  We characterise digraphs of directed treewidth one in terms of forbidden
butterfly minors. Moreover, we show that there is a linear relation between the
hypertree-width of the dual of the cycle hypergraph of D, i. e. the hypergraph
with vertices V (D) where every hyperedge corresponds to a directed cycle in D,
and the directed treewidth of D. Based on this we show that a digraph has
directed treewidth one if and only if its cycle hypergraph is a hypertree.
