  In the present paper, we study algorithmic questions for the arc-intersection
graph of directed paths on a tree. Such graphs are known to be perfect (proved
by Monma and Wei in 1986). We present faster algorithms than all previously
known algorithms for solving the minimum coloring and the minimum clique cover
problems. They both run in $O(np)$ time, where $n$ is the number of vertices of
the tree and $p$ the number of paths. Another result is a polynomial algorithm
computing a kernel in the intersection graph, when its edges are oriented in a
clique-acyclic way. Indeed, such a kernel exists for any perfect graph by a
theorem of Boros and Gurvich. Such algorithms computing kernels are known only
for few classes of perfect graphs.
