Uniquely Realisable Graphs in Analytic Normed Planes

IF 0.9 2区 数学 Q2 MATHEMATICS
Sean Dewar, John Hewetson, Anthony Nixon
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引用次数: 0

Abstract

A framework $(G,p)$ in Euclidean space $\mathbb{E}^{d}$ is globally rigid if it is the unique realisation, up to rigid congruences, of $G$ with the edge lengths of $(G,p)$. Building on key results of Hendrickson [28] and Connelly [14], Jackson and Jordán [29] gave a complete combinatorial characterisation of when a generic framework is global rigidity in $\mathbb{E}^{2}$. We prove an analogous result when the Euclidean norm is replaced by any norm that is analytic on $\mathbb{R}^{2} \setminus \{0\}$. Specifically, we show that a graph $G=(V,E)$ has an open set of globally rigid realisations in a non-Euclidean analytic normed plane if and only if $G$ is 2-connected and $G-e$ contains 2 edge-disjoint spanning trees for all $e\in E$. We also prove that the analogous necessary conditions hold in $d$-dimensional normed spaces.
解析规范平面中的唯一可实现图形
如果欧几里得空间 $\mathbb{E}^{d}$ 中的 $(G,p)$是 $G$ 边长为 $(G,p)$ 的唯一实现,那么它就是全局刚性的。在亨德里克森[28]和康奈利[14]的关键结果基础上,杰克逊和乔丹[29]给出了通用框架在 $\mathbb{E}^{2}$ 中具有全局刚性时的完整组合特征。当欧几里德规范被$\mathbb{R}^{2}$上任何解析规范取代时,我们证明了类似的结果。\setminus \{0\}$。具体地说,我们证明了当且仅当 $G$ 是 2 连接的并且 $G-e$ 包含 E$ 中所有 $e\$ 的 2 个边缘相交的生成树时,图 $G=(V,E)$ 在非欧几里得解析规范平面上有一个开放的全局刚性现实集。我们还证明了类似的必要条件在 $d$ 维规范空间中成立。
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来源期刊
CiteScore
2.00
自引率
10.00%
发文量
316
审稿时长
1 months
期刊介绍: International Mathematics Research Notices provides very fast publication of research articles of high current interest in all areas of mathematics. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics.
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