Skip to main content

Generic Rigidity with Forced Symmetry and Sparse Colored Graphs

  • Chapter
  • First Online:
Rigidity and Symmetry

Part of the book series: Fields Institute Communications ((FIC,volume 70))

Abstract

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    See, e.g., the recent conferences [9, 21, 40].

  2. 2.

    Infinite frameworks with no other assumptions can exhibit quite complicated behavior [33].

  3. 3.

    We are extending the terminology “Teichmüller space” from its more typical usage for the group \(\mathbb{Z}^{2}\) and lattices in \(\mathrm{PSL}(2, \mathbb{R})\). Our definition of \(\text{Teich}(\mathbb{Z}^{2})\) is non-standard since the usual one allows only unit-area fundamental domains.

  4. 4.

    Colored graphs are also known as “gain graphs” or “voltage graphs” [53]. The terminology of colored graphs originates from [36].

  5. 5.

    The sparsity counts we describe here are slightly different from what is stated in [38, Theorem 4.2.1], but they are equivalent by an argument similar to that in the proof of Proposition 4. This presentation highlights the connection to colored-Laman graphs.

References

  1. Berardi, M., Heeringa, B., Malestein, J., Theran, L.: Rigid components in fixed-latice and cone frameworks. In: Proceedings of the 23rd Annual Canadian Conference on Computational Geometry (CCCG), Toronto (2011). http://www.cccg.ca/proceedings/2011/papers/paper52.pdf

  2. Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume. Math. Ann. 70(3), 297–336 (1911). doi:10.1007/BF01564500

    Article  MATH  MathSciNet  Google Scholar 

  3. Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich. Math. Ann. 72(3), 400–412 (1912). doi:10.1007/BF01456724

    Google Scholar 

  4. Borcea, C.S., Streinu, I.: Periodic frameworks and flexibility. Proc. R. Soc. Lond. Ser. Math. Phys. Eng. Sci. 466(2121), 2633–2649 (2010). ISSN 1364-5021. doi:10.1098/rspa.2009.0676. http://dx.doi.org/10.1098/rspa.2009.0676

  5. Borcea, C.S., Streinu, I.: Minimally rigid periodic graphs. Bull. Lond. Math. Soc. 43(6), 1093–1103 (2011). http://dx.doi.org/10.1112/blms/bdr044

  6. Fekete, Z.: Source location with rigidity and tree packing requirements. Oper. Res. Lett. 34(6), 607–612 (2006). ISSN 0167-6377. doi:10.1016/j.orl.2005.10.005. http://dx.doi.org/10.1016/j.orl.2005.10.005

  7. Fekete, Z., Szegő, L.: A note on [k, l]-sparse graphs. In: Bondy, A., Fonlupt, J., Fouquet, J.-L., Fournier, J.-C., Ramrez Alfonsn, J.L. (eds.) Graph Theory in Paris. Trends Mathematics, pp. 169–177. Birkhäuser, Basel (2007). doi:10.1007/978-3-7643-7400-6_13. http://dx.doi.org/10.1007/978-3-7643-7400-6_13

  8. Fekete, Z., Jordán, T., Kaszanitzky, V.: Rigid two-dimensional frameworks with two coincident points. Talk at the Fields Institute Workshop of Rigidity (2011). http://www.fields.utoronto.ca/audio/11-12/wksp_rigidity/kaszanitzky/

  9. Conder, M., Connelly, R., Jordán, T., Monson, B., Schulte, E., Streinu, I., Weiss, A. I., Whiteley, W.: Fields Institute Workshop on Rigidity and Symmetry (2011). http://www.fields.utoronto.ca/programs/scientific/11-12/discretegeom/wksp_symmetry/

  10. Fowler, P.W., Guest, S.D.: A symmetry extension of Maxwell’s rule for rigidity of frames. Int. J. Solids Struct. 37(12), 1793–1804 (2000). ISSN 0020-7683. doi:10.1016/S0020-7683(98)00326-6. http://www.sciencedirect.com/science/article/pii/S0020768398003266

  11. Guest, S.D., Hutchinson, J.W.: On the determinacy of repetitive structures. J. Mech. Phys. Solids 51(3), 383–391 (2003). ISSN 0022-5096. doi:10.1016/S0022-5096(02)00107-2. http://www.sciencedirect.com/science/article/pii/S0022509602001072

  12. Haas, R.: Characterizations of arboricity of graphs. Ars Comb. 63, 129–137 (2002). ISSN 0381-7032

    Google Scholar 

  13. Haas, R., Lee, A., Streinu, I., Theran, L.: Characterizing sparse graphs by map decompositions. J. Comb. Math. Comb. Comput. 62, 3–11 (2007). ISSN 0835-3026

    Google Scholar 

  14. Jordán, T., Kaszanitzky, V., Tanigawa, S.: Gain-sparsity and symmetry-forced rigidity in the plane (2012, preprint). http://www.cs.elte.hu/egres/tr/egres-12-17.pdf

  15. Kangwai, R.D., Guest, S.D.: Detection of finite mechanisms in symmetric structures. Int. J. Solids Struct. 36(36), 5507–5527 (1999). ISSN 0020-7683. doi:10.1016/S0020-7683(98)00234-0. http://www.sciencedirect.com/science/article/pii/S0020768398002340

  16. Kangwai, R.D, Guest, S.D.: Symmetry-adapted equilibrium matrices. Int. J. Solids Struct. 37(11), 1525–1548 (2000). ISSN 0020-7683. doi:10.1016/S0020-7683(98)00318-7. http://www.sciencedirect.com/science/article/pii/S0020768398003187

  17. Katoh, N., Tanigawa, S.: A proof of the molecular conjecture. Discret. Comput. Geom. 45(4), 647–700 (2011). ISSN 0179-5376. doi:10.1007/s00454-011-9348-6. http://dx.doi.org/10.1007/s00454-011-9348-6

  18. Katoh, N., Tanigawa, S.: Rooted-tree decompositions with matroid constraints and the infinitesimal rigidity of frameworks with boundaries (2013). ISSN 0895-4801. http://dx.doi.org/10.1137/110846944

  19. Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970). ISSN 0022-0833

    Google Scholar 

  20. Lee, A., Streinu, I.: Pebble game algorithms and sparse graphs. Discret. Math. 308(8), 1425–1437 (2008). ISSN 0012-365X. doi:10.1016/j.disc.2007.07.104. http://dx.doi.org/10.1016/j.disc.2007.07.104

  21. London Mathematical Society Workshop: Rigidity of Frameworks and Applications (2010). http://www.maths.lancs.ac.uk/~power/LancRigidFrameworks.htm

  22. Lovász, L.: Matroid matching and some applications. J. Comb. Theory Ser. B 28(2), 208–236 (1980). ISSN 0095-8956. doi:10.1016/0095-8956(80)90066-0. http://dx.doi.org/10.1016/0095-8956(80)90066-0

  23. Lovász, L., Yemini, Y.: On generic rigidity in the plane. SIAM J. Algebr. Discret. Methods 3(1), 91–98 (1982). ISSN 0196-5212. doi:10.1137/0603009. http://dx.doi.org/10.1137/0603009

  24. Malestein, J., Theran, L.: Generic rigidity of frameworks with orientation-preserving crystallographic symmetry (2011, preprint). arXiv:1108.2518. http://arxiv.org/abs/1108.2518. The results of this preprint have been merged into [27, 28]

  25. Malestein, J., Theran, L.: Generic rigidity of frameworks with reflection symmetry (2012, preprint). arXiv:1203.2276. http://arxiv.org/abs/1203.2276. The results of this preprint have been merged into [28

  26. Malestein, J., Theran, L.: Generic combinatorial rigidity of periodic frameworks. Adv. Math. 233, 291–331 (2013). ISSN 0001-8708. doi:10.1016/j.aim.2012.10.007. http://dx.doi.org/10.1016/j.aim.2012.10.007

  27. Malestein, J., Theran, L.: Frameworks with forced symmetry II: orientation-preserving crystallographic groups. Geometriae Dedicata (2013). doi:10.1007/s10711-013-9878-6. http://dx.doi.org/10.1007/s10711-013-9878-6

  28. Malestein, J., Theran, L.: Frameworks with forced symmetry I: reflections and rotations (2013, preprint). arXiv:304.0398. http://arxiv.org/abs/1304.0398

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

    Google Scholar 

  30. Nixon, A., Owen, J.: An inductive construction of (2,1)-tight graphs (2011, preprint). arXiv:1103.2967. http://arxiv.org/abs/1103.2967

  31. Nixon, A., Owen, J.C., Power, S.C.: Rigidity of frameworks supported on surfaces. SIAM J. Discret. Math. 26(4), 1733–1757 (2012). ISSN 0895-4801. doi:10.1137/110848852. http://dx.doi.org/10.1137/110848852

  32. Owen, J.C., Power, S.C.: Frameworks symmetry and rigidity. Int. J. Comput. Geom. Appl. 20(6), 723–750 (2010). ISSN 0218-1959. doi:10.1142/S0218195910003505. http://dx.doi.org/10.1142/S0218195910003505

  33. Owen, J.C., Power, S.C.: Infinite bar-joint frameworks, crystals and operator theory. N. Y. J. Math. 17, 445–490 (2011). ISSN 1076-9803. http://nyjm.albany.edu:8000/j/2011/17_445.html

  34. Recski, A.: A network theory approach to the rigidity of skeletal structures. II. Laman’s theorem and topological formulae. Discret. Appl. Math. 8(1), 63–68 (1984). ISSN 0166-218X. doi:10.1016/0166-218X(84)90079-9. http://dx.doi.org/10.1016/0166-218X(84)90079-9

  35. Recski, A.: Matroid Theory and Its Applications in Electric Network Theory and in Statics. Volume 6 of Algorithms and Combinatorics. Springer, Berlin (1989). ISBN 3-540-15285-7

    Google Scholar 

  36. Rivin, I.: Geometric simulations – a lesson from virtual zeolites. Nat. Mater. 5(12), 931–932 (2006). doi:10.1038/nmat1792. http://dx.doi.org/10.1038/nmat1792

  37. Ross, E.: Periodic rigidity. Talk at the Spring AMS Sectional Meeting (2009). http://www.ams.org/meetings/sectional/1050-52-71.pdf

  38. Ross, E.: The Rigidity of Periodic Frameworks as Graphs on a Torus. PhD thesis, York University (2011). http://www.math.yorku.ca/~ejross/RossThesis.pdf

  39. Ross, E., Schulze, B., Whiteley, W.: Finite motions from periodic frameworks with added symmetry. Int. J. Solids Struct. 48(11–12), 1711–1729 (2011). ISSN 0020-7683. doi:10.1016/j.ijsolstr.2011.02.018. http://dx.doi.org/10.1016/j.ijsolstr.2011.02.018

  40. Guest, S., Fowler, P., Power S.: Royal Society Workshop on Rigidity of Periodic and Symmetric Structures in Nature and Engineering (2012). http://royalsociety.org/events/Rigidity-of-periodic-and-symmetric-structures/

  41. Schulze, B.: Symmetric Laman theorems for the groups \(\mathcal{C}_{2}\) and \(\mathcal{C}_{s}\). Electron. J. Comb. 17(1), Research Paper 154, 61 (2010). ISSN 1077-8926. http://www.combinatorics.org/Volume_17/Abstracts/v17i1r154.html

  42. Schulze, B.: Symmetric versions of Laman’s theorem. Discret. Comput. Geom. 44(4), 946–972 (2010). ISSN 0179-5376. doi:10.1007/s00454-009-9231-x. http://dx.doi.org/10.1007/s00454-009-9231-x

  43. Schulze, B., Whiteley, W.: The orbit rigidity matrix of a symmetric framework. Discret. Comput. Geom. 46(3), 561–598 (2011). ISSN 0179-5376. doi:10.1007/s00454-010-9317-5. http://dx.doi.org/10.1007/s00454-010-9317-5

  44. Servatius, B., Whiteley, W.: Constraining plane configurations in computer-aided design: combinatorics of directions and lengths. SIAM J. Discret. Math. 12(1), 136–153 (1999). (electronic) ISSN 0895-4801. doi:10.1137/S0895480196307342. http://dx.doi.org/10.1137/S0895480196307342

  45. Streinu, I., Theran, L.: Sparsity-certifying graph decompositions. Graphs Comb. 25(2), 219–238 (2009). ISSN 0911-0119. doi:10.1007/s00373-008-0834-4. http://dx.doi.org/10.1007/s00373-008-0834-4

  46. Streinu, I., Theran, L.: Slider-pinning rigidity: a Maxwell-Laman-type theorem. Discret. Comput. Geom. 44(4), 812–837 (2010). ISSN 0179-5376. doi: 10.1007/s00454-010-9283-y. http://dx.doi.org/10.1007/s00454-010-9283-y

  47. Streinu, I., Theran, L.: Natural realizations of sparsity matroids. Ars Math. Contemp. 4(1), 141–151 (2011). ISSN 1855-3966

    Google Scholar 

  48. Tanigawa, S.: Matroids of gain graphs in applied discrete geometry (2012, preprint). http://arxiv.org/abs/1207.3601

  49. Tay, T.-S.: Rigidity of multigraphs. I. Linking rigid bodies in n-space. J. Comb. Theory Ser. B 36(1), 95–112 (1984). ISSN 0095-8956. doi:10.1016/0095-8956(84)90016-9. http://dx.doi.org/10.1016/0095-8956(84)90016-9

  50. Treacy, M., Rivin, I., Balkovsky, E., Randall, K.: Enumeration of periodic tetrahedral frameworks. II. Polynodal graphs. Microporous Mesoporous Mater. 74, 121–132 (2004)

    Article  Google Scholar 

  51. Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM J. Discret. Math. 1(2), 237–255 (1988). ISSN 0895-4801. doi:10.1137/0401025. http://dx.doi.org/10.1137/0401025

  52. Whiteley, W.: Some matroids from discrete applied geometry. In: Bonin, J., Oxley, J.G., Servatius, B. (eds.) Matroid Theory. Volume 197 of Contemporary Mathematics, pp. 171–311. American Mathematical Society, Providence (1996)

    Google Scholar 

  53. Zaslavsky, T.: A mathematical bibliography of signed and gain graphs and allied areas. Electron. J. Comb. 5(Dynamic Surveys 8), 124pp (1998). (electronic) ISSN 1077-8926. http://www.combinatorics.org/Surveys/index.html. Manuscript prepared with Marge Pratt

Download references

Acknowledgements

We thank the Fields Institute for its hospitality during the Workshop on Rigidity and Symmetry, the workshop organizers for putting together the program, and the conference participants for many interesting discussions. LT is supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 247029-SDModels. JM partially supported by NSF CDI-I grant DMR 0835586 and (for finial preparation) the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 226135.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Justin Malestein .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

Malestein, J., Theran, L. (2014). Generic Rigidity with Forced Symmetry and Sparse Colored Graphs. In: Connelly, R., Ivić Weiss, A., Whiteley, W. (eds) Rigidity and Symmetry. Fields Institute Communications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0781-6_12

Download citation

Publish with us

Policies and ethics