Abstract. Given a graph with weights on its vertices, a subset of vertices is called a feedback vertex set if it intersects every cycle in the graph. The problem of finding a feedback vertex set with the minimum total weight is called the feedback vertex set problem, which is a well-known NP-hard problem and arises in a variety of applications. The purpose of this talk is to present a characterization of all graphs that enjoy a certain min-max relation on feedback vertex sets; the characterization leads to a polynomial-time solution of the feedback vertex set problem in this graph class.