Minimum factor for plane E1c extensions

Determine whether every 1-extension of type E1c applied to a minimally rigid plane graph increases the complex realization count by at least $12/7$.

Background

Experimental data up to 13 vertices shows a smallest observed multiplier of approximately 1.71, which equals 12/7. The authors suggest that this observed value may be the universal minimum for type E1c extensions, but do not prove it.

References

Since, we have a complete data set for up to 13 vertices, where this lowest factor is the same for all $7 n 12$ we might conjecture that it is indeed the lowest possible.

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Section 7.1, subsection “Extension Constructions,” paragraph following Table Laman_Increase_H2c