Double Roman Graphs in P(3k, k)

A double Roman dominating function on a graph G=(V,E) is f:V?{0,1,2,3} with the properties that if f(u)=0, then vertex u adjacent to at least one assigned 3 or two vertices 2, and f(u)=1, 2 3. The weight of f equals w(f)=?v?Vf(v). domination number ?dR(G) G minimum G. said be ?dR(G)=3?(G), where ?(G) We obtain sharp lower bound generalized Petersen graphs P(3k,k), we construct solutions providing upper bounds, which gives exact values for all P(3k,k). This implies P(3k,k) only either k?0 (mod 3) k?{1,4}.