Exponential lower bound for one-extensions

Prove a universal lower bound for the increase in the number of complex realizations under planar 1-extensions that grows by more than a constant factor.

Background

Planar minimally rigid graphs can be generated through extension constructions, but the realization-count increase associated with 1-extensions is not fully understood. The paper notes that known results provide lower bounds for certain subclasses, while a general lower bound exceeding a constant remains unresolved. Solving this problem would quantify how realization counts evolve along general Henneberg constructions.

References

little is known for the 1-extensions and hence even a general lower bound that is more than constant is an open problem (compare cite{LowerBounds}).

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Section 7, subsection “Plane,” subsection “Extension Constructions”