Characterization of Some Claw-Free Graphs in Co-Secure Domination Number

For a vertex subset S of a graph G, if each vertex of G is either in S or adjacent to some vertex in S, then S is a dominating set of G. Let S be a dominating set of a graph G. If each vertex v not in S has a neighbor u in S such that (S\{u})∪{v} is also a dominating set of G, then S is a secure dominating set of G. If each vertex u in S has a neighbor v not in S such that (S\{u})∪{v} is also a dominating set of G, then S is a co-secure dominating set of G. The minimum cardinality of a secure (resp. co-secure) dominating set of G is the secure (resp. co-secure) domination number of G. Arumugam et al. proposed the questions to characterize a graph G such that the co-secure domination number of G equals the independence number and the secure domination number of G, respectively. Inspired by those questions, in this paper, we obtain two classes of claw-free graphs such that the co-secure domination number equal the independence number and the secure domination number. Our results provide some theoretical basis of claw-free graphs for networks.