### Abstract

Given a directed graph D = (V, A) and a set of specified vertices S = {s_{1},... , S_{d}} ⊆ V with |S| = d and a function f: S → ℕ where ℕ denotes the set of natural numbers, we present a necessary and sufficient condition that there exist Σ_{si∈S} f(s_{i}) arc-disjoint in-trees denoted by T_{i,1},T _{i,2},...,T_{i,f}(s_{i}) for every i = 1,...,d such that T_{i,1},..., T_{i,f}(s_{i}) are rooted at s _{i} and each T_{i,j} spans vertices from which s_{i} is reachable. This generalizes the result of Edmonds [2], i.e., the necessary and sufficient condition that for a directed graph D = (V, A) with a specified vertex s ∈ V, there are k arc-disjoint in-trees rooted at s each of which spans V. Furthermore, we extend another characterization of packing in-trees of Edmonds [1] to the one in our case.

Original language | English |
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Title of host publication | Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms |

Pages | 518-526 |

Number of pages | 9 |

Publication status | Published - Dec 1 2008 |

Event | 19th Annual ACM-SIAM Symposium on Discrete Algorithms - San Francisco, CA, United States Duration: Jan 20 2008 → Jan 22 2008 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Other

Other | 19th Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country | United States |

City | San Francisco, CA |

Period | 1/20/08 → 1/22/08 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms*(pp. 518-526). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).

**Arc-disjoint in-trees in directed graphs.** / Kamiyama, Naoyuki; Katoh, Naoki; Takizawa, Atsushi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms.*Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 518-526, 19th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, United States, 1/20/08.

}

TY - GEN

T1 - Arc-disjoint in-trees in directed graphs

AU - Kamiyama, Naoyuki

AU - Katoh, Naoki

AU - Takizawa, Atsushi

PY - 2008/12/1

Y1 - 2008/12/1

N2 - Given a directed graph D = (V, A) and a set of specified vertices S = {s1,... , Sd} ⊆ V with |S| = d and a function f: S → ℕ where ℕ denotes the set of natural numbers, we present a necessary and sufficient condition that there exist Σsi∈S f(si) arc-disjoint in-trees denoted by Ti,1,T i,2,...,Ti,f(si) for every i = 1,...,d such that Ti,1,..., Ti,f(si) are rooted at s i and each Ti,j spans vertices from which si is reachable. This generalizes the result of Edmonds [2], i.e., the necessary and sufficient condition that for a directed graph D = (V, A) with a specified vertex s ∈ V, there are k arc-disjoint in-trees rooted at s each of which spans V. Furthermore, we extend another characterization of packing in-trees of Edmonds [1] to the one in our case.

AB - Given a directed graph D = (V, A) and a set of specified vertices S = {s1,... , Sd} ⊆ V with |S| = d and a function f: S → ℕ where ℕ denotes the set of natural numbers, we present a necessary and sufficient condition that there exist Σsi∈S f(si) arc-disjoint in-trees denoted by Ti,1,T i,2,...,Ti,f(si) for every i = 1,...,d such that Ti,1,..., Ti,f(si) are rooted at s i and each Ti,j spans vertices from which si is reachable. This generalizes the result of Edmonds [2], i.e., the necessary and sufficient condition that for a directed graph D = (V, A) with a specified vertex s ∈ V, there are k arc-disjoint in-trees rooted at s each of which spans V. Furthermore, we extend another characterization of packing in-trees of Edmonds [1] to the one in our case.

UR - http://www.scopus.com/inward/record.url?scp=48249147916&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=48249147916&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:48249147916

SN - 9780898716474

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 518

EP - 526

BT - Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms

ER -