## Abstract

Let G be a graph which is k-outconnected from a specified root node r, that is, G has k openly disjoint paths between r and v for every node v. We give necessary and sufficient conditions for the existence of a pair rv, rw of edges for which replacing these edges by a new edge vw gives a graph that is k-outconnected from r. This generalizes a theorem of Bienstock et al. on splitting off edges while preserving k-node-connectivity. We also prove that if C is a cycle in G such that each edge in C is critical with respect to k-outconnectivity from r, then C has a node v, distinct from r, which has degree k. This result is the rooted counterpart of a theorem due to Mader. We apply the above results to design approximation algorithms for the following problem: given a graph with nonnegative edge weights and node requirements c_{u} for each node u, find a minimum-weight subgraph that contains max(c_{u}, c_{v}) openly disjoint paths between every pair of nodes u, v. For metric weights, our approximation guarantee is 3. For uniform weights, our approximation guarantee is min[2, (k + 2q-1)/k]. Here k is the maximum node requirement, and q is the number of positive node requirements.

Original language | English |
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Pages (from-to) | 353-375 |

Number of pages | 23 |

Journal | Algorithmica |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2001 |

## Keywords

- Approximation algorithms
- Graph connectivity
- K-Outconnectivity
- Metric costs
- NP-hard problems
- Splitting-off theorems
- Uniform costs
- k-Connectivity