An outer independent double Italian dominating function on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2,3\}$ for which each vertex $x\in V(G)$ with $f(x)\in \{0,1\}$ then $\sum_{y\in N[x]}f(y)\geqslant 3$ and vertices assigned $0$ under $f$ are independent. The outer independent double Italian domination number $\gamma_{oidI}(G)$ is the minimum weight of an outer independent double Italian dominating function of graph $G$. In this work, we present some contributions to the study of outer independent double Italian domination of three graph products. We characterize the Cartesian product, lexicographic product and direct product of custom graphs in terms of this parameter. We also provide the best possible upper and lower bounds for these three products for arbitrary graphs.
