On the maximal number of real embeddings of spatial minimally rigid graphs

TL;DR

This paper develops a new method to determine the maximum number of real embeddings of minimally rigid graphs in three-dimensional space, achieving a full classification for 7-vertex graphs and improving lower bounds.

Contribution

It introduces a novel approach using coupler curves to optimize parameters for real embeddings, enabling the first complete classification for 7-vertex graphs in 3D.

Findings

Complete classification of real embeddings for 7-vertex graphs in R^3

Improved lower bounds on maximum real embeddings in R^3

New methodology using coupler curves for spatial rigidity problems

Abstract

The number of embeddings of minimally rigid graphs in $\mathbb{R}^D$ is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap between upper and lower bounds is still enormous. Specific values and its asymptotic behavior are major and fascinating open problems in rigidity theory. Our work considers the maximal number of real embeddings of minimally rigid graphs in $\mathbb{R}^3$. We modify a commonly used parametric semi-algebraic formulation that exploits the Cayley-Menger determinant to minimize the {\em a priori} number of complex embeddings, where the parameters correspond to edge lengths. To cope with the huge dimension of the parameter space and find specializations of the parameters that maximize the number of real embeddings, we introduce a method based on coupler curves that makes the sampling feasible for spatial minimally rigid graphs. Our methodology results in the first full classification of the number of real embeddings of graphs with 7 vertices in $\mathbb{R}^3$, which was the smallest open case. Building on this and certain 8-vertex graphs, we improve the previously known general lower bound on the maximum number of real embeddings in $\mathbb{R}^3$.