On the Complexity of Co-secure Dominating Set Problem

A set \(D \subseteq V\) of a graph \(G=(V, E)\) is a dominating set of \(G\) if every vertex \(v\in V\setminus D\) is adjacent to at least one vertex in \(D.\) A set \(S \subseteq V\) is a co-secure dominating set (CSDS) of a graph \(G\) if \(S\) is a dominating set of \(G\) and for each vertex \(u \in S\) there exists a vertex \(v \in V\setminus S\) such that \(uv \in E\) and \((S\setminus \{u\}) \cup \{v\}\) is a dominating set of \(G\). The minimum cardinality of a co-secure dominating set of \(G\) is the co-secure domination number and it is denoted by \(\gamma_{cs}(G)\). Given a graph \(G=(V, E)\), the minimum co-secure dominating set problem (Min Co-secure Dom) is to find a co-secure dominating set of minimum cardinality. In this paper, we strengthen the inapproximability result of Min Co-secure Dom for general graphs by showing that this problem can not be approximated within a factor of \((1- \epsilon)\ln |V|\) for perfect elimination bipartite graphs and star convex bipartite graphs unless P=NP. On the positive side, we show that Min Co-secure Dom can be approximated within a factor of \(O(\ln |V|)\) for any graph \(G\) with \(\delta(G)\geq 2\). For \(3\)-regular and \(4\)-regular graphs, we show that Min Co-secure Dom is approximable within a factor of \(\dfrac{8}{3}\) and \(\dfrac{10}{3}\), respectively. Furthermore, we prove that Min Co-secure Dom is APX-complete for \(3\)-regular graphs.