### Abstract

A graph G=(V,E) is called minimally (k,T)-edge-connected with respect to some T⊆V if there exist k-edge-disjoint paths between every pair u,vT but this property fails by deleting any edge of G. We show that |V| can be bounded by a (linear) function of k and |T| if each vertex in V-T has odd degree. We prove similar bounds in the case when G is simple and k≤3. These results are applied to prove structural properties of optimal solutions of the shortest k-edge-connected Steiner network problem. We also prove lower bounds on the corresponding Steiner ratio.

Original language | English |
---|---|

Pages (from-to) | 421-432 |

Number of pages | 12 |

Journal | Discrete Applied Mathematics |

Volume | 131 |

Issue number | 2 |

DOIs | |

Publication status | Published - Sep 12 2003 |

### Fingerprint

### Keywords

- Edge-connectivity of graphs
- Steiner networks
- Steiner ratio

### ASJC Scopus subject areas

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

### Cite this

**On minimally k-edge-connected graphs and shortest k-edge-connected Steiner networks.** / Jordán, T.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol. 131, no. 2, pp. 421-432. https://doi.org/10.1016/S0166-218X(02)00465-1

}

TY - JOUR

T1 - On minimally k-edge-connected graphs and shortest k-edge-connected Steiner networks

AU - Jordán, T.

PY - 2003/9/12

Y1 - 2003/9/12

N2 - A graph G=(V,E) is called minimally (k,T)-edge-connected with respect to some T⊆V if there exist k-edge-disjoint paths between every pair u,vT but this property fails by deleting any edge of G. We show that |V| can be bounded by a (linear) function of k and |T| if each vertex in V-T has odd degree. We prove similar bounds in the case when G is simple and k≤3. These results are applied to prove structural properties of optimal solutions of the shortest k-edge-connected Steiner network problem. We also prove lower bounds on the corresponding Steiner ratio.

AB - A graph G=(V,E) is called minimally (k,T)-edge-connected with respect to some T⊆V if there exist k-edge-disjoint paths between every pair u,vT but this property fails by deleting any edge of G. We show that |V| can be bounded by a (linear) function of k and |T| if each vertex in V-T has odd degree. We prove similar bounds in the case when G is simple and k≤3. These results are applied to prove structural properties of optimal solutions of the shortest k-edge-connected Steiner network problem. We also prove lower bounds on the corresponding Steiner ratio.

KW - Edge-connectivity of graphs

KW - Steiner networks

KW - Steiner ratio

UR - http://www.scopus.com/inward/record.url?scp=0041828747&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041828747&partnerID=8YFLogxK

U2 - 10.1016/S0166-218X(02)00465-1

DO - 10.1016/S0166-218X(02)00465-1

M3 - Article

AN - SCOPUS:0041828747

VL - 131

SP - 421

EP - 432

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 2

ER -