Vertex-disjoint directed and undirected cycles in general digraphs

The dicycle transversal number τ(D) of a digraph D is the minimum size of a dicycle transversal of D, i.e. a set T⊆. V(D) such that D T is acyclic. We study the 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)=θ. It is known that there is a polynomial time algorithm for this problem when restricted to strongly connected graphs, which actually finds B, C if they exist. We generalize this to any class of digraphs D with either τ(D). ≠. 1 or τ(D)=1 and a bounded number of dicycle transversals, and show that the problem is NP-complete for a special class of digraphs D with τ(D)=1 and, hence, in general.