Minimum factor for spherical E1c extensions

Prove that the minimum realization-count multiplier for a type E1c 1-extension on minimally rigid graphs on the sphere is $3/2$.

Background

For spherical E1c extensions, the smallest observed multiplier is $1.5$, and this value occurs throughout the complete computations for graphs with 6 through 12 vertices. The authors explicitly frame the claim that $3/2$ is universally minimal as a conjectural conclusion.

References

Again since we have complete data for graph with less than 13 vertices and since $1.5$ is the lowest factor for all graphs with $6 n 12$ we may conjecture that $k_1=1.5$.

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Section 7.2, subsection “Extension constructions,” paragraph following the spherical E1c proposition