### Abstract

Let G = (V,E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A⊆V. An A-path is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P. (If Γ is not abelian, we sum the labels in their order along the path.) We are interested in the maximum number of vertex-disjoint A-paths each of non-zero weight. When A = V this problem is equivalent to the maximum matching problem. The general case also includes Mader's S-paths problem. We prove that for any positive integer k, either there are k vertex-disjoint A-paths each of non-zero weight, or there is a set of at most 2k -2 vertices that meets each of the non-zero A-paths. This result is obtained as a consequence of an exact min-max theorem.

Original language | English (US) |
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Pages (from-to) | 521-532 |

Number of pages | 12 |

Journal | Combinatorica |

Volume | 26 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2006 |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

*Combinatorica*,

*26*(5), 521-532. https://doi.org/10.1007/s00493-006-0030-1