Disjoint directed and undirected paths and cycles in digraphs

We show that the following problem is NP-complete: Given a digraph D and distinct vertices s,t of D, decide whether the underlying graph of D contains two internally disjoint (s,t)-paths P and Q such that P is a directed (s,t)-path in D. Using this result we characterize those mixed linkage problems which are polynomially solvable (assuming P≠NP). This complements the classical dichotomy by Fortune, Hopcroft, and Wyllie classifying those directed linkage problems that are polynomially solvable. We furthermore show that, contrary to the case of directed linkages in digraphs, the mixed problem remains NP-complete for acyclic digraphs.