Graphs with Flexible Labelings

  • Georg Grasegger
  • Jan LegerskýEmail author
  • Josef Schicho


For a flexible labeling of a graph, it is possible to construct infinitely many non-equivalent realizations keeping the distances of connected points constant. We give a combinatorial characterization of graphs that have flexible labelings, possibly non-generic. The characterization is based on colorings of the edges with restrictions on the cycles. Furthermore, we give necessary criteria and sufficient ones for the existence of such colorings.


Graph realization Flexibility Rigidity Linkage Laman graph 

Mathematics Subject Classification

51K99 70B99 05C78 



This Project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant Agreement No. 675789. Partially supported by the Austrian Science Fund (FWF): P26607, W1214-N15 (Project DK9); and by the Upper Austrian Government.


  1. 1.
    Deuring, M.: Lectures on the Theory of Algebraic Functions of One Variable. Lecture Notes in Mathematics, vol. 314. Springer, Berlin (1973)zbMATHGoogle Scholar
  2. 2.
    Dixon, A.: On certain deformable frameworks. Messenger 29(2), 1–21 (1899)zbMATHGoogle Scholar
  3. 3.
    Fekete, Z., Jordán, T., Kaszanitzky, V.E.: Rigid two-dimensional frameworks with two coincident points. Graphs Comb. 31(3), 585–599 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Graver, J.E., Servatius, B., Servatius, H.: Combinatorial Rigidity. Graduate Studies in Mathematics, vol. 2. American Mathematical Society (1993)Google Scholar
  5. 5.
    Jackson, B., Jordán, T., Servatius, B., Servatius, H.: Henneberg moves on mechanisms. Beitr. Algebra Geom. 56(2), 587–591 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Maehara, H., Tokushige, N.: When does a planar bipartite framework admit a continuous deformation? Theor. Comput. Sci. 263(1–2), 345–354 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Pollaczek-Geiringer, H.: Über die Gliederung ebener Fachwerke. Z. Angew. Math. Mech. 7, 58–72 (1927)CrossRefzbMATHGoogle Scholar
  9. 9.
    Stachel, H.: On the flexibility and symmetry of overconstrained mechanisms. Philos. Trans. R. Soc. Lond. A. 372(2008), 20120040 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Streinu, I., Theran, L.: Combinatorial genericity and minimal rigidity. In: Computational Geometry (SCG’08), pp. 365–374. ACM, New York (2008)Google Scholar
  11. 11.
    Walter, D., Husty, M.L.: On a nine-bar linkage, its possible configurations and conditions for paradoxical mobility. In: 12th World Congress on Mechanism and Machine Science (IFToMM’07) (2007)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria
  2. 2.Research Institute for Symbolic Computation (RISC)Johannes Kepler University LinzLinzAustria

Personalised recommendations