Disjoint Perfect Secure Domination in the Cartesian Product and Lexicographic Product of Graphs

Let G be a graph. A dominating set D⊆V (G) is called a secure dominating set of G if for each vertex u∈V (G) ∖D, there exists a vertex v∈D such that uv∈ E (G) and the set (D∖v) ∪u is a dominating set of G. If every u∈V (G) ∖D is adjacent to exactly one vertex in D, then D is a perfect secure dominating set of G. Let D be a minimum perfect secure dominating set of G. If S⊆V (G) ∖D is a perfect secure dominating set of G, then S is called an inverse perfect secure dominating set of G with respect to D. A disjoint perfect secure dominating set of G is the set C=D∪S⊆V (G). Furthermore, the disjoint perfect secure domination number, denoted by γₚs γₚs (G), is the minimum cardinality of a disjoint perfect secure dominating set of G. A disjoint perfect secure dominating set of cardinality γₚs γₚs (G) is called γₚs γₚs-set. In this paper, we extended the study on the concept of disjoint perfect secure domination in graphs. Furthermore, we characterized the disjoint perfect secure domination in the Cartesian product and lexicographic product of two graphs.