  Let $k$ be an integer with $k\geq 2$. A digraph $D$ is $k$-quasi-transitive,
if for any path $x_0x_1\ldots x_k$ 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 $D[V(D)\setminus
V(P)]$ are both semicomplete digraphs. In this paper, we shall prove that if
$k$ is odd and $k\ge 5$, 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.
