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

We say that a digraph is a $(t,\lambda )$-liking digraph if every t vertices have exactly λ common out-neighbors. In 1975, Plesník (1975) [14] proved that any $(t,1)$-liking digraph is the complete digraph on $t+1$ vertices for each $t\ge 3$. Choi et al. (2025) [5] 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 $(t,\lambda )$-liking digraphs with $t\ge \lambda +2$ and giving some equivalent conditions for a $(t,\lambda )$-liking digraph being a complete digraph on $t+\lambda$ vertices.