Digraphs in which every t vertices have exactly common out-neighbors

We say that a digraph is a (t, ) -liking digraph if every t vertices have exactly common out-neighbors. In 1975, Plesn\'ik Graphs with a homogeneity, 1975. Glasnik Mathematicki 10: 9-23 proved that any (t, 1) -liking digraph is the complete digraph on t+1 vertices for each t 3. Choi et al. A digraph version of the Friendship Theorem, 2023. arXiv preprint arXiv: 2305. 04058 (to appear in Discrete mathematics) showed that a (2, 1) -liking digraph is a fancy wheel digraph or a k-diregular digraph for some positive integer k. In this paper, we extend these results by completely characterizing the (t, ) -liking digraphs with t +2 and giving some equivalent conditions for a (t, ) -liking digraph being a complete digraph on t+ vertices.