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Vertex Splitting, Coincident Realisations, and Global Rigidity of Braced Triangulations

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Abstract

We give a relatively short graph theoretic proof of a result of Jordán and Tanigawa that a 4-connected graph which has a spanning plane triangulation as a proper subgraph is generically globally rigid in \({\mathbb {R}}^3\). Our proof is based on a new sufficient condition for the so called vertex splitting operation to preserve generic global rigidity in \({\mathbb {R}}^d\).

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Notes

  1. They mistakenly state in [15] that a stronger form of Theorem 1.2 (in which the hypothesis that \(G'\) can be realised as a \(vv'\)-coincident infinitesimally rigid framework in \({\mathbb {R}}^d\) is replaced by the hypothesis that \(G'-vv'\) is rigid) is implied by [4].

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Acknowledgements

We would like to thank the referees for their careful reading and helpful comments which have greatly improved this paper. Our work was supported by JST ERATO Grant Number JPMJER1903, JSPS KAKENHI Grant Number 18K11155, and EPSRC overseas travel grant EP/T030461/1.

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, University of Galway, Galway, Ireland

    James Cruickshank

  2. School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, UK

    Bill Jackson

  3. Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

    Shin-ichi Tanigawa

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  1. James Cruickshank
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  2. Bill Jackson
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  3. Shin-ichi Tanigawa
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Correspondence to Bill Jackson.

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Cruickshank, J., Jackson, B. & Tanigawa, Si. Vertex Splitting, Coincident Realisations, and Global Rigidity of Braced Triangulations. Discrete Comput Geom 69, 192–208 (2023). https://doi.org/10.1007/s00454-022-00459-9

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