On the structure of strong 3-quasi-transitive digraphs

In 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-quasi-transitive digraphs distinct from strong semicomplete digraphs and strong semicomplete bipartite digraphs and provide a complete characterization of strong 3-quasi-transitive digraphs.