Hardness results and approximability of cosecure domination in graphs

Let G = (V, E) be a simple graph with no isolated vertices. A dominating set S of G is said to be a cosecure dominating set of G, if for every vertex v is an element of S there exists a vertex u is an element of V\S such that uv is an element of E and (S\{v}) boolean OR{u} is a dominating set of G. The Minimum Cosecure Domination problem is to find a minimum cardinality cosecure dominating set of G. In this paper, we show that the decision version of the problem is NP-complete for split graphs, undirected path graphs (subclasses of chordal graphs), and circle graphs. We also present a linear-time algorithm to compute the cosecure domination number of cographs (subclass of circle graphs). In addition, we present a few results on the approximation aspects of the problem.