Intersection properties of maximal directed cuts in digraphs

If DD is a finite digraph, a directed cut is a subset of arcs in DD having tail in some subset X⊆V(D)X⊆V(D) and head in V(D)∖XV(D)∖X. In this paper we prove two general results concerning intersections between maximal paths, cycles and maximal directed cuts in DD. As a direct consequence of these results, we deduce that there is a path, or a cycle, in DD that crosses each maximal directed cut.