### Abstract

Let G=(V+s,E) be a digraph with ρ(s)=δ(s) which is k-edge-connected in V. Mader (European J. Combin. 3 (1982) 63) proved that there exists a pair vs,st of edges which can be "split off", that is, which can be replaced by a new edge vt, preserving k-edge-connectivity in V. Such a pair is called admissible. We extend this theorem by showing that for more than ρ(s)/2 edges vs there exist at least ρ(s)/2 edges st such that vs and st form an admissible pair. We apply this result to the problem of splitting off edges in G when a prespecified bipartition V=A∪B is also given and no edge can be split off with both endvertices in A or both in B. We prove that an admissible pair satisfying the bipartition constraints exists if ρ(s)≥2k+1. Based on this result we develop a polynomial algorithm which gives an almost optimal solution to the bipartition constrained edge-connectivity augmentation problem. In this problem we are given a directed graph H=(V,E), a bipartition V=A∪B and a positive integer k; the goal is to find a smallest set F of edges for which H′=(V,E∪F) is k-edge-connected and no edge of the augmenting set F has both endvertices in A or both in B. Our algorithm adds at most k edges more than the optimum.

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

Pages (from-to) | 49-62 |

Number of pages | 14 |

Journal | Discrete Applied Mathematics |

Volume | 115 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Nov 15 2001 |

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### ASJC Scopus subject areas

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

### Cite this

*Discrete Applied Mathematics*,

*115*(1-3), 49-62. https://doi.org/10.1016/S0166-218X(01)00214-1

**Bipartition constrained edge-splitting in directed graphs.** / Gabow, Harold N.; Jordán, T.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol. 115, no. 1-3, pp. 49-62. https://doi.org/10.1016/S0166-218X(01)00214-1

}

TY - JOUR

T1 - Bipartition constrained edge-splitting in directed graphs

AU - Gabow, Harold N.

AU - Jordán, T.

PY - 2001/11/15

Y1 - 2001/11/15

N2 - Let G=(V+s,E) be a digraph with ρ(s)=δ(s) which is k-edge-connected in V. Mader (European J. Combin. 3 (1982) 63) proved that there exists a pair vs,st of edges which can be "split off", that is, which can be replaced by a new edge vt, preserving k-edge-connectivity in V. Such a pair is called admissible. We extend this theorem by showing that for more than ρ(s)/2 edges vs there exist at least ρ(s)/2 edges st such that vs and st form an admissible pair. We apply this result to the problem of splitting off edges in G when a prespecified bipartition V=A∪B is also given and no edge can be split off with both endvertices in A or both in B. We prove that an admissible pair satisfying the bipartition constraints exists if ρ(s)≥2k+1. Based on this result we develop a polynomial algorithm which gives an almost optimal solution to the bipartition constrained edge-connectivity augmentation problem. In this problem we are given a directed graph H=(V,E), a bipartition V=A∪B and a positive integer k; the goal is to find a smallest set F of edges for which H′=(V,E∪F) is k-edge-connected and no edge of the augmenting set F has both endvertices in A or both in B. Our algorithm adds at most k edges more than the optimum.

AB - Let G=(V+s,E) be a digraph with ρ(s)=δ(s) which is k-edge-connected in V. Mader (European J. Combin. 3 (1982) 63) proved that there exists a pair vs,st of edges which can be "split off", that is, which can be replaced by a new edge vt, preserving k-edge-connectivity in V. Such a pair is called admissible. We extend this theorem by showing that for more than ρ(s)/2 edges vs there exist at least ρ(s)/2 edges st such that vs and st form an admissible pair. We apply this result to the problem of splitting off edges in G when a prespecified bipartition V=A∪B is also given and no edge can be split off with both endvertices in A or both in B. We prove that an admissible pair satisfying the bipartition constraints exists if ρ(s)≥2k+1. Based on this result we develop a polynomial algorithm which gives an almost optimal solution to the bipartition constrained edge-connectivity augmentation problem. In this problem we are given a directed graph H=(V,E), a bipartition V=A∪B and a positive integer k; the goal is to find a smallest set F of edges for which H′=(V,E∪F) is k-edge-connected and no edge of the augmenting set F has both endvertices in A or both in B. Our algorithm adds at most k edges more than the optimum.

UR - http://www.scopus.com/inward/record.url?scp=0345821674&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0345821674&partnerID=8YFLogxK

U2 - 10.1016/S0166-218X(01)00214-1

DO - 10.1016/S0166-218X(01)00214-1

M3 - Article

AN - SCOPUS:0345821674

VL - 115

SP - 49

EP - 62

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 1-3

ER -