Further results on secure restrained domination in graphs
P. Roushini Leely Pushpam, Chitra Suseendran · Journal of Discrete Mathematical Sciences and Cryptography · 2016
Let G=(V, E) be a graph and let S ⊆ V. The set S is a dominating set of G if every vertex in V \ S is adjacent to some vertex in S. The set S is a restrained dominating set if every vertex in V \ S is adjacent to a vertex in S and to a vertex in V \ S. The minimum cardinality of a restrained dominating set is called restrained domination number of G and it is denoted by γr(G). A set S ⊆ V(G) is called a secure (restrained) dominating set if S is (restrained) dominating and for all u ϵ V \ S there exists v ϵ S ∩ N(u) such that (S \ {v}) ∪ {u} is (restrained) dominating. The minimum cardinality of a secure (restrained) dominating set is a secure (restrained) domination number of G and it is denoted by γs(G) (γsr(G)). In this paper we characterize few classes of graphs for which γr(G)=γsr(G) and γs(G)=γsr(G). Specific values of certain graphs are determined and Nordhaus Gaddum type results are discussed.