On the Complexity of Co-secure Dominating Set Problem
share
A set D ⊆ V of a graph G=(V, E) is a dominating set of G if every vertex v∈ V∖ D is adjacent to at least one vertex in D. A set S ⊆ 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 ∈ S there exists a vertex v ∈ V∖ S such that uv ∈ E and (S∖{u}) ∪{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 γ_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- ϵ)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 δ(G)≥ 2. For 3-regular and 4-regular graphs, we show that Min Co-secure Dom is approximable within a factor of 83 and 103, respectively. Furthermore, we prove that Min Co-secure Dom is APX-complete for 3-regular graphs.
READ FULL TEXT