perfect roman 3-domination of graphs

For a graph g = (v; e) with v = v (g) and e = e(g), a perfect roman f3g-dominating function isa function f : v ! f0; 1; 2; 3g having the property that 3 ≤ pu2ng[v] f(u) ≤ 4, if f(v) 2 f0; 1g for anyvertex v 2 g. the weight of a perfect roman f3g-dominating function f is the sum f(v ) = pv2v (g) f(v)and the minimum weight of a perfect roman f3g-dominating function on g is the perfect roman f3gdomination number of g, denoted by γfp r3g(g). in this manuscript we study the perfect roman f3gdomination of some graphs.