Exact factor for planar E1a extensions

Prove that every 1-extension of type E1a applied to a minimally rigid planar graph exactly doubles the number of complex realizations.

Background

Type E1a is one of three subclasses of planar 1-extensions distinguished by the number of edges among the selected vertices. Existing results give a lower bound of two for the realization-count ratio, and computations suggest that the ratio is always exactly two.

The conjecture has been verified in the paper for graphs with at most 12 vertices but remains unresolved in general.

References

We might therefore conjecture \begin{conjecture} Let $G$ be a minimally rigid graph and $G'$ be obtained from $G$ by a 1-extension of type E1a. Then $\frac{\lambda_2(G')}{\lambda_2(G)}=2$. \end{conjecture}

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Conjecture, Section 8.1.1, Extension Constructions

Let $G$ be a minimally rigid graph and $G'$ be obtained from $G$ by a 1-extension of type E1a. Then $\frac{2{G'}{2{G}=2$.

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

We might therefore conjecture

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