A min-max relation on packing feedback vertex sets

Let C be a graph with a nonnegative integral function w defined on V(G). A family F of subsets of V(G) (repetition is allowed) is called a feedback vertex set Packing in G if the removal of any member of F from G leaves a forest, and every vertex v is an element of V(G) is contained in at most w(v) members of T. The weight of a cycle C in G is the sum of w(v), over all vertices v of C. In this paper we characterize all graphs with the property that, for any nonnegative integral function w, the maximum cardinality of a feedback vertex set packing is equal to the minimum weight of a cycle.