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arXiv · 2309.16416

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

Sean Dewar·Georg Grasegger
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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.

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Sean Dewar, Georg Grasegger. 2023-09-28. The number of realisations of a rigid graph in Euclidean and spherical geometries. https://doi.org/10.5802/alco.390

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