Fixed-parameter tractability by relationship-graph vertex cover for the size-constrained variant

Determine whether IR-FO-SCC is fixed-parameter tractable when parameterized by the vertex cover number of the relationship graph $G$.

Background

The paper develops an FPT algorithm for IR-FO-FSC parameterized by the vertex cover number of the relationship graph and gives an XP algorithm for IR-FO-SCC parameterized by the combined parameter $k+\tw(G)$. However, it does not establish fixed-parameter tractability for the more general size-constrained coalition variant under the vertex-cover parameter. The conclusion identifies this as the most intriguing unresolved case remaining in the results table.

References

Beyond resolving the remaining open cases in \Cref{tab:results}---the most intriguing one, in our view, being whether the size-constrained (SCC) variant is fixed-parameter tractable with respect to the vertex cover number of the relationship graph---we see two natural directions for future research.

— Individual Rationality in Constrained Hedonic Games: Friends, Enemies, and Neutrals  (2608.14461 - Schierrreich et al., 14 Aug 2026) in Conclusion, Section 5; also marked by “?” in Table 1 for the row $\vc$ and column IR-FO-SCC.