The satisfactory partition problem Original Research Article

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,