The number of realisations of a rigid graph in Euclidean and spherical geometries

TL;DR

This paper proves that for any dimension, the number of complex realisations of a rigid graph in Euclidean space is always less than or equal to the number of spherical realisations, extending previous results and introducing coning techniques.

Contribution

It establishes a universal inequality between Euclidean and spherical realisation counts for all dimensions using new coning methods.

Findings

The inequality c_d(G) ≤ c_d^*(G) holds for all d-rigid graphs.

Introduces coning as a technique to relate Euclidean and spherical realisations.

Confirms previous partial results for higher dimensions.

Abstract

A graph is $d$-rigid if for any generic realisation of the graph in $\mathbb{R}^d$ (equivalently, the $d$-dimensional sphere $\mathbb{S}^d$), there are only finitely many non-congruent realisations in the same space with the same edge lengths. By extending this definition to complex realisations in a natural way, we define $c_d(G)$ to be the number of equivalent $d$-dimensional complex realisations of a $d$-rigid graph $G$ for a given generic realisation, and $c^*_d(G)$ to be the number of equivalent $d$-dimensional complex spherical realisations of $G$ for a given generic spherical realisation. Somewhat surprisingly, these two realisation numbers are not always equal. Recently developed algorithms for computing realisation numbers determined that the inequality $c_2(G) \leq c_2^*(G)$ holds for any minimally 2-rigid graph $G$ with 12 vertices or less. In this paper we confirm that, for any dimension $d$, the inequality $c_d(G) \leq c_d^*(G)$ holds for every $d$-rigid graph $G$. This result is obtained via new techniques involving coning, the graph operation that adds an extra vertex adjacent to all original vertices of the graph.