Domination in digraphs and their direct and Cartesian products

A dominating (respectively, total dominating) set S of a digraph D is a set of vertices in D such that the union of the closed (respectively, open) out-neighborhoods of vertices in S equals the vertex set of D. The minimum size of a dominating (respectively, total dominating) set of D is the domination (respectively, total domination) number of D, denoted gamma(D) (respectively, gamma t(D)). The maximum number of pairwise disjoint closed (respectively, open) in-neighborhoods of D is denoted by rho(D) (respectively, rho o(D)). We prove that in digraphs whose underlying graphs have girth at least 7, the closed (respectively, open) in-neighborhoods enjoy the Helly property, and use these two results to prove that in any ditree T (i.e., a digraph whose underlying graph is a tree), gamma t(T)=rho o(T) and gamma(T)=rho(T). By using the former equality we then prove that gamma t(GxT)=gamma t(G)gamma t(T), where G is any digraph and T is any ditree, each without a source vertex, and GxT is their direct product. From the equality gamma(T)=rho(T) we derive the bound gamma(GT)>=gamma(G)gamma(T), where G is an arbitrary digraph, T an arbitrary ditree and GT is their Cartesian product. In general digraphs this Vizing-type bound fails, yet we prove that for any digraphs G and H, where gamma(G)>=gamma(H), we have gamma(GH)>= 12 gamma(G)(gamma(H)+1). This inequality is sharp as demonstrated by an infinite family of examples. Ditrees T and digraphs H enjoying gamma(TH)=gamma(T)gamma(H) are also investigated.