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Universal Rigidity of Complete Bipartite Graphs

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Abstract

We describe a very simple condition that is necessary for the universal rigidity of a complete bipartite framework \((K(n,m),\mathbf{p},\mathbf{q})\). This condition is also sufficient for universal rigidity under a variety of weak assumptions, such as general position. Even without any of these assumptions, in complete generality, we extend these ideas to obtain an efficient algorithm, based on a sequence of linear programs, that determines whether an input framework of a complete bipartite graph is universally rigid or not.

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References

  1. Alfakih, A.Y.: On bar frameworks, stress matrices and semidefinite programming. Math. Program. 129(1, Ser. B), 113–128 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alfakih, A.Y.: Local, dimensional and universal rigidities: a unified Gram matrix approach. In: Rigidity and Symmetry, pp. 41–60. Springer, New York (2014)

  3. Alfakih, A.Y., Ye, Y.: On affine motions and bar frameworks in general position. Linear Algebra Appl. 438(1), 31–36 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Bolker, E.D., Roth, B.: When is a bipartite graph a rigid framework? Pac. J. Math. 90(1), 27–44 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  6. Connelly, R.: Rigidity and energy. Invent. Math. 66(1), 11–33 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Connelly, R.: On generic global rigidity. In: Applied Geometry and Discrete Mathematics DIMACS Discrete Mathematics & Theoretical Computer Science, vol. 4, pp. 147–155. American Mathematical Society, Providence (1991)

  8. Connelly, R., Gortler, S.J.: Iterative universal rigidity. Discrete Comput. Geom. 53(4), 847–877 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Connelly, R., Whiteley, W.: Second-order rigidity and prestress stability for tensegrity frameworks. SIAM J. Discrete Math. 9(3), 453–491 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Connelly, R., Whiteley, W.: Global rigidity: the effect of coning. Discrete Comput. Geom. 43(4), 717–735 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Eckhoff, J.: Helly, Radon, and Carathéodory type theorems. In: Handbook of Convex Geometry, vol. A, B, pp. 389–448. North-Holland, Amsterdam (1993)

  12. Gortler, S.J., Thurston, D.P.: Characterizing the universal rigidity of generic frameworks. Discrete Comput. Geom. 51(4), 1017–1036 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gribling, S., de Laat, D., Laurent, M.: Matrices with high completely positive semidefinite rank. arXiv:1605.00988 (2016)

  14. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  15. Jordán, T., Nguyen, V.-H.: On universally rigid frameworks on the line. Technical report TR-2012-10, Egerváry Research Group on Combinatorial Optimization, Budapest, Hungary (2012)

  16. Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)

    Book  Google Scholar 

  17. Ramana, M.V.: An exact duality theory for semidefinite programming and its complexity implications. Math. Program. 77(2, Ser. B), 129–162 (1997)

    MathSciNet  MATH  Google Scholar 

  18. So, A.M.-C., Ye, Y.: Theory of semidefinite programming for sensor network localization. Math. Program. 109(2–3, Ser. B), 367–384 (2007)

  19. Tsirelson, B.: Quantum bell-type inequalities. Hadronic. J. Suppl. 8, 329–345 (1993)

    MATH  Google Scholar 

  20. Whiteley, W.: Infinitesimal motions of a bipartite framework. Pac. J. Math. 110(1), 233–255 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The impetus for this paper is the result in [15] for K(n, m) on a line. It was a desire to generalize that result, which was the starting point for this paper. The elephant in the room is the paper by E. Bolker and B. Roth [5]. This paper was constantly in the background leading us to what was true and what was not. It gives a reasonably complete picture of which configurations of complete bipartite graphs are infinitesimally rigid. Also, one can see stress matrices there quite naturally. Their basic tool was the tensor product of a vector with itself, where instead we think of it as using the Veronese map. Other work we did not formally use, but is still lurking in the background, is the very insightful paper [20] by W. Whiteley. The idea there is that an infinitesimal flex \(\mathbf{p}'\) of a bipartite framework with corresponding configuration \(\mathbf{p}\) on a quadric can be easily described. Furthermore, the two configurations \(\mathbf{p}+\mathbf{p}'\) and \(\mathbf{p}-\mathbf{p}'\) describe equivalent frameworks. Thus they are not even globally rigid, and they are separated by a quadric surface. This is the basis in [7] to show that K(5, 5) is not globally rigid (thus not universally rigid) in \({\mathbb {R}}^3\). But on the other hand, there are many examples of complete bipartite graphs in any \({\mathbb {R}}^d\) that are globally rigid, but not universally, as we have shown here. The main result of [12] applied to complete bipartite graphs, shows that when the configuration is generic, the rank and positive semi-definiteness of the stress matrix determines when the configuration is universally rigid. What we have done here, for complete bipartite graphs, is to replace the condition of being generic, which is problematic to determine in general, with the more precise condition of being in quadric general position in Corollary 4.8. We would like to thank Deborah Alves whose experiments kept on suggesting the correctness of Theorem 4.4, long before we knew how to prove it. This work was partially supported by NSF Grants DMS-1564493 and DMS-1564473.

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Authors and Affiliations

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

    Robert Connelly

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

    Steven J. Gortler

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  1. Robert Connelly
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  2. Steven J. Gortler
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Correspondence to Robert Connelly.

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Connelly, R., Gortler, S.J. Universal Rigidity of Complete Bipartite Graphs. Discrete Comput Geom 57, 281–304 (2017). https://doi.org/10.1007/s00454-016-9836-9

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