Hitting Cycles through Prescribed Vertices or Edges

We prove that for every set [Formula: see text] of vertices of a directed graph [Formula: see text], the maximum number of vertices in [Formula: see text] contained in a collection of vertex-disjoint cycles in [Formula: see text] is at least the minimum size of a set of vertices that hits all cycles containing a vertex of [Formula: see text]. As a consequence, the directed tree-width of a directed graph is linearly bounded in its cycle-width, which improves the previously known quadratic upper bound. We further show that the corresponding statement in bidirected graphs is true and that its edge-variant holds in both undirected and directed graphs, but fails in bidirected graphs. The vertex-version in undirected graphs remains an open problem.