**1. Introduction**

In this work we join two natural concepts. The first one is operations on digraphs (under some rules) that result in a bigger digraph than the starting ones. The second one is partitions of sets. There exist many digraph products for which the vertex set is the Cartesian product of vertex sets of its factors (there are also several operations which have (di)graph product in their name, but the vertex set is defined in a different manner). They differ by the definitions of the edge sets. Among them, four are called standard products. These are the Cartesian product, the strong product, the direct product and the lexicographic product. One can find a rich bibliography about them (see [1]). One standard approach of studying the digraph products is to study their structure and how to recognize them. Another approach is to deduce the properties of (di)graph products with respect to some properties of their factors. The later is also the topic of this work.

Partitions of objects are always interesting and useful as a mathematical concept, as every partition yields an equivalence relation. This further enables a factor structure of starting objects, which often brings simplification and deeper insight. Therefore, it is natural to study different kinds of partitions and the existence of them. Unfortunately, we are often not in the position to describe the mentioned relation with the properties of the investigated objects. This often disables further studies.

Graph theory offers a wide range of possibilities for partitions, one of them being the partitions of vertices. Open neighborhoods are a natural example for partitioning the set of vertices. Among graphs this was initiated in 1993 by Cockayne et al. in [2], where such partitions were named total perfect codes. The terminology efficient open domination graphs was introduced by Gavlas and Schultz in 2002 (see [3]). The study of efficient open domination of Cayley graphs can be found in [4]. Grid graphs, that is Cartesian products of two paths, were investigated in [5–7] and direct products of graphs with such a partition were characterized in [8]. Characterizations of efficient open domination graphs among lexicographic, strong and disjunctive product of two graphs can be found in [9]. In the same paper [9] the Cartesian products of some known families of graphs with respect to efficient open domination were also investigated. Later, in [10], one factor of a Cartesian product was fixed while the other factor was characterized in such a way that its Cartesian product is an efficient open domination graph.

Existence of a partition of vertices of a graph into closed neighborhoods was initiated even earlier by Biggs in 1976 (see [11]) under the name 1-perfect graphs. The name efficient (closed) domination graphs was proposed later by Bange et al. in [12]. This subject became quite popular and throughout the years several combinatorial and computational results were presented. One of the latest results of this type is that the problem of efficient closed domination is solvable in polynomial time for the class of *P*6-free graphs, as shown in [13] and independently in [14]. This was further investigated in [15] for some subclasses of *P*6-free graphs. The authors use the maximum weight independent set problem of a square graph *G*<sup>2</sup> to which the efficient closed domination of *G* can be reduced. Among products the strong product was treated in [16] and the direct product of (an arbitrary number of) cycles was covered in a series of papers [17–19]. For the lexicographic product the topic was covered in [20], while Mollard deals with the efficient closed domination Cartesian product in [21]. Recently, graphs that are both efficient open and efficient closed domination at the same time were considered in [22].

In the case of digraphs one can also distinguish between in- and out-neighborhoods besides open and closed neighborhoods. However, this dilemma is artificial because if we reverse the orientation of the digraph, then in-neighborhoods become out-neighborhoods and vice versa. Hence, we can deal with efficient open and efficient closed domination digraphs. Efficient open domination digraphs were introduced in [23] and studied further in [24–27]. In [28] Schaudt presented a useful characterization under the name of efficient total domination digraphs. See also [29] for more recent results. As in the case of graphs, there is more literature concerning efficient closed domination digraphs than that of efficient open domination digraphs. Here we mention only [30], a recent work that brings the results on the efficient closed domination among standard products of digraphs.

The paper is organized as follows. In the coming section we first settle the terminology. A section with several results on efficient open domination Cartesian products of digraphs follows. There we present a method for constructing an efficient open domination Cartesian product of digraphs with one fixed factor. Section four is devoted to the efficient open domination strong products of digraphs. We characterize those for which the factors have uni-cyclic graphs as their underlying graphs. Moreover, we conjecture that these are the only efficient open domination digraphs among strong products. The last section brings characterizations of the efficient open domination direct and lexicographic products of digraphs.
