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Uniquely realisable graphs in analytic normed planes

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Uniquely realisable graphs in analytic normed planes. / Dewar, Sean; Hewetson, John; Nixon, Anthony.
In: International Mathematics Research Notices, Vol. 2024, No. 17, 30.09.2024, p. 12269-12302.

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Harvard

Dewar, S, Hewetson, J & Nixon, A 2024, 'Uniquely realisable graphs in analytic normed planes', International Mathematics Research Notices, vol. 2024, no. 17, pp. 12269-12302. https://doi.org/10.1093/imrn/rnae162

APA

Dewar, S., Hewetson, J., & Nixon, A. (2024). Uniquely realisable graphs in analytic normed planes. International Mathematics Research Notices, 2024(17), 12269-12302. https://doi.org/10.1093/imrn/rnae162

Vancouver

Dewar S, Hewetson J, Nixon A. Uniquely realisable graphs in analytic normed planes. International Mathematics Research Notices. 2024 Sept 30;2024(17):12269-12302. Epub 2024 Jul 22. doi: 10.1093/imrn/rnae162

Author

Dewar, Sean ; Hewetson, John ; Nixon, Anthony. / Uniquely realisable graphs in analytic normed planes. In: International Mathematics Research Notices. 2024 ; Vol. 2024, No. 17. pp. 12269-12302.

Bibtex

@article{07d15b89e1ac4848a74cbc6ebedda02c,
title = "Uniquely realisable graphs in analytic normed planes",
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 Jord{\'a}n [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.",
author = "Sean Dewar and John Hewetson and Anthony Nixon",
year = "2024",
month = sep,
day = "30",
doi = "10.1093/imrn/rnae162",
language = "English",
volume = "2024",
pages = "12269--12302",
journal = "International Mathematics Research Notices",
issn = "1073-7928",
publisher = "Oxford University Press",
number = "17",

}

RIS

TY - JOUR

T1 - Uniquely realisable graphs in analytic normed planes

AU - Dewar, Sean

AU - Hewetson, John

AU - Nixon, Anthony

PY - 2024/9/30

Y1 - 2024/9/30

N2 - 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.

AB - 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.

U2 - 10.1093/imrn/rnae162

DO - 10.1093/imrn/rnae162

M3 - Journal article

VL - 2024

SP - 12269

EP - 12302

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 17

ER -