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. MSC 2000 subject classification. Primary: 90C10, 90C27, 90C57. OR/MS subject classification. Primary: Programming/graphs.