### Abstract

A subset J of edges of a connected undirected graph G=(V, E) is called a join if |C∩J|≤|C|/2 for every circuit C of G. Answering a question of P. Solé and Th. Zaslavsky, we derive a min-max formula for the maximum cardinality μ of a joint of G. Namely, μ=(φ+|V|-1)/2 where φ denotes the minimum number of edges whose contraction leaves a factor-critical graph. To study these parameters we introduce a new decomposition of G, interesting for its own sake, whose building blocks are factor-critical graphs and matching-covered bipartite graphs. We prove that the length of such a decomposition is always φ and show how an optimal join can be constructed as the union of perfect matchings in the building blocks. The proof relies on the Gallai-Edmonds structure theorem and gives rise to a polynomial time algorithm to construct the optima in question.

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

Pages (from-to) | 65-81 |

Number of pages | 17 |

Journal | Combinatorica |

Volume | 13 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1993 |

### Fingerprint

### Keywords

- AMS subject classification code (1991): 05C70, 05C75, 94B60

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)

### Cite this

**Conservative weightings and ear-decompositions of graphs.** / Frank, A.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 13, no. 1, pp. 65-81. https://doi.org/10.1007/BF01202790

}

TY - JOUR

T1 - Conservative weightings and ear-decompositions of graphs

AU - Frank, A.

PY - 1993/3

Y1 - 1993/3

N2 - A subset J of edges of a connected undirected graph G=(V, E) is called a join if |C∩J|≤|C|/2 for every circuit C of G. Answering a question of P. Solé and Th. Zaslavsky, we derive a min-max formula for the maximum cardinality μ of a joint of G. Namely, μ=(φ+|V|-1)/2 where φ denotes the minimum number of edges whose contraction leaves a factor-critical graph. To study these parameters we introduce a new decomposition of G, interesting for its own sake, whose building blocks are factor-critical graphs and matching-covered bipartite graphs. We prove that the length of such a decomposition is always φ and show how an optimal join can be constructed as the union of perfect matchings in the building blocks. The proof relies on the Gallai-Edmonds structure theorem and gives rise to a polynomial time algorithm to construct the optima in question.

AB - A subset J of edges of a connected undirected graph G=(V, E) is called a join if |C∩J|≤|C|/2 for every circuit C of G. Answering a question of P. Solé and Th. Zaslavsky, we derive a min-max formula for the maximum cardinality μ of a joint of G. Namely, μ=(φ+|V|-1)/2 where φ denotes the minimum number of edges whose contraction leaves a factor-critical graph. To study these parameters we introduce a new decomposition of G, interesting for its own sake, whose building blocks are factor-critical graphs and matching-covered bipartite graphs. We prove that the length of such a decomposition is always φ and show how an optimal join can be constructed as the union of perfect matchings in the building blocks. The proof relies on the Gallai-Edmonds structure theorem and gives rise to a polynomial time algorithm to construct the optima in question.

KW - AMS subject classification code (1991): 05C70, 05C75, 94B60

UR - http://www.scopus.com/inward/record.url?scp=0013438007&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0013438007&partnerID=8YFLogxK

U2 - 10.1007/BF01202790

DO - 10.1007/BF01202790

M3 - Article

AN - SCOPUS:0013438007

VL - 13

SP - 65

EP - 81

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 1

ER -