Symmetric one-recluse case for the k-coalition variant

Determine the computational complexity of IR-FO-$k$ for symmetric friend-oriented hedonic games with exactly one recluse.

Background

A recluse is an agent with no friends and at least one enemy. The paper proves that, for the kk-coalition variant under friend-oriented preferences, symmetric instances with no recluses are solvable in polynomial time, while asymmetric instances remain NP-hard even when there are no recluses. It also establishes hardness when at least two recluses are present. Consequently, the symmetric case with exactly one recluse is the remaining unresolved value in this parameterization.

References

\Cref{thm:FO:k:symmetric:atLeastOneFriend:poly,thm:FO:k:asymmetric:atLeastOneFriend:NPh} leave open only the symmetric case with exactly one recluse.

— Individual Rationality in Constrained Hedonic Games: Friends, Enemies, and Neutrals  (2608.14461 - Schierrreich et al., 14 Aug 2026) in End of Section 4.3, immediately after Theorem 4.7 (the theorem labeled thm:FO:k:asymmetric:atLeastOneFriend:NPh); also marked by “?” in Table 1 for the row $r=1$ and column IR-FO-$k$.