A min‐max relation on packing feedback vertex sets

Let G be a graph with a nonnegative integral function w defined on V(G). A collection 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 ∈ V(G) is contained in at most w(v) members of ℱ. The weight of a cycle C in G is the sum of w(v), over all vertices v of C. The purpose of this paper is to 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.