Characterize the applicability of the clique-based GMNL construction for larger graphs

Characterize whether graph states associated with larger graphs contain, up to local complementation, a representative to which the clique-based genuine-multipartite-nonlocality construction of Theorem \ref{thm: clique} applies.

Background

Theorem \ref{thm: clique} constructs a GMNL-detecting inequality when some vertex has a neighborhood inducing a complete graph. Local complementation can transform many graph states into representatives satisfying this condition.

The authors verify applicability for nearly all local-complementation classes of connected graphs with at most eight vertices, but identify one exceptional class where the construction is inapplicable. They leave unresolved whether further exceptional graph classes occur for larger graphs.