**6. Conclusions**

In this work we treated the four standard products of digraphs (the Cartesian, the strong, the direct and the lexicographic) with respect to the efficient open domination. The idea is to describe which digraphs among these products are efficient open domination digraphs and to describe them with the properties of their factors. We completely characterized such digraphs among the direct product (Theorem 8) and among the lexicographic product (Theorem 9). For the efficient open domination Cartesian product digraphs the characterizations are given for those for which the first factor has an underlying graph that is a path (Theorems 1, 3 and 4), a cycle (Theorem 2) or a star (Theorem 5). This yields an idea on how to deal with the Cartesian product of digraphs with one fixed factor and an arbitrary second one. Among the efficient open domination strong product of digraphs we characterized those in which both factors have uni-cyclic graphs as their underlying graphs (Theorems 6 and 7). We also conjecture that this are the only strong product digraphs that are the efficient open domination digraphs.

**Author Contributions:** All authors contributed equally to this work. Conceptualization, D.B. and I.P.; methodology, D.B. and I.P.; formal analysis, D.B. and I.P.; validation, D.B. and I.P.; writing—original draft preparation, D.B. and I.P.; writing—review and editing, D.B. and I.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially funded by Javna Agencija za Raziskovalno Dejavnost RS under gran<sup>t</sup> numbers P1-0297 and J1-9109.

**Conflicts of Interest:** The authors declare no conflict of interest.
