Exact factor for three-dimensional E2Xs511 and E2Vs511 extensions

Prove that every type E2Xs511 or E2Vs511 2-extension applied to a minimally 3-rigid graph multiplies the number of complex realizations in three-dimensional space by exactly two.

Background

The paper examines selected two-edge-deletion extension steps in dimension three. The E2Xs511 and E2Vs511 classes have multiplier two in all computations on graphs with at most nine vertices, and the authors formulate the corresponding universal assertion as a conjecture.

References

\begin{conjecture} Let $G$ be a minimally 3-rigid graph and $G'$ be obtained from $G$ by a 1-extension of type E2Xs511 or E2Vs511. Then $\frac{3{G'}{3{G}=2$. \end{conjecture}

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Section 7.3, subsection “Space,” paragraph introducing the E2Xs511 and E2Vs511 conjecture

Steps of type E2Xs511 and E2Vs511 on graphs with at most nine vertices always increase the number of realizations by a factor of two. \begin{conjecture} Let $G$ be a minimally 3-rigid graph and $G'$ be obtained from $G$ by a 1-extension of type E2Xs511 or E2Vs511. Then $\frac{\lambda_3(G')}{\lambda_3(G)}=2$. \end{conjecture}

— Explorations on the number of realizations of minimally rigid graphs  (2502.04736 - Grasegger, 7 Feb 2025) in Conjecture, Section 8.3.1, paragraph after the E2Xs511 and E2Vs511 discussion