Bowler, NathanNathanBowlerGhorbani, EbrahimEbrahimGhorbaniGut, FlorianFlorianGutJacobs, Raphael W.Raphael W.JacobsReich, FlorianFlorianReich2026-10-192026-10-192026-07-07SIAM Journal on Discrete Mathematics 40 (3): 988-999 (2026)https://hdl.handle.net/11420/65354We prove that for every set S of vertices of a directed graph D, the maximum number of vertices in S contained in a collection of vertex-disjoint cycles in D is at least the minimum size of a set of vertices that hits all cycles containing a vertex of S. 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.en1095-714620263988999Society for Industrial & Applied Mathematics (SIAM)Cycle-widthCyclesDirected tree-widthHitting setsNatural Sciences and Mathematics::510: MathematicsJournal Article10.1137/24M1716604