Universal lower bound beyond constant factors for planar 1-extensions

Establish a universal lower bound greater than a constant for the increase in the complex realization count caused by every planar 1-extension of a minimally rigid graph.

Background

Planar minimally rigid graphs can be constructed using extension operations. Although some subclasses of 1-extensions are known to increase realization counts by at least specific factors, the paper states that a general lower bound exceeding a constant remains unresolved for arbitrary planar 1-extensions.

References

While it is known and easy to see that 0-extensions always increase the number of realizations by a factor of two , 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 7, subsection “Plane,” paragraph preceding “Extension Constructions”