1. Hitting cycles through prescribed vertices or edges
- Author
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Bowler, Nathan, Ghorbani, Ebrahim, Gut, Florian, Jacobs, Raphael W., and Reich, Florian
- Subjects
Mathematics - Combinatorics ,05C38 - Abstract
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., Comment: 12 pages, 2 figures
- Published
- 2024