Arc-disjoint directed and undirected cycles in digraphs

The dicycle transversal number τ(D) of a digraph D is the minimum size of a dicycle transversal of D, that is a set of vertices of D, whose removal from D makes it acyclic. An arc a of a digraph D with at least one cycle is a transversal arc if a is in every directed cycle of D (making D-a acyclic). In [3] and [4], we completely characterized the complexity of following problem: Given a digraph D, decide if there is a dicycle B in D and a cycle C in its underlying undirected graph UG(D) such that V(B)∩V(C)=Ø. It turns out that the problem is polynomially solvable for digraphs with a constantly bounded number of transversal vertices (including cases where τ(D)≥2). In the remaining case (allowing arbitrarily many transversal vertices) the problem is NP-complete. In this article, we classify the complexity of the arc-analog of this problem, where we ask for a dicycle B and a cycle C that are arc-disjoint, but not necessarily vertex-disjoint. We prove that the problem is polynomially solvable for strong digraphs and for digraphs with a constantly bounded number of transversal arcs (but possibly an unbounded number of transversal vertices). In the remaining case (allowing arbitrarily many transversal arcs) the problem is NP-complete.