Rank-width lower bound for Perfect Code beyond split graphs

Develop a rank-width lower bound for Perfect Code on a graph class other than split graphs, since split graphs admit a linear-time characterization, implicit representation, and counting algorithm for all perfect codes.

Background

Perfect Code imposes both exact-one domination for unselected vertices and independence of the selected vertices. The paper identifies it as outside the current lower-bound framework because the constructions test only whether a clause or equality-checker vertex has at least one selected assignment neighbor.

Split graphs cannot serve as hard instances for the desired lower bound: the appendix gives a complete characterization of perfect codes on split graphs, including implicit representation, counting, and optimization in linear time. Any extension must therefore use a different host graph class.

References

Perfect Code is likewise an open case for the present framework. Split graphs, however, cannot provide hard instances: a split partition can be found in linear time, and Appendix~\ref{app:perfect-code} gives a complete characterization that permits implicit representation and counting of all perfect codes, as well as finding minimum- and maximum-cardinality perfect codes, in linear time. Any extension of our lower-bound approach to Perfect Code must therefore use a different host graph class.

— Lower Bounds for Domination-Type Problems Parameterized by Rank-Width  (2608.18854 - Liu et al., 19 Aug 2026) in Section 5, subsection “Perfect Dominating Set and Perfect Code”