A Min-Max Relation on Packing Feedback Vertex Sets

Abstract

Let G be a graph with a nonnegative integral function w defined on V(G). A family $\mathcal{F}$ of subsets of V(G) (repetition is allowed) is called a feedback vertex set packing in G if the removal of any member of $\mathcal{F}$ from G leaves a forest, and every vertex v∈ V(G) is contained in at most w(v) members of $\mathcal{F}$. 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.

Supported in part by: ¹The NSF of China under Grant No. 70221001 and 60373012, ²NSA grant H98230-05-1-0081, NSF grant ITR-0326387, and AFOSR grant: F49620-03-1-0239-0241, and ³The Research Grants Council of Hong Kong.