### Abstract

Given a directed graph D = (V,A) with a set of d specified vertices S = {s_{1},...,s _{d} }⊆ V and a function f: S → ℤ_{+} where ℤ_{+} denotes the set of non-negative integers, we consider the problem which asks whether there exist ∑_{i=1}^{d} f (s_{i}) 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}, each T_{i,j} spans vertices from which s _{i} is reachable and the union of all arc sets of T_{i,j} for i = 1,...,d and j = 1,...,f(s_{i} ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in ∑_{i=1}^{d} f (s_{i}) and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.

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

Title of host publication | Computing and Combinatorics - 14th Annual International Conference, COCOON 2008, Proceedings |

Pages | 444-457 |

Number of pages | 14 |

DOIs | |

Publication status | Published - Aug 4 2008 |

Event | 14th Annual International Conference on Computing and Combinatorics, COCOON 2008 - Dalian, China Duration: Jun 27 2008 → Jun 29 2008 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 5092 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 14th Annual International Conference on Computing and Combinatorics, COCOON 2008 |
---|---|

Country | China |

City | Dalian |

Period | 6/27/08 → 6/29/08 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Computing and Combinatorics - 14th Annual International Conference, COCOON 2008, Proceedings*(pp. 444-457). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5092 LNCS). https://doi.org/10.1007/978-3-540-69733-6_44

**Covering directed graphs by in-trees.** / Kamiyama, Naoyuki; Katoh, Naoki.

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

*Computing and Combinatorics - 14th Annual International Conference, COCOON 2008, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 5092 LNCS, pp. 444-457, 14th Annual International Conference on Computing and Combinatorics, COCOON 2008, Dalian, China, 6/27/08. https://doi.org/10.1007/978-3-540-69733-6_44

}

TY - GEN

T1 - Covering directed graphs by in-trees

AU - Kamiyama, Naoyuki

AU - Katoh, Naoki

PY - 2008/8/4

Y1 - 2008/8/4

N2 - Given a directed graph D = (V,A) with a set of d specified vertices S = {s1,...,s d }⊆ V and a function f: S → ℤ+ where ℤ+ denotes the set of non-negative integers, we consider the problem which asks whether there exist ∑i=1d f (si) in-trees denoted by T i, 1, Ti, 2, ..., Ti f(si) for every i = 1,...,d such that Ti, 1,...,Ti, f(si) are rooted at s i, each Ti,j spans vertices from which s i is reachable and the union of all arc sets of Ti,j for i = 1,...,d and j = 1,...,f(si ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in ∑i=1d f (si) and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.

AB - Given a directed graph D = (V,A) with a set of d specified vertices S = {s1,...,s d }⊆ V and a function f: S → ℤ+ where ℤ+ denotes the set of non-negative integers, we consider the problem which asks whether there exist ∑i=1d f (si) in-trees denoted by T i, 1, Ti, 2, ..., Ti f(si) for every i = 1,...,d such that Ti, 1,...,Ti, f(si) are rooted at s i, each Ti,j spans vertices from which s i is reachable and the union of all arc sets of Ti,j for i = 1,...,d and j = 1,...,f(si ) covers A. In this paper, we prove that such set of in-trees covering A can be found by using an algorithm for the weighted matroid intersection problem in time bounded by a polynomial in ∑i=1d f (si) and the size of D. Furthermore, for the case where D is acyclic, we present another characterization of the existence of in-trees covering A, and then we prove that in-trees covering A can be computed more efficiently than the general case by finding maximum matchings in a series of bipartite graphs.

UR - http://www.scopus.com/inward/record.url?scp=48249139377&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=48249139377&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-69733-6_44

DO - 10.1007/978-3-540-69733-6_44

M3 - Conference contribution

AN - SCOPUS:48249139377

SN - 3540697322

SN - 9783540697329

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 444

EP - 457

BT - Computing and Combinatorics - 14th Annual International Conference, COCOON 2008, Proceedings

ER -