Hitting cycles through prescribed vertices or edges

We 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.

Content

Subjects

Cycle-width

Cycles

Directed tree-width

Hitting sets

DDC Class

510: Mathematics