New upper bounds for the number of embeddings of minimally rigid graphs

TL;DR

This paper introduces new combinatorial upper bounds for the number of embeddings of minimally rigid graphs in various dimensions, significantly improving previous exponential bounds and providing a novel approach based on graph orientations.

Contribution

It presents the first substantial improvement over long-standing bounds using combinatorial methods instead of algebraic root counting.

Findings

New bounds for 2D and 3D embeddings: O(3.7764^{|V|}) and O(6.8399^{|V|})

Improved asymptotic bounds by a factor of 1/√2

First approach using combinatorial arguments rather than algebraic solutions

Abstract

By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system. Naturally, the complex solutions of such systems extend the notion of rigidity to $\mathbb{C}^d$. A major open problem has been to obtain tight upper bounds on the number of embeddings in $\mathbb{C}^d$, for a given number $|V|$ of vertices, which obviously also bound their number in $\mathbb{R}^d$. Moreover, in most known cases, the maximal numbers of embeddings in $\mathbb{C}^d$ and $\mathbb{R}^d$ coincide. For decades, only the trivial bound of $O(2^{d\cdot |V|})$ was known on the number of embeddings.Recently, matrix permanent bounds have led to a small improvement for $d\geq 5$. This work improves upon the existing upper bounds for the number of embeddings in $\mathbb{R}^d$ and $S^d$, by exploiting outdegree-constrained orientations on a graphical construction, where the proof iteratively eliminates vertices or vertex paths. For the most important cases of $d=2$ and $d=3$, the new bounds are $O(3.7764^{|V|})$ and $O(6.8399^{|V|})$, respectively. In general, the recent asymptotic bound mentioned above is improved by a factor of $1/ \sqrt{2}$. Besides being the first substantial improvement upon a long-standing upper bound, our method is essentially the first general approach relying on combinatorial arguments rather than algebraic root counts.