AbstractA dominating (respectively, total dominating) set of a digraph is a set of vertices in such that the union of the closed (respectively, open) out‐neighborhoods of vertices in equals the vertex set of . The minimum size of a dominating (respectively, total dominating) set of is the domination (respectively, total domination) number of , denoted (respectively, ). The maximum number of pairwise disjoint closed (respectively, open) in‐neighborhoods of is denoted by (respectively, ). 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 (i.e., a digraph whose underlying graph is a tree), and . By using the former equality we then prove that , where is any digraph and is any ditree, each without a source vertex, and is their direct product. From the equality we derive the bound , where is an arbitrary digraph, an arbitrary ditree and is their Cartesian product. In general digraphs this Vizing‐type bound fails, yet we prove that for any digraphs and , where , we have . This inequality is sharp as demonstrated by an infinite family of examples. Ditrees and digraphs enjoying are also investigated.