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 oidI (G) is the minimum weight taken over all outer independent double Italian dominating functions of G. In this work, we characterize Outer independent double Italian domination of some graph Cartesian products. Finally, we investigate the families of all graphs G such that $\gamma_{oidI} (G) = |V (G)|$ and for $\delta(G) \ge 2$, the graphs with this property are characterized.
