Minimal Edge-Coverings of Pairs of Sets

Abstract

We derive a new min-max formula for the minimum number of new edges to be added to a given directed graph to make it k-node-connected. This gives rise to a polynomial time algorithm (via the ellipsoid method) to compute the augmenting edge set of minimum cardinality. (Such an algorithm or formula was previously known only for k = 1). Our main result is actually a new min-max theorem concerning “bisupermodular” functions on pairs of sets. This implies the node-connectivity augmentation theorem mentioned above as well as a generalization of an earlier result of the first author on the minimum number of new directed edges whose addition makes a digraph k-edge-connected. As further special cases of the main theorem, we derive an extension of (Lubiw’s extension of) Gyo{combining double acute accents theorem on intervals, Mader’s theorem on splitting off edges in directed graphs, and Edmonds’ theorem on matroid partitions. © 1995 by Academic Press, Inc.