Vertex-disjoint directed and undirected cycles in general digraphs

6 The dicycle transversal number τ(D) of a digraph D is the mini7 mum size of a dicycle transversal of D, i. e. a set T ⊆ V (D) such that 8 D − T is acyclic. We study the following problem: Given a digraph D, 9 decide if there is a dicycle B in D and a cycle C in its underlying undi10 rected graph UG(D) such that V (B)∩V (C) = ∅. It is known that there is 11 a polynomial time algorithm for this problem when restricted to strongly 12 connected graphs, which actually finds B,C if they exist. We generalize 13 this to any class of digraphs D with either τ(D) 6= 1 or τ(D) = 1 and 14 a bounded number of dicycle transversals, and show that the problem is 15 NP-complete for a special class of digraphs D with τ(D) = 1 and, hence, 16 in general. 17 AMS classification: 05c38, 05c20, 05c85. 18