Algebraic methods for counting Euclidean embeddings of rigid graphs

TL;DR

This paper develops algebraic methods to estimate the number of Euclidean embeddings of rigid graphs, providing new bounds and computational results that advance understanding in robotics and bioinformatics.

Contribution

It introduces polynomial system-based bounds for embeddings in 2D and 3D, including the first lower bound in 3D and improved upper bounds for small graphs.

Findings

Established a lower bound of about 2.52^n in D.

Provided upper bounds for graphs with up to 10 vertices.

Achieved tight bounds for graphs with up to 7 vertices in D.

Abstract

The study of (minimally) rigid graphs is motivated by numerous applications, mostly in robotics and bioinformatics. A major open problem concerns the number of embeddings of such graphs, up to rigid motions, in Euclidean space. We capture embeddability by polynomial systems with suitable structure, so that their mixed volume, which bounds the number of common roots, to yield interesting upper bounds on the number of embeddings. We focus on $\RR^2$ and $\RR^3$, where Laman graphs and 1-skeleta of convex simplicial polyhedra, respectively, admit inductive Henneberg constructions. We establish the first lower bound in $\RR^3$ of about $2.52^n$, where $n$ denotes the number of vertices. Moreover, our implementation yields upper bounds for $n \le 10$ in $\RR^2$ and $\RR^3$, which reduce the existing gaps, and tight bounds up to $n=7$ in $\RR^3$.