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The dicycle transversal number t(D) of a digraph D is the minimum size of adicycle transversal of D, i. e. a set T of vertices of D such that D-T isacyclic. We study the following problem: Given a digraph D, decide if there isa dicycle B in D and a cycle C in the underlying undirected graph of D suchsuch that B,C are disjoint. It is known that there is a polynomial timealgorithm for this problem when restricted to strongly connected graphs, whichactually finds B,C if they exist. We generalize this to any class of digraphs Dwith either t(D) not equal to 1 or t(D)=1 and a bounded number of dicycletransversals, and show that the problem is NP-complete for a special class ofdigraphs D with t(D)=1 and, hence, in general.

Vertex-disjoint directed and undirected cycles in general digraphs