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.