### 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 |
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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 |

### All Science Journal Classification (ASJC) codes

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

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## Cite this

*Journal of Combinatorial Optimization*,

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