Part of this research has been supported by the Netherlands Organisation for Scientific Research (NWO), grant 639.033.403. A preliminary version of this article appeared in the Proceedings of ESA 2010 ‡.
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Original Article
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Feedback Vertex Sets in Tournaments†
Article first published online: 26 JUN 2012
DOI: 10.1002/jgt.21631
© 2012 Wiley Periodicals, Inc.
Additional Information
How to Cite
Gaspers, S. and Mnich, M. (2013), Feedback Vertex Sets in Tournaments. J. Graph Theory, 72: 72–89. doi: 10.1002/jgt.21631
- †
Publication History
- Issue published online: 23 OCT 2012
- Article first published online: 26 JUN 2012
- Manuscript Revised: 19 OCT 2011
- Manuscript Received: 15 AUG 2010
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Keywords:
- algorithms and data structures;
- tournaments;
- feedback vertex set;
- polynomial delay;
- combinatorial bounds
Abstract
We study combinatorial and algorithmic questions around minimal feedback vertex sets (FVS) in tournament graphs. On the combinatorial side, we derive upper and lower bounds on the maximum number of minimal FVSs in an n-vertex tournament. We prove that every tournament on n vertices has at most 1.6740n minimal FVSs, and that there is an infinite family of tournaments, all having at least 1.5448n minimal FVSs. This improves and extends the bounds of Moon (1971). On the algorithmic side, we design the first polynomial space algorithm that enumerates the minimal FVSs of a tournament with polynomial delay. The combination of our results yields the fastest known algorithm for finding a minimum-sized FVS in a tournament.