  Many of the tools developed for the theory of tree-decompositions of graphs
do not work for directed graphs. In this paper we show that some of the most
basic tools do work in the case where the model digraph is a directed path.
Using these tools we define a notion of a directed blockage in a digraph and
prove a min-max theorem for directed path-width analogous to the result of
Bienstock, Roberston, Seymour and Thomas for blockages in graphs. Furthermore,
we show that every digraph with directed path width $\geq k$ contains each
arboresence of order $\leq k + 1$ as a butterfly minor. Finally we also show
that every digraph admits a linked directed path-decomposition of minimum
width, extending a result of Kim and Seymour on semi-complete digraphs.
