Disjoint Paths in Acyclic Digraphs

Let D be an acyclic digraph with sources X = {x1, x2} and sinks Y = {y1, y2} such that every vertex u ∈ V(D)\(X ∪ Y) has in- and outdegree at least two. A theorem of Thomassen states that either D has a pair of disjoint directed paths (x1, y2,) and (x2, y1), or D has a planar representation with x1, x2, y2, y1 around the boundary of the outside face in this cyclic order. We generalize this theorem by relaxing the conditions on the vertices of sets X and Y as much as possible. For this generalization we present a simple, algorithmic proof.