Complexity of the Feedback Vertex Set Problem in Tournaments with Forbidden Subtournaments

TL;DR

This paper investigates the computational complexity of the minimum feedback vertex set problem in tournaments with certain forbidden subtournaments, identifying cases where the problem is polynomial-time solvable and others where it remains NP-complete.

Contribution

It establishes the complexity status for MFBVS in tournaments with specific forbidden subtournaments and provides a necessary condition for polynomial solvability.

Findings

MFBVS is in P for W_5-free and U_5-free tournaments.

MFBVS remains NP-complete for T_5-free tournaments.

A necessary condition for polynomial solvability is identified, but it is not sufficient.

Abstract

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.