*Article* **Efficient Open Domination in Digraph Products**

**Dragana Božovi´c 1,2 and Iztok Peterin 1,2,\***


Received: 6 February 2020; Accepted: 27 March 2020; Published: 2 April 2020

**Abstract:** A digraph *D* is an efficient open domination digraph if there exists a subset *S* of *V*(*D*) for which the open out-neighborhoods centered in the vertices of *S* form a partition of *<sup>V</sup>*(*D*). In this work we deal with the efficient open domination digraphs among four standard products of digraphs. We present a method for constructing the efficient open domination Cartesian product of digraphs with one fixed factor. In particular, we characterize those for which the first factor has an underlying graph that is a path, a cycle or a star. We also characterize the efficient open domination strong product of digraphs that have factors whose underlying graphs are uni-cyclic graphs. The full characterizations of the efficient open domination direct and lexicographic product of digraphs are also given.

**Keywords:** efficient open domination; digraphs; products of digraphs
