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Globally Linked Pairs of Vertices in Rigid Frameworks

  • Bill Jackson
  • Tibor JordánEmail author
  • Zoltán Szabadka
Chapter
Part of the Fields Institute Communications book series (FIC, volume 70)

Abstract

A 2-dimensional framework (G, p) is a graph G = (V, E) together with a map \(p: V \rightarrow \mathbb{R}^{2}\). We consider the framework to be a straight line realization of G in \(\mathbb{R}^{2}\). Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length. A pair of vertices {u, v} is globally linked in G if the distance between the points corresponding to u and v is the same in all pairs of equivalent generic realizations of G.In this paper we extend our previous results on globally linked pairs and complete the characterization of globally linked pairs in minimally rigid graphs. We also show that the Henneberg 1-extension operation on a non-redundant edge preserves the property of being not globally linked, which can be used to identify globally linked pairs in broader families of graphs. We prove that if (G, p) is generic then the set of globally linked pairs does not change if we perturb the coordinates slightly. Finally, we investigate when we can choose a non-redundant edge e of G and then continuously deform a generic realization of Ge to obtain equivalent generic realizations of G in which the distances between a given pair of vertices are different.

Keywords

Global rigidity Rigid framework Bar-and-joint framework Globally linked pair Minimally rigid graph 

Subject Classifications

52C25 05C10 

Notes

Acknowledgements

Section 3 is based on an earlier manuscript of one of the authors [25]. We thank Viet-Hang Nguyen for pointing out an error in [25] and for several useful discussions. We are grateful to the anonymous referee for several very useful comments. We also thank Balázs Csikós and András Szűcs for interesting discussions on the differential topology aspects of our problems and Oleg Karpenkov for suggestions which led to the example given in Fig. 4.

This work was supported by the MTA-ELTE Egerváry Research Group on Combinatorial Optimization, the Hungarian Scientific Research Fund grant no. K81472, and the National Development Agency of Hungary, grant no. CK 80124, based on a source from the Research and Technology Innovation Fund.

References

  1. 1.
    Abbot, T.G.: Generalizations of Kempe’s universality theorem. MSc thesis, MIT (2008). http://web.mit.edu/tabbott/www/papers/mthesis.pdf
  2. 2.
    Asimow, L., Roth, B.: The rigidity of graphs. Trans. Am. Math. Soc. 245, 279–289 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Berg, A.R., Jordán, T.: A proof of Connelly’s conjecture on 3-connected circuits of the rigidity matroid. J. Comb. Theory Ser. B. 88, 77–97 (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Di Battista, G., Zwick, U. (eds.) Proceedings of 11th Annual European Symposium on Algorithms (ESA) 2003, Budapest. Springer Lecture Notes in Computer Science, vol. 2832, pp. 78–89 (2003)CrossRefGoogle Scholar
  5. 5.
    Connelly, R.: On generic global rigidity. In: Applied Geometry and Discrete Mathematics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 147–155. American Mathematical Society, Providence (1991)Google Scholar
  6. 6.
    Connelly, R., Whiteley, W.: Global rigidity: the effect of coning. Discret. Comput. Geom. 43, 717–735 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Frank, S., Jiang, J.: New classes of counterexamples to Hendrickson’s global rigidity conjecture. Discret. Comput. Geom. 45(3), 574–591 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Fulton, W.: Algebraic Curves (2008). http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf
  9. 9.
    Gluck, H.: Almost all simply connected closed surfaces are rigid. In: Geometric Topology. Volume 438 of Lecture Notes in Mathematics, pp. 225–239. Springer, Berlin/New York (1975)Google Scholar
  10. 10.
    Gortler, S., Healy, A., Thurston, D.: Characterizing generic global rigidity. Am. J. Math. 132(4), 897–939 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Graver, J., Servatius, B., Servatius, H.: Combinatorial Rigidity. AMS Graduate Studies in Mathematics, vol. 2. American Mathematical Society, Providence (1993)Google Scholar
  12. 12.
    Hendrickson, B.: Conditions for unique graph realizations. SIAM J. Comput. 21, 65–84 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Henneberg, L.: Die Graphische Statik Der Starren Systeme. B.G. Teubner, Leipzig/Berlin (1911)zbMATHGoogle Scholar
  14. 14.
    Hodge, W.V.D., Pedoe, D.: Methods of Algebraic Geometry, vol. 2. Cambridge University Press, Cambridge (1952)zbMATHGoogle Scholar
  15. 15.
    Jackson, B., Jordán, T.: Connected rigidity matroids and unique realizations of graphs. J. Comb. Theory Ser. B 94, 1–29 (2005)CrossRefzbMATHGoogle Scholar
  16. 16.
    Jackson, B., Jordán, T.: Graph theoretic techniques in the analysis of uniquely localizable sensor networks. In: Mao, G., Fidan, B. (eds.) Localization Algorithms and Strategies for Wireless Sensor Networks. IGI Global, Hershey (2009)Google Scholar
  17. 17.
    Jackson, B., Keevash, P.: Necessary conditions for the global rigidity of direction-length frameworks. Discret. Comput. Geom. 46, 72–85 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Jackson, B., Jordán, T., Szabadka, Z.: Globally linked pairs of vertices in equivalent realizations of graphs. Discret. Comput. Geom. 35, 493–512 (2006)CrossRefzbMATHGoogle Scholar
  19. 19.
    Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4(4), 331–340 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Owen, J.C., Power, S.C.: Infinite bar-joint frameworks, crystals and operator theory. N. Y. J. Math. 17, 445–490 (2011)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Oxley, J.G.: Matroid Theory. Oxford Science Publications, pp. xii+532. Clarendon/Oxford University Press, New York (1992)Google Scholar
  22. 22.
    Saxe, J.B.: Embeddability of weighted graphs in k-space is strongly NP-hard. Technical report, Computer Science Department, Carnegie-Mellon University, Pittsburgh (1979)Google Scholar
  23. 23.
    Seidenberg, A.: A new decision method for elementary algebra. Ann. Math. 60, 365–374 (1954)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Servatius, B., Servatius, H.: Henneberg moves on mechanisms (2013, preprint)Google Scholar
  25. 25.
    Szabadka, Z.: Globally linked pairs of vertices in minimally rigid graphs. Technical reports series of the Egerváry Research Group, Budapest, TR-2010-02 (2010)Google Scholar
  26. 26.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry. Manuscript. RAND Corporation, Santa Monica (1948). Republished as A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1951)Google Scholar
  27. 27.
    Tay, T.S., Whiteley, W.: Generating isostatic frameworks. Structural Topology 11, 21–69 (1985)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Whiteley, W.: Some matroids from discrete applied geometry. In: Bonin, J.E., Oxley, J.G., Servatius, B. (eds.) Matroid Theory, Seattle, 1995. Contemporary Mathematics, vol. 197, pp. 171–311. American Mathematical Society, Providence (1996)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesQueen Mary University of LondonLondonUK
  2. 2.Department of Operations ResearchEotvos University, and the MTA-ELTE Egervary Research Group on Combinatorial optimizationBudapestHungary
  3. 3.Department of Operations ResearchEötvös UniversityBudapestHungary

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