a b s t r a c tIn this paper, D = (V (D), A(D)) denotes a loopless directed graph (digraph) with at most one arc from u to v for every pair of vertices u and v of V (D). Given a digraph D, we say that D is 3-quasi-transitive if, whenever u → v → w → z in D, then u and z are adjacent or u = z.In Bang-Jensen (2004) [3], Bang-Jensen introduced 3-quasi-transitive digraphs and claimed that the only strong 3-quasi-transitive digraphs are the strong semicomplete digraphs and strong semicomplete bipartite digraphs. In this paper, we exhibit a family of strong 3-quasitransitive digraphs distinct from strong semicomplete digraphs and strong semicomplete bipartite digraphs and provide a complete characterization of strong 3-quasi-transitive digraphs.Classes of graphs characterized by forbidding families of induced graphs play an important role in Graph Theory. Given a family of graphs F , we say that a graph G is F -free if G has no induced subgraph in F . Perfect graphs are probably the best known class of such graphs.The class of connected graphs with no induced 2-paths are the complete graphs. The class of connected graphs with no induced 3-paths are the so-called cographs or complement-reducible graphs which were characterized in [7].In this paper we study a directed version of F -free graphs. In order to do this, given F , a family of oriented graphs, we say that D is an orientedly F -free digraph if there is no digraph in F isomorphic to any induced subdigraph of any orientation of D (an orientation of a digraph D is a spanning subdigraph of D in which we choose only one arc between any two adjacent vertices of D; for example, any orientation of any semicomplete digraph is a tournament). If F = {F }, we say orientedly F -free instead of orientedly F -free.There are three different possible orientations of the 2-path, see Fig. 1. In J 1 , J 2 , and J 3 any arc between the two vertices with a dotted edge between them is forbidden. Orientedly J 1 -free digraphs (respectively orientedly J 2 -free digraphs) were introduced by Bang-Jensen in [1] as a generalization of semicomplete digraphs. They are known as locally in-semicomplete (resp. locally out-semicomplete) digraphs. Orientedly {J 1 , J 2 }-free digraphs were introduced in the same paper and were characterized by Guo, Gutin, and Volkmann in [4]. Orientedly J 3 -free digraphs were introduced by GhouilaHouri in [9]. Observe that orientedly J 3 -free digraphs are the same family which were characterized by Bang-Jensen and Huang in [6].There are four different possible orientations of the 3-path, see Fig. 2. In H 1 , H 2 , H 3 , and H 4 , any arc between the two vertices with a dotted edge between them is forbidden. Orientedly {H 1 , H 2 }-free digraphs were introduced by BangJensen as a common generalization of both semicomplete digraphs and semicomplete bipartite digraphs in [2]. They are the arc-locally semicomplete digraphs which were characterized by . Orientedly H 1 -free digraphs (resp. orientedly H 2 -free digraphs) were introduced by Wang and Wang as arc-locall...