### 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→ℕ where ℕ denotes the set of positive integers, we consider the problem which asks whether there exist Σ_{i=1} ^{d} f(s _{i} ) in-trees denoted by T_{i,1}, T_{i,2},...,Ti,f(s_{i}) for every i=1,...,d such that T_{i,1},...,Ti,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 Σ _{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 |
---|---|

Pages (from-to) | 2-18 |

Number of pages | 17 |

Journal | Journal of Combinatorial Optimization |

Volume | 21 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2011 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Journal of Combinatorial Optimization*,

*21*(1), 2-18. https://doi.org/10.1007/s10878-009-9242-9

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

Research output: Contribution to journal › Article

*Journal of Combinatorial Optimization*, vol. 21, no. 1, pp. 2-18. https://doi.org/10.1007/s10878-009-9242-9

}

TY - JOUR

T1 - Covering directed graphs by in-trees

AU - Kamiyama, Naoyuki

AU - Katoh, Naoki

PY - 2011/1/1

Y1 - 2011/1/1

N2 - 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→ℕ where ℕ denotes the set of positive integers, we consider the problem which asks whether there exist Σi=1 d f(s i ) in-trees denoted by Ti,1, Ti,2,...,Ti,f(si) for every i=1,...,d such that Ti,1,...,Ti,f(si) 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 Σ 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.

AB - 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→ℕ where ℕ denotes the set of positive integers, we consider the problem which asks whether there exist Σi=1 d f(s i ) in-trees denoted by Ti,1, Ti,2,...,Ti,f(si) for every i=1,...,d such that Ti,1,...,Ti,f(si) 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 Σ 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.

UR - http://www.scopus.com/inward/record.url?scp=79951809774&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79951809774&partnerID=8YFLogxK

U2 - 10.1007/s10878-009-9242-9

DO - 10.1007/s10878-009-9242-9

M3 - Article

AN - SCOPUS:79951809774

VL - 21

SP - 2

EP - 18

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

IS - 1

ER -