The number of realisations of a rigid graph in Euclidean and spherical geometries
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914062894366720 |
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| author | Dewar, Sean Grasegger, Georg |
| author_facet | Dewar, Sean Grasegger, Georg |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_16416 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The number of realisations of a rigid graph in Euclidean and spherical geometries Dewar, Sean Grasegger, Georg Combinatorics Algebraic Geometry 52C25 (Primary) 05C30, 68R10, 68R12 (Secondary) 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. |
| title | The number of realisations of a rigid graph in Euclidean and spherical geometries |
| topic | Combinatorics Algebraic Geometry 52C25 (Primary) 05C30, 68R10, 68R12 (Secondary) |
| url | https://arxiv.org/abs/2309.16416 |