k-quasi-transitive digraphs of large diameter

A recursive characterization (the so-called Canonical Decomposition) is known for quasi-transitive digraphs, but no characterization is known for k-quasi-transitive digraphs in the general case. Recently, Wang and Zhang proved that if k is an even integer, then a k-quasi-transitive digraph of diameter at least k+2 admits a partition of its vertex set into two parts, each of them inducing a semicomplete digraph. In this work, we will present an analogous result for the case when k is an odd integer and discuss some of its consequences and future lines of research.