An outer independent double Italian dominating function (OIDIDF) on a graph G is a function f : V (G) ! f0; 1; 2; 3g for which each vertex x 2 V (G) with f(x) 2 f0; 1g then Py2N[x] f(x) > 3 and vertices assigned 0 under f is independent. The outer independent double Italian domination number γoidI(G) is the minimum weight of an OIDIDF on graph G. In this work, we present some contributions to the study of outer independent double Italian domination of Lexicographic graph product. We give some sharp upper and lower bounds on Lexicographic product of graphs.
