The structure of strong $k$-quasi-transitive digraphs with large diameters

Let $k$ be an integer with $k\ge 2$. A digraph $D$ is $k$-quasi-transitive, if for any path $x_0{x}_1\dots x_k$ of length $k$, $x_0$ and $x_k$ are adjacent. Suppose that there exists a path of length at least $k+2$ in $D$. Let $P$ be a shortest path of length $k+2$ in $D$. Wang and Zhang [Hamiltonian paths in $k$-quasi-transitive digraphs, Discrete Mathematics, 339(8) (2016) 2094–2099] proved that if $k$ is even and $k\ge 4$, then $D[V(P)]$ and $D[V(D)\setminus V(P)]$arebothsemicompletedigraphs\mathrm{.}Inthispaper,weshallprovethatif$k$isoddand$k\ge 5$,then$D[V(P)]$ is either a semicomplete digraph or a semicomplete bipartite digraph and $D[V(D)\setminus V(P)]$ is either a semicomplete digraph, a semicomplete bipartite digraph or an empty digraph.