### Abstract

The SATISFACTORY PARTITION problem consists in deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler [Partitioning a graph to satisfy all vertices, Technical report, Swiss Federal Institute of Technology, Lausanne, 1998; Algorithmic approach to the satisfactory graph partitioning problem, European J. Oper. Res. 125 (2000) 283-291] and further studied by other authors but its complexity remained open until now. We prove in this paper that SATISFACTORY PARTITION, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. However, for graphs with maximum degree at most 4 the problem is polynomially solvable. We also study generalizations and variants of this problem where a partition into k nonempty parts ( k {greater than or slanted equal to} 3) is requested.

Original language | English |
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Pages (from-to) | 1236-1245 |

Number of pages | 10 |

Journal | Discrete Applied Mathematics |

Volume | 154 |

Issue number | 8 |

DOIs | |

Publication status | Published - May 15 2006 |

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### Keywords

- Complexity
- Graph
- Polynomial algorithm
- Satisfactory partition

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

*154*(8), 1236-1245. https://doi.org/10.1016/j.dam.2005.10.014