### 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.

Original language | English |
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Pages (from-to) | 73-110 |

Number of pages | 38 |

Journal | Journal of Combinatorial Theory, Series B |

Volume | 65 |

Issue number | 1 |

DOIs | |

Publication status | Published - Sep 1995 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics