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Feedback vertex sets in tournaments. (English) Zbl 1259.05070
Summary: 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 $$1.6740^{n}$$ minimal FVSs, and that there is an infinite family of tournaments, all having at least $$1.5448^{n}$$ minimal FVSs. This improves and extends the bounds of J. W. Moon [Proc. Edinb. Math. Soc., II. Ser. 17, 345–349 (1971; Zbl 0234.05108)]. 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.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments 05C85 Graph algorithms (graph-theoretic aspects) 05C30 Enumeration in graph theory 68P05 Data structures
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