Universal lower bound for planar realization counts

Prove that every minimally rigid graph with n vertices has at least 2^{n-2} complex realizations in the plane.

Background

Earlier work conjectured the lower bound 2{n-2} for the complex realization count of every minimally rigid planar graph with n vertices. The paper verifies the conjecture computationally through 13 vertices, but explicitly states that the conjecture remains unproved in general.

References

In cite{LowerBounds,Jackson2018} it was conjectured that $2{G} 2{n-2}$. While this conjecture remains unproven, recent computations have shown that it is true at least for graphs with at most 13 vertices and non of the larger examples we computed would contradict it either.

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Section 2, subsection “Realizations in the plane”