Toward the complexity of the existence of wonderfully stable partitions and strictly core stable coalition structures in enemy-oriented hedonic games

We study the computational complexity of the existence and the verification problem for wonderfully stable partitions (WSPE and WSPV) and of the existence problem for strictly core stable coalition structures (SCSCS) in enemy-oriented hedonic games. In this note, we show that WSPV is NP-complete and both WSPE and SCSCS are DP-hard, where DP is the second level of the boolean hierarchy, and we discuss an approach for classifying the latter two problems in terms of their complexity.