### Abstract

As a vertex-disjoint analogue of Edmonds' arc-disjoint arborescences theorem, it was conjectured that given a directed graph D with a specified vertex r, there are k spanning arborescences rooted at r such that for every vertex v of D the k directed walks from r to v in these arborescences are internally vertex-disjoint if and only if for every vertex v of D there are k internally vertex-disjoint directed walks from r to v. Whitty (1987) [10] affirmatively settled this conjecture for k≤2, and Huck (1995) [6] constructed counterexamples for k≥3, and Huck (1999) [7] proved that the conjecture is true for every k when D is acyclic. In this paper, we generalize these results by using the concept of "convexity" which is introduced by Fujishige (2010) [4].

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

Pages (from-to) | 453-459 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 313 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2013 |

### Fingerprint

### Keywords

- Arborescences
- Convex sets
- Packing
- Vertex-disjoint paths

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*313*(4), 453-459. https://doi.org/10.1016/j.disc.2012.11.006

**Independent arborescences in directed graphs.** / Frank, A.; Fujishige, Satoru; Kamiyama, Naoyuki; Katoh, Naoki.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 313, no. 4, pp. 453-459. https://doi.org/10.1016/j.disc.2012.11.006

}

TY - JOUR

T1 - Independent arborescences in directed graphs

AU - Frank, A.

AU - Fujishige, Satoru

AU - Kamiyama, Naoyuki

AU - Katoh, Naoki

PY - 2013

Y1 - 2013

N2 - As a vertex-disjoint analogue of Edmonds' arc-disjoint arborescences theorem, it was conjectured that given a directed graph D with a specified vertex r, there are k spanning arborescences rooted at r such that for every vertex v of D the k directed walks from r to v in these arborescences are internally vertex-disjoint if and only if for every vertex v of D there are k internally vertex-disjoint directed walks from r to v. Whitty (1987) [10] affirmatively settled this conjecture for k≤2, and Huck (1995) [6] constructed counterexamples for k≥3, and Huck (1999) [7] proved that the conjecture is true for every k when D is acyclic. In this paper, we generalize these results by using the concept of "convexity" which is introduced by Fujishige (2010) [4].

AB - As a vertex-disjoint analogue of Edmonds' arc-disjoint arborescences theorem, it was conjectured that given a directed graph D with a specified vertex r, there are k spanning arborescences rooted at r such that for every vertex v of D the k directed walks from r to v in these arborescences are internally vertex-disjoint if and only if for every vertex v of D there are k internally vertex-disjoint directed walks from r to v. Whitty (1987) [10] affirmatively settled this conjecture for k≤2, and Huck (1995) [6] constructed counterexamples for k≥3, and Huck (1999) [7] proved that the conjecture is true for every k when D is acyclic. In this paper, we generalize these results by using the concept of "convexity" which is introduced by Fujishige (2010) [4].

KW - Arborescences

KW - Convex sets

KW - Packing

KW - Vertex-disjoint paths

UR - http://www.scopus.com/inward/record.url?scp=84870696714&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84870696714&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2012.11.006

DO - 10.1016/j.disc.2012.11.006

M3 - Article

AN - SCOPUS:84870696714

VL - 313

SP - 453

EP - 459

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 4

ER -