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Sufficient Conditions for the Global Rigidity of Periodic Graphs

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Sufficient Conditions for the Global Rigidity of Periodic Graphs. / Kaszanitzky, Viktoria; Kiraly, Csaba; Schulze, Bernd.

In: Discrete and Computational Geometry, Vol. 67, 31.01.2022, p. 1-16.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Harvard

Kaszanitzky, V, Kiraly, C & Schulze, B 2022, 'Sufficient Conditions for the Global Rigidity of Periodic Graphs', Discrete and Computational Geometry, vol. 67, pp. 1-16. https://doi.org/10.1007/s00454-021-00346-9

APA

Kaszanitzky, V., Kiraly, C., & Schulze, B. (2022). Sufficient Conditions for the Global Rigidity of Periodic Graphs. Discrete and Computational Geometry, 67, 1-16. https://doi.org/10.1007/s00454-021-00346-9

Vancouver

Kaszanitzky V, Kiraly C, Schulze B. Sufficient Conditions for the Global Rigidity of Periodic Graphs. Discrete and Computational Geometry. 2022 Jan 31;67:1-16. https://doi.org/10.1007/s00454-021-00346-9

Author

Kaszanitzky, Viktoria ; Kiraly, Csaba ; Schulze, Bernd. / Sufficient Conditions for the Global Rigidity of Periodic Graphs. In: Discrete and Computational Geometry. 2022 ; Vol. 67. pp. 1-16.

Bibtex

@article{5343fa2c8884453f8978d4f8e0a61b35,
title = "Sufficient Conditions for the Global Rigidity of Periodic Graphs",
abstract = "AbstractTanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic frameworks under fixed lattice representations. That is, we show that if a generic periodic framework is vertex-redundantly rigid, in the sense that the deletion of a single vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. Our proof is similar to the one of Tanigawa, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property in dimension $$d>2$$ d > 2 . We work around this issue by using slight modifications of recent results of Kaszanitzky et al. (2021). Secondly, while the rigidity of finite frameworks in $${\mathbb {R}}^d$$ R d on at most d vertices obviously implies their global rigidity, it is non-trivial to prove a similar result for periodic frameworks. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion between two equivalent d-dimensional realisations of a single graph in $${\mathbb {R}}^{2d}$$ R 2 d to periodic frameworks. As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic body-bar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly et al. (2013) regarding the global rigidity of generic finite body-bar frameworks.",
keywords = "Rigidity, Global rigidity, Body-bar framework, Periodic framework",
author = "Viktoria Kaszanitzky and Csaba Kiraly and Bernd Schulze",
year = "2022",
month = jan,
day = "31",
doi = "10.1007/s00454-021-00346-9",
language = "English",
volume = "67",
pages = "1--16",
journal = "Discrete and Computational Geometry",
issn = "0179-5376",
publisher = "Springer New York",

}

RIS

TY - JOUR

T1 - Sufficient Conditions for the Global Rigidity of Periodic Graphs

AU - Kaszanitzky, Viktoria

AU - Kiraly, Csaba

AU - Schulze, Bernd

PY - 2022/1/31

Y1 - 2022/1/31

N2 - AbstractTanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic frameworks under fixed lattice representations. That is, we show that if a generic periodic framework is vertex-redundantly rigid, in the sense that the deletion of a single vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. Our proof is similar to the one of Tanigawa, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property in dimension $$d>2$$ d > 2 . We work around this issue by using slight modifications of recent results of Kaszanitzky et al. (2021). Secondly, while the rigidity of finite frameworks in $${\mathbb {R}}^d$$ R d on at most d vertices obviously implies their global rigidity, it is non-trivial to prove a similar result for periodic frameworks. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion between two equivalent d-dimensional realisations of a single graph in $${\mathbb {R}}^{2d}$$ R 2 d to periodic frameworks. As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic body-bar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly et al. (2013) regarding the global rigidity of generic finite body-bar frameworks.

AB - AbstractTanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic frameworks under fixed lattice representations. That is, we show that if a generic periodic framework is vertex-redundantly rigid, in the sense that the deletion of a single vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. Our proof is similar to the one of Tanigawa, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property in dimension $$d>2$$ d > 2 . We work around this issue by using slight modifications of recent results of Kaszanitzky et al. (2021). Secondly, while the rigidity of finite frameworks in $${\mathbb {R}}^d$$ R d on at most d vertices obviously implies their global rigidity, it is non-trivial to prove a similar result for periodic frameworks. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion between two equivalent d-dimensional realisations of a single graph in $${\mathbb {R}}^{2d}$$ R 2 d to periodic frameworks. As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic body-bar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly et al. (2013) regarding the global rigidity of generic finite body-bar frameworks.

KW - Rigidity

KW - Global rigidity

KW - Body-bar framework

KW - Periodic framework

U2 - 10.1007/s00454-021-00346-9

DO - 10.1007/s00454-021-00346-9

M3 - Journal article

VL - 67

SP - 1

EP - 16

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

ER -