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Coincident Rigidity of 2-Dimensional Frameworks

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Abstract

Fekete, Jordán and Kaszanitzky (Graphs Combin 31:585–599, 2015) characterised the graphs which can be realised as 2-dimensional, infinitesimally rigid, bar-joint frameworks in which two given vertices are coincident. We formulate a conjecture which would extend their characterisation to an arbitrary set T of vertices and verify our conjecture when \(|T|=3\).

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References

  1. Abbott, T.: Generalizations of Kempe’s universality theorem. Master’s thesis, Massachusetts Institute of Technology (2008)

  2. Asimow, L., Roth, B.: The rigidity of graphs. Trans. Am. Math. Soc. 245, 279–289 (1978)

    Article  MathSciNet  Google Scholar 

  3. Bolker, E.D., Roth, B.: When is a bipartite graph a rigid framework? Pac. J. Math. 90, 27–44 (1980)

    Article  MathSciNet  Google Scholar 

  4. Fekete, Z., Jordán, T., Kaszanitzky, V.E.: Rigid two-dimensional frameworks with two coincident points. Graphs Combin. 31, 585–599 (2015)

    Article  MathSciNet  Google Scholar 

  5. Guler, H.: Rigidity of Frameworks. PhD thesis, Queen Mary University of London (2018)

  6. Jackson, B., Jordán, T.: Rigid two-dimensional frameworks with three collinear points. Graphs Combin. 21, 427–444 (2005)

    Article  MathSciNet  Google Scholar 

  7. Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970)

    Article  MathSciNet  Google Scholar 

  8. Lovász, L., Yemini, Y.: On generic rigidity in the plane. SIAM J. Algebr. Discrete Methods 21, 91–98 (1982)

    Article  MathSciNet  Google Scholar 

  9. Maxwell, J.C.: On the calculation of the equilibrium and stiffness of frames. Philos. Mag. 27, 294–299 (1864)

    Article  Google Scholar 

  10. Pollaczek-Geiringer, H.: Über die Gliederung ebener Fachwerke. Zeitschrift für. Angewandte Mathematik und Mechanik (ZAMM) 7, 58–72 (1927)

    Article  Google Scholar 

  11. Whiteley, W.: Infinitesimal rigidity of a bipartite framework. Pac. J. Math. 110, 233–255 (1984)

    Article  Google Scholar 

  12. Whiteley, W.: Some matroids from discrete applied geometry. In: Bonin, J.E., Oxley, J.G., Servatius, B. (eds.) Matroid theory, pp. 171–311. Contemporary Mathematics 197 (1996)

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Acknowledgements

We would like to thank Shin-ichi Tanigawa for a helpful converstaion which gave rise to the \(K_{5,5}\) example in Sect. 5.2.

Funding

The first author would also like to thank the Ministry of National Education of Turkey for PhD funding through a YLSY grant.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Arts and Sciences, Kastamonu University, Kastamonu, Turkey

    Hakan Guler

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

    Bill Jackson

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  1. Hakan Guler
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  2. Bill Jackson
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Correspondence to Hakan Guler.

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Guler, H., Jackson, B. Coincident Rigidity of 2-Dimensional Frameworks. Graphs and Combinatorics 38, 128 (2022). https://doi.org/10.1007/s00373-022-02540-9

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  • DOI: https://doi.org/10.1007/s00373-022-02540-9

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