  A graph G is said to be 1-perfectly orientable (1-p.o. for short) if it
admits an orientation such that the out-neighborhood of every vertex is a
clique in G. The class of 1-p.o. graphs forms a common generalization of the
classes of chordal and circular arc graphs. Even though 1-p.o. graphs can be
recognized in polynomial time, no structural characterization of 1-p.o. graphs
is known. In this paper we consider the four standard graph products: the
Cartesian product, the strong product, the direct product, and the
lexicographic product. For each of them, we characterize when a nontrivial
product of two graphs is 1-p.o.
