Decomposing graphs under degree constraints
Journal of Graph Theory
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Algorithms for vertex-partitioning problems on graphs with fixed clique-width
Theoretical Computer Science
Efficient algorithms for decomposing graphs under degree constraints
Discrete Applied Mathematics
Approximation of satisfactory bisection problems
Journal of Computer and System Sciences
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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 ≥ 3) is requested.