Disjoint Perfect Domination in the Cartesian Product of Two Graphs

Abstract

Let G be a connected simple graph. A dominating set S ⊆ V (G) is called a perfect dominating set of G if every u ∈ V (G)\S is dominated by exactly one element of S. Let D be a minimum perfect dominating set of G. A perfect dominating set S ⊂ (V (G) \ D) is called an inverse perfect dominating set of G with respect to D. A disjoint perfect dominating set of G is the set C = D ∪ S ⊆ V (G). Furthermore, the disjoint perfect domination number, denoted by γpγp(G), is the minimum cardinality of a disjoint perfect dominating set of G. A disjoint perfect dominating set of cardinality γpγp(G) is called γpγp-set. In this paper, we give some property of the disjoint perfect dominating set in the Cartesian products of two graphs.