\) and prove that MDDP can not be approximated within \((1-\epsilon ) \ln (|V|)\) for any \(\epsilon >0\) unless NP \(\subseteq \) DTIME\((|V|^{O(\log \log |V|)})\). Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree 3.