Co-Secure Domination Number in Some Graphs

Let G be a graph and S⊆V(G). If S is a dominating set of G, and for each vertex u∈V(G)−S, there is a neighbor of u in S, denoted by v, such that (S∖{v})∪{u} is a dominating set of G, then S is a secure dominating set (SDS) of G. Conversely, S is a co-secure dominating set (CSDS) of G if S is a dominating set of G and for each vertex v in S, V(G)−S contains a neighbor of v, denoted by u, such that (S∖{v})∪{u} is a dominating set of G. The minimum cardinality of a CSDS (resp. SDS) of G is the co-secure (resp. secure) domination number of G. We use γcs(G) and γs(G) to denote the co-secure domination number and secure domination number of G, respectively. Arumugam et al. proposed two questions: (1) Characterize a graph G with γcs(G)=α(G), where α(G) is the independence number of G; (2) Characterize a graph G with γcs(G)=γs(G). In this paper, we characterize some forbidden induced subgraphs for a graph G with γcs(G)=α(G); moreover, we obtain that γcs(G)=γs(G) for each {K3,C5,P5}-free graph G with δ(G)≥2. Our conclusions can generalize some known results.