Generically Globally Rigid Graphs Have Generic Universally Rigid Frameworks

Combinatorica volume 40, pages1–37(2020)Cite this article

Abstract

We show that any graph that is generically globally rigid in ℝd has a realization in ℝd that is both generic and universally rigid. This also implies that the graph also must have a realization in ℝd that is both infinitesimally rigid and universally rigid; such a realization serves as a certificate of generic global rigidity.

Our approach involves an algorithm by Lovász, Saks and Schrijver that, for a sufficiently connected graph, constructs a general position orthogonal representation of the vertices, and a result of Alfakih that shows how this representation leads to a stress matrix and a universally rigid framework of the graph.

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References

  1. [1]

    A. Y. Alfakih: Graph connectivity and universal rigidity of bar frameworks, Discrete Appl. Math.217 (part 3) (2017), 707–710.

    MathSciNet  Article  Google Scholar 

  2. [2]

    A. Y. Alfakih and Y. Ye: On affine motions and bar frameworks in general position, Linear Algebra Appl.438 (2013), 31–36.

    MathSciNet  Article  Google Scholar 

  3. [3]

    A. Y. Alfakih, A. Khandani and H. Wolkowicz: Solving Euclidean distance matrix completion problems via semidefinite programming, Comput. Optim. Appl.12 (1999), 13–30. (Computational optimization — a tribute to Olvi Mangasarian, Part I.)

    MathSciNet  Article  Google Scholar 

  4. [4]

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

    MathSciNet  Article  Google Scholar 

  5. [5]

    L. Asimow and B. Roth: The rigidity of graphs, II, Journal of Mathematical Analysis and Applications68 (1979), 171–190.

    MathSciNet  Article  Google Scholar 

  6. [6]

    S. Basu, R. Pollack and M.-F. Roy: Algorithms in real algebraic geometry, volume 10 of Algorithms and Computation in Mathematics, Springer-Verlag, Berlin, second edition, 2006.

    Google Scholar 

  7. [7]

    A. R. Berg and T. Jordán: A proof of Connelly’s conjecture on 3-connected circuits of the rigidity matroid, J. Combin. Theory Ser. B88 (2003), 77–97.

    MathSciNet  Article  Google Scholar 

  8. [8]

    J. Bochnak, M. Coste and M.-F. Roy: Real algebraic geometry, volume 36 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1998.

    Google Scholar 

  9. [9]

    R. Connelly: Rigidity and energy, Invent. Math.66 (1982), 11–33.

    MathSciNet  Article  Google Scholar 

  10. [10]

    R. Connelly: On generic global rigidity, in: Applied geometry and discrete mathematics, volume 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 147–155, Amer. Math. Soc., Providence, RI, 1991.

    Google Scholar 

  11. [11]

    R. Connelly: Generic global rigidity, Discrete Comput. Geom.33 (2005), 549–563.

    MathSciNet  Article  Google Scholar 

  12. [12]

    R. Connelly and A. Back: Mathematics and tensegrity: Group and representation theory make it possible to form a complete catalogue of “strut-cable” constructions with prescribed symmetries, American Scientist86 (1998), 142–151.

    Article  Google Scholar 

  13. [13]

    R. Connelly and S. J. Gortler: Universal rigidity of complete bipartite graphs, Discrete Comput. Geom.57 (2017), 281–304.

    MathSciNet  Article  Google Scholar 

  14. [14]

    R. Connelly and W. J. Whiteley: Global rigidity: the effect of coning, Discrete Comput. Geom.43 (2010), 717–735.

    MathSciNet  Article  Google Scholar 

  15. [15]

    R. Connelly, S. J. Gortler and L. Theran: Generic global and universal rigidity, preprint, 2016, arXiv: 1604.07475v2.

  16. [16]

    D. Eisenbud: Linear sections of determinantal varieties, Amer. J. Math.110 (1988), 541–575.

    MathSciNet  Article  Google Scholar 

  17. [17]

    M. Giusti and M. Merle: Singularités isolées et sections planes de variétés déterminantielles. II. Sections de variétés déterminantielles par les plans de coordonnées, in: Algebraic geometry (La Rábida, 1981), volume 961 of Lecture Notes in Math., 103–118, Springer, Berlin, 1982. [Isolated singularities and plane sections of determinantal varieties. II. Sections of determinantal varieties by the coordinate planes].

  18. [18]

    S. J. Gortler and D. P. Thurston: Characterizing the universal rigidity of generic frameworks, Discrete Comput. Geom.51 (2014), 1017–1036.

    MathSciNet  Article  Google Scholar 

  19. [19]

    S. J. Gortler and D. P. Thurston: Generic global rigidity in complex and pseudo-Euclidean spaces, in: Rigidity and symmetry, volume 70 of Fields Inst. Commun., 131–154, Springer, New York, 2014.

    Google Scholar 

  20. [20]

    S. J. Gortler, A. D. Healy and D. P. Thurston: Characterizing generic global rigidity. Amer. J. Math.132 (2010), 897–939.

    MathSciNet  Article  Google Scholar 

  21. [21]

    B. Hendrickson: Conditions for unique graph realizations, SIAM J. Comput.21 (1992), 65–84.

    MathSciNet  Article  Google Scholar 

  22. [22]

    B. Hendrickson: The molecule problem: exploiting structure in global optimization, SIAM J. Optim.5 (1995), 835–857.

    MathSciNet  Article  Google Scholar 

  23. [23]

    J. Herzog, A. Macchia, S. S. Madani and V. Welker: On the ideal of orthogonal representations of a graph in ℝ2, Advances in Applied Mathematics71 (2015), 146–173.

    MathSciNet  Article  Google Scholar 

  24. [24]

    B. Jackson and T. Jordán: Connected rigidity matroids and unique realizations of graphs, J. Combin. Theory Ser. B94 (2005), 1–29.

    MathSciNet  Article  Google Scholar 

  25. [25]

    M. Jacobs: Connecting global and universal rigidity, preprint, 2010, arXiv:1011.4122.

  26. [26]

    C. R. Johnson and P. Tarazaga: Connections between the real positive semidefinite and distance matrix completion problems, Linear Algebra Appl.223/224 (1995), 375–391. (Special issue honoring Miroslav Fiedler and Vlastimil Pták.)

    MathSciNet  Article  Google Scholar 

  27. [27]

    T. Jordán and V.-H. Nguyen: On universally rigid frameworks on the line, Contrib. Discrete Math.10 (2015), 10–21.

    MathSciNet  MATH  Google Scholar 

  28. [28]

    T. Jordán and Z. Szabadka: Operations preserving the global rigidity of graphs and frameworks in the plane, Comput. Geom.42 (2009), 511–521.

    MathSciNet  Article  Google Scholar 

  29. [29]

    L. Liberti, C. Lavor, N. Maculan and A. Mucherino: Euclidean distance geometry and applications, SIAM Rev.56 (2014), 3–69.

    MathSciNet  Article  Google Scholar 

  30. [30]

    N. Linial: Finite metric-spaces—combinatorics, geometry and algorithms, in: Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 573–586, Higher Ed. Press, Beijing, 2002, arXiv: math/0304466.

    Google Scholar 

  31. [31]

    N. Linial, E. London and Y. Rabinovich: The geometry of graphs and some of its algorithmic applications, Combinatorica15 (1995), 215–245.

    MathSciNet  Article  Google Scholar 

  32. [32]

    L. Lovász, M. Saks and A. Schrijver: Orthogonal representations and connectivity of graphs, Linear Algebra Appl.114/115 (1989), 439–454. (See also, [33].)

    MathSciNet  Article  Google Scholar 

  33. [33]

    L. Lovász, M. Saks and A. Schrijver: A correction: “Orthogonal representations and connectivity of graphs”, Linear Algebra Appl.313 (2000), 101–105.

    MathSciNet  Article  Google Scholar 

  34. [34]

    J. Matoušek: Lectures on discrete geometry, volume 212 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2002.

    Google Scholar 

  35. [35]

    F. A. Potra and S. J. Wright: Interior-point methods, J. Comput. Appl. Math.124 (2000), 281–302. Numerical analysis 2000, Vol. IV, Optimization and nonlinear equations.

    MathSciNet  Article  Google Scholar 

  36. [36]

    J. B. Saxe: Embeddability of weighted graphs in k-space is strongly NP-hard, in: Proc. 17th Allerton Conf. in Communications, Control, and Computing, 480–489, 1979.

  37. [37]

    I. J. Schoenberg: Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espace distanciés vectoriellement applicable sur l’espace de Hilbert” [MR1503246], Ann. of Math. (2), 36 (1935), 724–732.

    MathSciNet  Article  Google Scholar 

  38. [38]

    A. M.-C. So and Y. Ye: Theory of semidefinite programming for sensor network localization, Math. Program., Ser. B.109 (2007), 367–384.

    MathSciNet  Article  Google Scholar 

  39. [39]

    J. Stasica: Smooth points of a semialgebraic set, in: Annales Polonici Mathematici, volume 82, 149–153, Instytut Matematyczny Polskiej Akademii Nauk, 2003.

  40. [40]

    The Stacks Project Authors: Stacks Project, http://stacks.math.columbia.edu, 2018.

  41. [41]

    L. Vandenberghe and S. Boyd: Semidefinite programming, SIAM Rev.38 (1996), 49–95.

    MathSciNet  Article  Google Scholar 

  42. [42]

    D. Veenendaal and P. Block: An overview and comparison of structural form finding methods for general networks, International Journal of Solids and Structures49 (2012), 3741–3753.

    Article  Google Scholar 

  43. [43]

    H. Whitney: Elementary structure of real algebraic varieties. Ann. of Math. (2)66 (1957), 545–556.

    MathSciNet  Article  Google Scholar 

  44. [44]

    Z. Zhu, A. M.-C. So and Y. Ye: Universal rigidity and edge sparsification for sensor network localization. SIAM J. Optim.20 (2010), 3059–3081.

    MathSciNet  Article  Google Scholar 

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Author information

Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Robert Connelly

  2. School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, Scotland, UK

    Louis Theran

  3. School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138, USA

    Steven J. Gortler

Authors
  1. Robert Connelly
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  2. Steven J. Gortler
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  3. Louis Theran
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Corresponding author

Correspondence to Louis Theran.

Additional information

An earlier version of this paper [15] had the title “Generic global and universal rigidity”.

Partially supported by NSF grant DMS-1564493.

Partially supported by NSF grant DMS-1564473.

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Connelly, R., Gortler, S.J. & Theran, L. Generically Globally Rigid Graphs Have Generic Universally Rigid Frameworks. Combinatorica 40, 1–37 (2020). https://doi.org/10.1007/s00493-018-3694-4

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Mathematics Subject Classification (2010)

  • 52C25
  • 05C62