### Abstract

Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to find a smallest set F of new edges for which G + F is k-vertex-connected. Polynomial algorithms for this problem have been found only for k ≤ 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general. In this paper, we develop an algorithm which delivers an optimal solution in polynomial time for every fixed k. In the case when the size of an optimal solution is large compared to k, our algorithm is polynomial for all k. We also derive a min-max formula for the size of a smallest augmenting set in this case. A key step in our proofs is a complete solution of the augmentation problem for a new family of graphs which we call k-independence free graphs. We also prove new splitting off theorems for vertex connectivity.

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

Pages (from-to) | 31-77 |

Number of pages | 47 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 94 |

Issue number | 1 |

DOIs | |

Publication status | Published - May 2005 |

### Fingerprint

### Keywords

- Algorithms
- Connectivity of graphs
- Vertex-connectivity augmentation

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

**Independence free graphs and vertex connectivity augmentation.** / Jackson, Bill; Jordán, T.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 94, no. 1, pp. 31-77. https://doi.org/10.1016/j.jctb.2004.01.004

}

TY - JOUR

T1 - Independence free graphs and vertex connectivity augmentation

AU - Jackson, Bill

AU - Jordán, T.

PY - 2005/5

Y1 - 2005/5

N2 - Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to find a smallest set F of new edges for which G + F is k-vertex-connected. Polynomial algorithms for this problem have been found only for k ≤ 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general. In this paper, we develop an algorithm which delivers an optimal solution in polynomial time for every fixed k. In the case when the size of an optimal solution is large compared to k, our algorithm is polynomial for all k. We also derive a min-max formula for the size of a smallest augmenting set in this case. A key step in our proofs is a complete solution of the augmentation problem for a new family of graphs which we call k-independence free graphs. We also prove new splitting off theorems for vertex connectivity.

AB - Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to find a smallest set F of new edges for which G + F is k-vertex-connected. Polynomial algorithms for this problem have been found only for k ≤ 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general. In this paper, we develop an algorithm which delivers an optimal solution in polynomial time for every fixed k. In the case when the size of an optimal solution is large compared to k, our algorithm is polynomial for all k. We also derive a min-max formula for the size of a smallest augmenting set in this case. A key step in our proofs is a complete solution of the augmentation problem for a new family of graphs which we call k-independence free graphs. We also prove new splitting off theorems for vertex connectivity.

KW - Algorithms

KW - Connectivity of graphs

KW - Vertex-connectivity augmentation

UR - http://www.scopus.com/inward/record.url?scp=15444372084&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=15444372084&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2004.01.004

DO - 10.1016/j.jctb.2004.01.004

M3 - Article

AN - SCOPUS:15444372084

VL - 94

SP - 31

EP - 77

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -