### Abstract

This paper solves the problem of increasing the edge-connectivity of a bipartite digraph by adding the smallest number of new edges that preserve bipartiteness. A natural application arises when we wish to reinforce a 2-dimensional square grid framework with cables. We actually solve the more general problem of covering a crossing family of sets with the smallest number of directed edges, where each new edge must join the blocks of a given bipartition of the elements. The smallest number of new edges is given by a min-max formula that has six infinite families of exceptional cases. We discuss a problem on network flows whose solution has a similar formula with three infinite families of exceptional cases. We also discuss a problem on arborescences whose solution has five infinite families of exceptions. We give an algorithm that increases the edge-connectivity of a bipartite digraph in the same time as the best-known algorithm for the problem without the bipartite constraint: O (km log n) for unweighted digraphs and O(nm log(n^{2}/m)) for weighted digraphs, where n, m and k are the number of vertices and edges of the given graph and the target connectivity, respectively.

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

Pages (from-to) | 449-486 |

Number of pages | 38 |

Journal | Journal of Combinatorial Optimization |

Volume | 4 |

Issue number | 4 |

Publication status | Published - 2000 |

### Fingerprint

### Keywords

- Connectivity augmentation
- Crossing family
- Graph algorithms
- Min-max theorems
- Rigidity
- Square grid frame-work

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computer Science Applications
- Mathematics(all)
- Applied Mathematics
- Control and Optimization
- Discrete Mathematics and Combinatorics

### Cite this

*Journal of Combinatorial Optimization*,

*4*(4), 449-486.

**Incrementing Bipartite Digraph Edge-Connectivity.** / Gabow, Harold N.; Jordán, T.

Research output: Contribution to journal › Article

*Journal of Combinatorial Optimization*, vol. 4, no. 4, pp. 449-486.

}

TY - JOUR

T1 - Incrementing Bipartite Digraph Edge-Connectivity

AU - Gabow, Harold N.

AU - Jordán, T.

PY - 2000

Y1 - 2000

N2 - This paper solves the problem of increasing the edge-connectivity of a bipartite digraph by adding the smallest number of new edges that preserve bipartiteness. A natural application arises when we wish to reinforce a 2-dimensional square grid framework with cables. We actually solve the more general problem of covering a crossing family of sets with the smallest number of directed edges, where each new edge must join the blocks of a given bipartition of the elements. The smallest number of new edges is given by a min-max formula that has six infinite families of exceptional cases. We discuss a problem on network flows whose solution has a similar formula with three infinite families of exceptional cases. We also discuss a problem on arborescences whose solution has five infinite families of exceptions. We give an algorithm that increases the edge-connectivity of a bipartite digraph in the same time as the best-known algorithm for the problem without the bipartite constraint: O (km log n) for unweighted digraphs and O(nm log(n2/m)) for weighted digraphs, where n, m and k are the number of vertices and edges of the given graph and the target connectivity, respectively.

AB - This paper solves the problem of increasing the edge-connectivity of a bipartite digraph by adding the smallest number of new edges that preserve bipartiteness. A natural application arises when we wish to reinforce a 2-dimensional square grid framework with cables. We actually solve the more general problem of covering a crossing family of sets with the smallest number of directed edges, where each new edge must join the blocks of a given bipartition of the elements. The smallest number of new edges is given by a min-max formula that has six infinite families of exceptional cases. We discuss a problem on network flows whose solution has a similar formula with three infinite families of exceptional cases. We also discuss a problem on arborescences whose solution has five infinite families of exceptions. We give an algorithm that increases the edge-connectivity of a bipartite digraph in the same time as the best-known algorithm for the problem without the bipartite constraint: O (km log n) for unweighted digraphs and O(nm log(n2/m)) for weighted digraphs, where n, m and k are the number of vertices and edges of the given graph and the target connectivity, respectively.

KW - Connectivity augmentation

KW - Crossing family

KW - Graph algorithms

KW - Min-max theorems

KW - Rigidity

KW - Square grid frame-work

UR - http://www.scopus.com/inward/record.url?scp=0041686615&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041686615&partnerID=8YFLogxK

M3 - Article

VL - 4

SP - 449

EP - 486

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

IS - 4

ER -