In this paper, we consider the complexity of the minimum feedback vertex set problem (MFBVS) for tournaments with forbidden subtournaments. The MFBVS problem in general tournaments is known to be NP-complete. We prove that the MFBVS problem for $W_5$-free and $U_5$-free tournaments is in P, and for $T_5$-free tournaments it remains NP-complete. Moreover, we prove a necessary condition for all $H$ such that the MFBVS problem for $H$-free tournaments is in P. We also show that the necessary condition is not sufficient.

翻译：本文研究了禁止特定子竞赛图的最小反馈顶点集问题的计算复杂度。在一般竞赛图中，该问题已知为NP完全问题。我们证明了在$W_5$-free和$U_5$-free竞赛图中，最小反馈顶点集问题属于P类；而在$T_5$-free竞赛图中，该问题仍保持NP完全性。此外，我们推导出所有满足$H$-free竞赛图中最小反馈顶点集问题属于P类的图$H$所需具备的必要条件，并证明该条件并非充分条件。