  We examine $t$-colourings of oriented graphs in which, for a fixed integer $k
\geq 1$, vertices joined by a directed path of length at most $k$ must be
assigned different colours. A homomorphism model that extends the ideas of
Sherk for the case $k=2$ is described. Dichotomy theorems for the complexity of
the problem of deciding, for fixed $k$ and $t$, whether there exists such a
$t$-colouring are proved.
