Establish a nonconstant lower bound for 1-extension factors

Establish a universal lower bound, growing by more than a constant factor, for the ratio of complex realization counts under 1-extensions of minimally rigid plane graphs.

Background

The paper studies how rigidity-preserving construction steps change realization counts. Although some subclasses of 1-extensions have known lower bounds, the authors state that a general lower bound exceeding a constant remains unresolved.

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 ).

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Section 8, subsection “Plane,” paragraph preceding “Extension Constructions”