## Abstract

Given a digraph D, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in D an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree zero. Gutin, Razgon and Kim [G. Gutin, I. Razgon, E.J. Kim, Minimum leaf out-branching problems, in: Proc. 4th International Conference on Algorithmic Aspects in Information and Management, AAIM'08, in: Lect. Notes Comput. Sci., vol. 5034 2008, pp. 235-246] proved that MinLOB is polynomial time solvable for acyclic digraphs which are exactly the digraphs of directed path-width (DAG-width, directed tree-width, respectively) 0. We investigate how much one can extend this polynomiality result. We prove that already for digraphs of directed path-width (directed tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand, we show that for digraphs of restricted directed tree-width (directed path-width, DAG-width, respectively) and a fixed integer k, the problem of checking whether there is an out-branching with at most k leaves is polynomial time solvable.

Original language | English |
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Pages (from-to) | 3000-3004 |

Number of pages | 5 |

Journal | Discrete Applied Mathematics |

Volume | 157 |

Issue number | 13 |

DOIs | |

Publication status | Published - 6 Jul 2009 |

Externally published | Yes |

## Keywords

- Computational complexity
- DAG-width
- Directed tree-width
- Leaves
- Out-branchings

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics