# Minimum leaf out-branching and related problems

Gutin, G. and Razgon, Igor and Kim, E.J.
(2009)
Minimum leaf out-branching and related problems.
*Theoretical Computer Science* 410
(45),
pp. 4571-4579.
ISSN 0304-3975.

## 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 0. We prove that MinLOB is polynomial-time solvable for acyclic digraphs. In general, MinLOB is NP-hard and we consider three parameterizations of MinLOB. We prove that two of them are NP-complete for every value of the parameter, but the third one is fixed-parameter tractable (FPT). The FPT parameterization is as follows: given a digraph D of order n and a positive integral parameter k, check whether D contains an out-branching with at most n−k leaves (and find such an out-branching if it exists). We find a problem kernel of order O(k2) and construct an algorithm of running time O(2O(klogk)+n6), which is an ‘additive’ FPT algorithm. We also consider transformations from two related problems, the minimum path covering and the maximum internal out-tree problems into MinLOB, which imply that some parameterizations of the two problems are FPT as well.

## Metadata

Item Type: | Article |
---|---|

Keyword(s) / Subject(s): | directed graphs, out-branchings, minimum number of leaves, fixed-parameter tractable, acyclic directed graphs |

School: | School of Business, Economics & Informatics > Computer Science and Information Systems |

Depositing User: | Sarah Hall |

Date Deposited: | 01 Aug 2013 15:54 |

Last Modified: | 01 Aug 2013 15:54 |

URI: | https://eprints.bbk.ac.uk/id/eprint/7927 |

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