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