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    Rights statement: This is the peer reviewed version of the following article: Grasegger, G., Guler, H., Jackson, B., and Nixon, A., Flexible circuits in the d-dimensional rigidity matroid, J. Graph. Theory. 2022; 100: 315– 330. https://doi.org/10.1002/jgt.22780 which has been published in final form at https://onlinelibrary.wiley.com/doi/10.1002/jgt.22780 This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving.

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Flexible circuits in the d-dimensional rigidity matroid

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Flexible circuits in the d-dimensional rigidity matroid. / Grasegger, Georg; Guler, Hakan; Jackson, Bill et al.
In: Journal of Graph Theory, Vol. 100, No. 2, 30.06.2022, p. 315-360.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

Harvard

Grasegger, G, Guler, H, Jackson, B & Nixon, A 2022, 'Flexible circuits in the d-dimensional rigidity matroid', Journal of Graph Theory, vol. 100, no. 2, pp. 315-360. https://doi.org/https://arxiv.org/abs/2003.06648, https://doi.org/10.1002/jgt.22780

APA

Vancouver

Grasegger G, Guler H, Jackson B, Nixon A. Flexible circuits in the d-dimensional rigidity matroid. Journal of Graph Theory. 2022 Jun 30;100(2):315-360. Epub 2021 Dec 6. doi: https://arxiv.org/abs/2003.06648, 10.1002/jgt.22780

Author

Grasegger, Georg ; Guler, Hakan ; Jackson, Bill et al. / Flexible circuits in the d-dimensional rigidity matroid. In: Journal of Graph Theory. 2022 ; Vol. 100, No. 2. pp. 315-360.

Bibtex

@article{84071971f8ec4b9d911a5081dde012e4,
title = "Flexible circuits in the d-dimensional rigidity matroid",
abstract = "A bar-joint framework (퐺,푝) in ℝ푑 is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of ℝ푑 . It is known that, when (퐺,푝) is generic, its rigidity depends only on the underlying graph 퐺 , and is determined by the rank of the edge set of 퐺 in the generic 푑 -dimensional rigidity matroid ℛ푑 . Complete combinatorial descriptions of the rank function of this matroid are known when 푑=1,2 , and imply that all circuits in ℛ푑 are generically rigid in ℝ푑 when 푑=1,2 . Determining the rank function of ℛ푑 is a long standing open problem when 푑≥3 , and the existence of nonrigid circuits in ℛ푑 for 푑≥3 is a major contributing factor to why this problem is so difficult. We begin a study of nonrigid circuits by characterising the nonrigid circuits in ℛ푑 which have at most 푑+6 vertices.",
keywords = "bar-joint framework, flexible circuit, rigid graph, rigidity matroid",
author = "Georg Grasegger and Hakan Guler and Bill Jackson and Anthony Nixon",
note = "This is the peer reviewed version of the following article: Grasegger, G., Guler, H., Jackson, B., and Nixon, A., Flexible circuits in the d-dimensional rigidity matroid, J. Graph. Theory. 2022; 100: 315– 330. https://doi.org/10.1002/jgt.22780 which has been published in final form at https://onlinelibrary.wiley.com/doi/10.1002/jgt.22780 This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving. ",
year = "2022",
month = jun,
day = "30",
doi = "https://arxiv.org/abs/2003.06648",
language = "English",
volume = "100",
pages = "315--360",
journal = "Journal of Graph Theory",
issn = "0364-9024",
publisher = "Wiley-Liss Inc.",
number = "2",

}

RIS

TY - JOUR

T1 - Flexible circuits in the d-dimensional rigidity matroid

AU - Grasegger, Georg

AU - Guler, Hakan

AU - Jackson, Bill

AU - Nixon, Anthony

N1 - This is the peer reviewed version of the following article: Grasegger, G., Guler, H., Jackson, B., and Nixon, A., Flexible circuits in the d-dimensional rigidity matroid, J. Graph. Theory. 2022; 100: 315– 330. https://doi.org/10.1002/jgt.22780 which has been published in final form at https://onlinelibrary.wiley.com/doi/10.1002/jgt.22780 This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving.

PY - 2022/6/30

Y1 - 2022/6/30

N2 - A bar-joint framework (퐺,푝) in ℝ푑 is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of ℝ푑 . It is known that, when (퐺,푝) is generic, its rigidity depends only on the underlying graph 퐺 , and is determined by the rank of the edge set of 퐺 in the generic 푑 -dimensional rigidity matroid ℛ푑 . Complete combinatorial descriptions of the rank function of this matroid are known when 푑=1,2 , and imply that all circuits in ℛ푑 are generically rigid in ℝ푑 when 푑=1,2 . Determining the rank function of ℛ푑 is a long standing open problem when 푑≥3 , and the existence of nonrigid circuits in ℛ푑 for 푑≥3 is a major contributing factor to why this problem is so difficult. We begin a study of nonrigid circuits by characterising the nonrigid circuits in ℛ푑 which have at most 푑+6 vertices.

AB - A bar-joint framework (퐺,푝) in ℝ푑 is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of ℝ푑 . It is known that, when (퐺,푝) is generic, its rigidity depends only on the underlying graph 퐺 , and is determined by the rank of the edge set of 퐺 in the generic 푑 -dimensional rigidity matroid ℛ푑 . Complete combinatorial descriptions of the rank function of this matroid are known when 푑=1,2 , and imply that all circuits in ℛ푑 are generically rigid in ℝ푑 when 푑=1,2 . Determining the rank function of ℛ푑 is a long standing open problem when 푑≥3 , and the existence of nonrigid circuits in ℛ푑 for 푑≥3 is a major contributing factor to why this problem is so difficult. We begin a study of nonrigid circuits by characterising the nonrigid circuits in ℛ푑 which have at most 푑+6 vertices.

KW - bar-joint framework

KW - flexible circuit

KW - rigid graph

KW - rigidity matroid

U2 - https://arxiv.org/abs/2003.06648

DO - https://arxiv.org/abs/2003.06648

M3 - Journal article

VL - 100

SP - 315

EP - 360

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 2

ER -