Abstract: | We study feedback vertex sets (FVS) in tournaments, which are orientations of complete graphs. As our main result, we show that any tournament on n nodes has at most 1.5949n minimal FVS. This significantly improves the previously best upper bound of 1.6667n by Fomin et al. STOC 2016] and 1.6740n by Gaspers and Mnich J. Graph Theory 72 (1):72–89, 2013]. Our new upper bound almost matches the best‐known lower bound of , due to Gaspers and Mnich. Our proof is algorithmic, and shows that all minimal FVS of tournaments can be enumerated in time . |