Skip to main content

New Upper Bounds for the Number of Embeddings of Minimally Rigid Graphs

Abstract

By definition, a rigid graph in \(\mathbb {R}^d\) (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system. Naturally, the complex solutions of such systems extend the notion of rigidity to \(\mathbb {C}^d\). A major open problem has been to obtain tight upper bounds on the number of embeddings in \(\mathbb {C}^d\), for a given number |V| of vertices, which obviously also bound their number in \(\mathbb {R}^d\). Moreover, in most known cases, the maximal numbers of embeddings in \(\mathbb {C}^d\) and \(\mathbb {R}^d\) coincide. For decades, only the trivial bound of \(O(2^{d|V|})\) was known on the number of embeddings. Recently, matrix permanent bounds have led to a small improvement for \(d\ge 5\). This work improves upon the existing upper bounds for the number of embeddings in \(\mathbb {R}^d\) and \(S^d\), by exploiting outdegree-constrained orientations on a graphical construction, where the proof iteratively eliminates vertices or vertex paths. For the most important cases of \(d=2\) and \(d=3\), the new bounds are \(O(3.7764^{|V|})\) and \(O(6.8399^{|V|})\), respectively. In general, we improve the exponent basis in the asymptotic behavior with respect to the number of vertices of the recent bound mentioned above by the factor of \(\sqrt{2}\). Besides being the first substantial improvement upon a long-standing upper bound, our method is essentially the first general approach relying on combinatorial arguments rather than algebraic root counts.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. In [4], indegree constraints were used but here, following [35], we use (equivalently) outdegrees.

  2. A similar discussion in [4, Sect. 2.2] offers details on the algebraic systems.

  3. Hanging edges are reminiscent of "directed loops" in hypergraphs [35]; "half-edges" also have a single endpoint.

  4. In [19, Chap. 3] these subgraphs are called blocks; "biconnected component" is used equivalently, e.g. [22, Chap. 8].

References

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

    MathSciNet  Article  Google Scholar 

  2. Baglivo, J.A., Graver, J.E.: Incidence and Symmetry in Design and Architecture. Cambridge Urban and Architectural Studies, vol. 7. Cambridge University Press, Cambridge (1983)

    Google Scholar 

  3. Bartzos, E., Emiris, I.Z., Legerský, J., Tsigaridas, E.: On the maximal number of real embeddings of minimally rigid graphs in \({\mathbb{R}}^2\), \({\mathbb{R}}^3\) and \(S^2\). J. Symbolic Comput. 102, 189–208 (2021)

    MathSciNet  Article  Google Scholar 

  4. Bartzos, E., Emiris, I.Z., Schicho, J.: On the multihomogeneous Bézout bound on the number of embeddings of minimally rigid graphs. Appl. Algebra Eng. Commun. Comput. 31(5–6), 325–357 (2020)

    Article  Google Scholar 

  5. Bartzos, E., Emiris, I.Z., Tzamos, Ch.: The m-Bézout bound and distance geometry. In: Computer Algebra in Scientific Computing—23rd Intern. Workshop (Sochi 2021). Lecture Notes in Computer Science, vol. 12865, pp. 6–20. Springer, Cham (2021)

  6. Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: 11th Annual European Symposium on Algorithms (Budapest 2003). Lecture Notes in Computer Science, vol. 2832, pp. 78–89. Springer, Berlin (2003)

  7. Bernstein, D.N.: The number of roots of a system of equations. Funct. Anal. Appl. 9(3), 183–185 (1975)

    MathSciNet  Article  Google Scholar 

  8. Billinge, S.J.L., Duxbury, P.M., Gonçalves, D.S., Lavor, C., Mucherino, A.: Assigned and unassigned distance geometry: applications to biological molecules and nanostructures. 4OR 14(4), 337–376 (2016)

    MathSciNet  Article  Google Scholar 

  9. Blumenthal, L.M.: Theory and Applications of Distance Geometry. Chelsea, New York (1970)

    MATH  Google Scholar 

  10. Borcea, C., Streinu, I.: The number of embeddings of minimally rigid graphs. Discrete Comput. Geom. 31(2), 287–303 (2004)

    MathSciNet  Article  Google Scholar 

  11. Bregman, L.M.: Some properties of nonnegative matrices and their permanents. Dokl. Akad. Nauk SSSR 211(1), 27–30 (1973). (in Russian)

  12. Capco, J., Gallet, M., Grasegger, G., Koutschan, Ch., Lubbes, N., Schicho, J.: The number of realizations of a Laman graph. SIAM J. Appl. Algebra Geom. 2(1), 94–125 (2018)

    MathSciNet  Article  Google Scholar 

  13. Emiris, I.Z., Tsigaridas, E.P., Varvitsiotis, A.: Mixed volume and distance geometry techniques for counting Euclidean embeddings of rigid graphs. In: Distance Geometry, pp. 23–45. Springer, New York (2013)

  14. Emmerich, D.G.: Structures Tendues et Autotendantes. Editions de La Villette, Paris (1988)

    Google Scholar 

  15. Felsner, S., Zickfeld, F.: On the number of planar orientations with prescribed degrees. Electron. J. Combin. 15(1), # 77 (2008)

  16. Gallet, M., Grasegger, G., Schicho, J.: Counting realizations of Laman graphs on the sphere. Electron. J. Combin. 27(2), # 2.5 (2020)

  17. Gáspár, M.E., Csermely, P.: Rigidity and flexibility of biological networks. Brief. Funct. Genom. 11(6), 443–456 (2012)

    Article  Google Scholar 

  18. Grasegger, G., Koutschan, Ch., Tsigaridas, E.: Lower bounds on the number of realizations of rigid graphs. Exp. Math. 29(2), 125–136 (2020)

    MathSciNet  Article  Google Scholar 

  19. Harary, F.: Graph Theory. Addison-Wesley, London (1969)

    Book  Google Scholar 

  20. Harris, J., Tu, L.W.: On symmetric and skew-symmetric determinantal varieties. Topology 23(1), 71–84 (1984)

    MathSciNet  Article  Google Scholar 

  21. Jackson, B., Owen, J.C.: Equivalent realisations of a rigid graph. Discrete Appl. Math. 256, 42–58 (2019)

    MathSciNet  Article  Google Scholar 

  22. Jungnickel, D.: Graphs, Networks and Algorithms. Algorithms and Computation in Mathematics, vol. 5. Springer, Berlin (2005)

  23. Khovanskii, A.G.: Newton polyhedra and the genus of complete intersections. Funct. Anal. Appl. 12(1), 38–46 (1978)

    MathSciNet  Article  Google Scholar 

  24. Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32(1), 1–31 (1976)

    MathSciNet  Article  Google Scholar 

  25. Krick, L., Broucke, M.E., Francis, B.A.: Stabilisation of infinitesimally rigid formations of multi-robot networks. Int. J. Control 82(3), 423–439 (2009)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  27. Maxwell, J.C.: On the calculation of the equilibrium and stiffness of frames. Philos. Mag. 27(182), 294–299 (1864)

    Article  Google Scholar 

  28. Minc, H.: Upper bounds for permanents of \((0,1)\)-matrices. Bull. Am. Math. Soc. 69, 789–791 (1963)

    MathSciNet  Article  Google Scholar 

  29. Pollaczek-Geiringer, H.: Über die Gliederung ebener Fachwerke. Zeitschrift für Angewandte Mathematik und Mechanik 7(1), 58–72 (1927)

    Article  Google Scholar 

  30. Pollaczek-Geiringer, H.: Zur Gliederungstheorie räumlicher Fachwerke. Zeitschrift für Angewandte Mathematik und Mechanik 12(6), 369–376 (1932)

    Article  Google Scholar 

  31. Schulze, B., Whiteley, W.: Rigidity and scene analysis. In: Handbook of Discrete and Computational Geometry. CRC Press Ser. Discrete Math. Appl., pp. 1593–1632. CRC Press, Boca Raton (1997)

  32. Shafarevich, I.R.: Basic Algebraic Geometry 1. Varieties in Projective Space. Springer, Heidelberg (2013)

    Book  Google Scholar 

  33. Shai, O., Sljoka, A., Whiteley, W.: Directed graphs, decompositions, and spatial linkages. Discrete Appl. Math. 161(18), 3028–3047 (2013)

    MathSciNet  Article  Google Scholar 

  34. Steffens, R., Theobald, T.: Mixed volume techniques for embeddings of Laman graphs. Comput. Geom. 43(2), 84–93 (2010)

    MathSciNet  Article  Google Scholar 

  35. Streinu, I., Theran, L.: Sparse hypergraphs and pebble game algorithms. Eur. J. Combin. 30(8), 1944–1964 (2009)

    MathSciNet  Article  Google Scholar 

  36. Tay, T.-S., Whiteley, W.: Generating isostatic frameworks. Struct. Topol. 11, 21–69 (1985)

    MathSciNet  MATH  Google Scholar 

  37. Whiteley, W.: Cones, infinity and \(1\)-story buildings. Struct. Topol. 8, 53–70 (1983)

    MathSciNet  MATH  Google Scholar 

  38. Zelazo, D., Franchi, A., Allgöwer, F., Bülthoff, H.H., Giordano, P.R.: Rigidity maintenance control for multi-robot systems. In: Robotics: Science and Systems VIII (Sydney 2012), # 60. MIT Press, Cambridge (2013)

  39. Zhu, Z., So, A.M.-C., Ye, Y.: Universal rigidity and edge sparsification for sensor network localization. SIAM J. Optim. 20(6), 3059–3081 (2010)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

EB was fully supported and IZE was partially supported by project ARCADES which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No 675789. EB and IZE are members of team AROMATH, joint between INRIA Sophia-Antipolis, France, and NKUA.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Evangelos Bartzos.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Editor in Charge: Kenneth Clarkson

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bartzos, E., Emiris, I.Z. & Vidunas, R. New Upper Bounds for the Number of Embeddings of Minimally Rigid Graphs. Discrete Comput Geom 68, 796–816 (2022). https://doi.org/10.1007/s00454-022-00370-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-022-00370-3

Keywords

  • Distance geometry
  • Minimally rigid graph
  • Rigid embedding
  • Upper bound
  • Laman graph
  • Oriented graph

Mathematics Subject Classification

  • 52C25
  • 14N10