Abstract
A perfect Roman dominating function on a graph G is a function f:VG?0,1,2 for which every vertex v with fv=0 is adjacent to exactly one neighbor u with fu=2. The weight of f is the sum of the weights of the vertices. The perfect Roman domination number of a graph G, denoted by gamma RpG, is the minimum weight of a perfect Roman dominating function on G. In this paper, we prove that if G is the Cartesian product of a path Pr and a path Ps, a path Pr and a cycle Cs, or a cycle Cr and a cycle Cs, where r,s > 5, then gamma RpG & LE;2/3G.