### Abstract

Let G = (V + s, E) be a graph with a designated vertex s of degree d(s), and let f(s) = (d_{1}, d_{2}, ..., d_{p}) be a partition of d(s) into p positive integers. An f(s)-detachment of G is a graph G′ obtained by "splitting" s into p vertices, called the pieces of s, such that the degrees of the pieces of s in G′ are given by f(s). Thus every edge sw ∈ E corresponds to an edge of G′ connecting some piece of s to w. We give necessary and sufficient conditions for the existence of an f(s)-detachment of G in which the local edge-connectivities between pairs of vertices in V satisfy prespecified lower bounds. Our result is a common generalization of a theorem of Mader on edge splittings preserving local edge-connectivities and a result of Fleiner on f(s)-detachments satisfying uniform lower bounds. It implies a conjecture of Fleiner on f(s)-detachments preserving local edge-connectivities. By using our characterization we extend a theorem of Frank on local edge-connectivity augmentation of graphs to the case when stars of given degrees are added, and we also solve the local edge-connectivity augmentation problem for 3-uniform hypergraphs.

Original language | English |
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Pages (from-to) | 72-87 |

Number of pages | 16 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 17 |

Issue number | 1 |

DOIs | |

Publication status | Published - Sep 1 2003 |

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### Keywords

- Detachments of graphs
- Edge splitting
- Edge-connectivity

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*17*(1), 72-87. https://doi.org/10.1137/S0895480199363933