Perfect secure domination in graphs

Let G=(V,E) be a graph. A subset S of V is a dominating set of G if every vertex in V∖S is adjacent to a vertex in S. A dominating set S is called a secure dominating set if for each v∈V∖S there exists u∈S such that v is adjacent to u and S1=(S∖{u})∪{v} is a dominating set. If further the vertex u∈S is unique, then S is called a perfect secure dominating set. The minimum cardinality of a perfect secure dominating set of G is called the perfect secure domination number of G and is denoted by γps(G). In this paper we initiate a study of this parameter and present several basic results.