On the structure of strong 3-quasi-transitive digraphs

In this paper, D=(V(D),A(D))D=(V(D),A(D)) denotes a loopless directed graph   (digraph) with at most one arc from uu to vv for every pair of vertices uu and vv of V(D)V(D). Given a digraph DD, we say that DD is 3-quasi-transitive   if, whenever u→v→w→zu→v→w→z in DD, then uu and zz are adjacent or u=zu=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.