The separated box product of two digraphs

A new product construction of graphs and digraphs, called the separated box product, is presented, and several of its properties are discussed. The construction is based on the standard box product of digraphs. However, unlike the standard box product, it preserves the valence of the factor digraphs: If every vertex in both factors has in-valence and out-valence k for some fixed k, then so does every vertex of the separated box product. Questions about the symmetries of the product and their relations to symmetries of the factor graphs are considered. An application of this construction to the case of tetravalent edge-transitive graphs is discussed in detail.