The diverging unidirectional star graph is the unidirectional star graph with n+1 nodes(vertices) and n arcs, n nodes of in-degree 1 and out-degree 0. The converging unidirectional graph consisting of n+1 vertices and n arcs, where n vertices have out-degree one and in-degree zero, with all arcs oriented towards the single central vertex. A diverging unidirectional perfect binary tree is a directed tree where each internal vertex, except for the root, has an out-degree of two and in-degree of one, with all leaves at the same depth; conversely, a converging unidirectional perfect tree has the opposite structure. Here, we investigate the in-domination and out-domination numbers associated with these classes of directed graph. Explicit expressions for the same numbers are derived for the afore mentioned families of graphs as well as for unidirectional non-homogeneous super caterpillar graphs.