An outer independent double Italian dominating function is a double Italian dominating function f for which the set of vertices assigned 0 under f is independent. The outer independent double Italian domination number γoidI(G) is the minimum weight taken over all outer independent double Italian dominating functions of G. In this work, we present some contributions to the study of outer independent double Italian domination in graphs. We show that the outer independent double Italian domination number is NP-complete even when restricted to planner graphs with maximum degree at most four. We characterize the families of all connected graphs G with γoidI(G) = 5. We also investigate the families of all graphs G such that γoidI(G) = |V (G)| and for δ(G) ≥ 2, the graphs with this property are characterized. Finally, we characterize the families of all connected graphs G of order n, for which γoidI(G) ∈ {2n − k| 0 ≤ k ≤ 5}
