### Abstract

Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are nonnegative integers. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such an "almost uniform" partition is called an (l,u)-partition. We deal with three problems to find an (l,u)-partition of a given graph; the minimum partition problem is to find an (l,u)-partition with the minimum number of components; the maximum partition problem is defined analogously; and the p-partition problem is to find an (l,u)-partition with a fixed number p of components. All these problems are NP-complete or NP-hard, respectively, even for series-parallel graphs. In this paper we show that both the minimum partition problem and the maximum partition problem can be solved in time O(u4n) and the p-partition problem can be solved in time O(p2u4n) for any series-parallel graph with n vertices. The algorithms can be extended for partial k-trees, that is, graphs with bounded tree-width.

Original language | English |
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Pages (from-to) | 142-154 |

Number of pages | 13 |

Journal | Journal of Discrete Algorithms |

Volume | 4 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 Mar 1 |

### Keywords

- (l,u)-partition
- Algorithm
- Lower bound
- Maximum partition problem
- Minimum partition problem
- Partial k-tree
- Series-parallel graph
- Upper bound

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Journal of Discrete Algorithms*,

*4*(1), 142-154. https://doi.org/10.1016/j.jda.2005.01.005