In 1999, Huang (Australas. J. Combin., 19, 203–208) proved that every round digraph is locally semicomplete, and characterized the round digraph. In this paper, a generalization of the round digraph, the positive-round digraph, is introduced. We note that positive-round digraphs form subclasses of the class of locally in-semicomplete digraphs and give a sufficient condition, so that a strong locally in-semicomplete digraph is positive-round. Immediately, we obtain a characterization of strong positive-round local in-tournaments.