  In this article we consider the Directed Steiner Path Cover problem on
directed co-graphs. Given a directed graph G=(V,E) and a subset T of V of
so-called terminal vertices, the problem is to find a minimum number of
vertex-disjoint simple directed paths, which contain all terminal vertices and
a minimum number of non-terminal vertices (Steiner vertices). The primary
minimization criteria is the number of paths. We show how to compute in linear
time a minimum Steiner path cover for directed co-graphs. This leads to a
linear time computation of an optimal directed Steiner path on directed
co-graphs, if it exists. Since the Steiner path problem generalizes the
Hamiltonian path problem, our results imply the first linear time algorithm for
the directed Hamiltonian path problem on directed co-graphs. We also give
binary integer programs for the (directed) Hamiltonian path problem, for the
(directed) Steiner path problem, and for the (directed) Steiner path cover
problem. These integer programs can be used to minimize change-over times in
pick-and-place machines used by companies in electronic industry.
