Disjoint directed and undirected paths and cycles in digraphs

Authors:
Jørgen Bang-Jensen;Matthias Kriesell
Affiliations:
Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark;Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
Venue:
Theoretical Computer Science
Year:
2009

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Abstract

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 PNP). 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.