We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our interpretation of realizations allows us to provide a combinatorial characterization of these graphs in terms of the existence of particular colorings of the edges. Moreover, we determine necessary relations for flexibility between the spherical lengths of the edges. We conclude by classifying all possible motions on the sphere of the complete bipartite graph with 3+3 vertices where no two vertices coincide or are antipodal.


  1. graph
  2. sphere
  3. Dixon
  4. flexibility
  5. moduli space

MSC codes

  1. 05C99
  2. 53A17
  3. 70B99
  4. 51F99

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V. Alexandrov, An example of a flexible polyhedron with nonconstant volume in the spherical space, Beitr. Algebra Geom., 38 (1997), pp. 11--18.
O. Bottema, Die Bahnkurven eines merkwürdigen Zwölfstabgetriebes, Österreichisches Ingenieur-Archiv, 14 (1960), pp. 218--222.
D. Corinaldi, M. Callegari, and J. Angeles, Singularity-free path-planning of dexterous pointing tasks for a class of spherical parallel mechanisms, Mech. Mach. Theory, 128 (2018), pp. 47--57, https://doi.org/10.1016/j.mechmachtheory.2018.05.006.
A. C. Dixon, On certain deformable frameworks, Messenger, 29 (1899), pp. 1--21.
Y. Eftekhari, B. Jackson, A. Nixon, B. Schulze, S.-i. Tanigawa, and W. Whiteley, Point-hyperplane frameworks, slider joints, and rigidity preserving transformations, J. Combin. Theory Ser. B, 135 (2019), pp. 48--74, https://doi.org/10.1016/j.jctb.2018.07.008.
T. Essomba and L. N. Vu, Kinematic analysis of a new five-bar spherical decoupled mechanism with two-degrees of freedom remote center of motion, Mech. Mach. Theory, 119 (2018), pp. 184--197, https://doi.org/10.1016/j.mechmachtheory.2017.09.010.
A. A. Gaifullin, Flexible cross-polytopes in spaces of constant curvature, Proc. Steklov Inst. Math., 286 (2014), pp. 77--113, https://doi.org/10.1134/S0081543814060066.
A. A. Gaifullin, Embedded flexible spherical cross-polytopes with nonconstant volumes, Proc. Steklov Inst. Math., 288 (2015), pp. 56--80, https://doi.org/10.1134/S0081543815010058.
M. Gallet, G. Grasegger, and J. Schicho, Counting realizations of Laman graphs on the sphere, Electron. J. Combin., 27 (2020), P2.5, https://doi.org/10.37236/8548.
C. G. Gibson and J. M. Selig, Movable hinged spherical quadrilaterals---I, Mech. Mach. Theory, 23 (1988), pp. 13--18, https://doi.org/10.1016/0094-114X(88)90004-3.
C. G. Gibson and J. M. Selig, Movable hinged spherical quadrilaterals---II Singularities and reductions, Mech. Mach. Theory, 23 (1988), pp. 19--24, https://doi.org/10.1016/0094-114X(88)90005-5.
G. Grasegger, J. Legerský, and J. Schicho, Graphs with Flexible Labelings, Discrete Comput. Geom., 62 (2019), pp. 461--480, https://doi.org/10.1007/s00454-018-0026-9.
G. Grasegger, J. Legerský, and J. Schicho, Graphs with flexible labelings allowing injective realizations, Discrete Math., 343 (2020), 111713, https://doi.org/10.1016/j.disc.2019.111713.
G. Hegedüs, Z. Li, J. Schicho, and H.-P. Schröcker, The theory of bonds II: Closed 6R linkages with maximal genus, J. Symbolic Comput., 68 (2015), pp. 167--180, https://doi.org/10.1016/j.jsc.2014.09.035.
G. Hegedüs, J. Schicho, and H.-P. Schröcker, The theory of bonds: A new method for the analysis of linkages, Mech. Mach. Theory, 70 (2013), pp. 407--424, https://doi.org/10.1016/j.mechmachtheory.2013.08.004.
I. Izmestiev, Projective background of the infinitesimal rigidity of frameworks, Geom. Dedicata, 140 (2009), pp. 183--203, https://doi.org/10.1007/s10711-008-9339-9.
S. Keel, Intersection theory of moduli space of stable $n$-pointed curves of genus zero, Trans. Amer. Math. Soc., 330 (1992), pp. 545--574, https://doi.org/10.2307/2153922.
F. F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks $M_{g,n}$, Math. Scand., 52 (1983), pp. 161--199, https://doi.org/10.7146/math.scand.a-12001.
G. Laman, On graphs and rigidity of plane skeletal structures, J. Engrg. Math., 4 (1970), pp. 331--340, https://doi.org/10.1007/BF01534980.
Z. Li, J. Schicho, and H.-P. Schröcker, A survey on the theory of bonds, IMA J. Math. Control Inform., 35 (2018), pp. 279--295, https://doi.org/10.1093/imamci/dnw048.
G. Nawratil, Reducible compositions of spherical four-bar linkages with a spherical coupler component, Mech. Mach. Theory, 46 (2011), pp. 725--742, https://doi.org/10.1016/j.mechmachtheory.2010.12.004.
G. Nawratil, Reducible compositions of spherical four-bar linkages without a spherical coupler component, Mech. Mach. Theory, 49 (2012), pp. 87--103, https://doi.org/10.1016/j.mechmachtheory.2011.11.003.
G. Nawratil and H. Stachel, Composition of spherical four-bar-mechanisms, in New Trends in Mechanism Science, D. Pisla, M. Ceccarelli, M. Husty, and B. Corves, eds., Dordrecht, Amsterdam, 2010, pp. 99--106.
A. Nixon, J. C. Owen, and S. C. Power, Rigidity of frameworks supported on surfaces, SIAM J. Discrete Math., 26 (2012), pp. 1733--1757, https://doi.org/10.1137/110848852.
A. Nixon, J. C. Owen, and S. C. Power, A characterization of generically rigid frameworks on surfaces of revolution, SIAM J. Discrete Math., 28 (2014), pp. 2008--2028, https://doi.org/10.1137/130913195.
A. Nixon and E. Ross, One brick at a time: A survey of inductive constructions in rigidity theory, in Rigidity and Symmetry, Springer, New York, 2014, pp. 303--324, https://doi.org/10.1007/978-1-4939-0781-6_15.
H. Pollaczek-Geiringer, Über die Gliederung ebener Fachwerke, ZAMM Z. Angew. Math. Mech., 7 (1927), pp. 58--72, https://doi.org/10.1002/zamm.19270070107.
E. Ross, The rigidity of periodic body-bar frameworks on the three-dimensional fixed torus, Philos. Trans. A, 372 (2014), https://doi.org/10.1098/rsta.2012.0112.
E. Ross, Inductive constructions for frameworks on a two-dimensional fixed torus, Discrete Comput. Geom., 54 (2015), pp. 78--109, https://doi.org/10.1007/s00454-015-9697-7.
F. V. Saliola and W. Whiteley, Some Notes on the Equivalence of First-Order Rigidity in Various Geometries, Tech. report, https://arxiv.org/abs/0709.3354, 2007.
H. Stachel, Flexible cross-polytopes in the Euclidean 4-space, J. Geom. Graph., 4 (2001), pp. 159--167.
H. Stachel, Flexible octahedra in the hyperbolic space, in Non-Euclidean Geometries: János Bolyai Memorial Volume, A. Prékopa and E. Molnár, eds., Springer, New York, 2006, pp. 209--225, https://doi.org/10.1007/0-387-29555-0_11.
H. Stachel, On the flexibility and symmetry of overconstrained mechanisms, Philos. Trans. A, 372 (2013), https://doi.org/10.1098/rsta.2012.0040.
J. Sun, W. Liu, and J. Chu, Synthesis of spherical four-bar linkage for open path generation using wavelet feature parameters, Mech. Mach. Theory, 128 (2018), pp. 33--46, https://doi.org/10.1016/j.mechmachtheory.2018.05.008.
D. Walter and M. L. Husty, On a nine-bar linkage, its possible configurations and conditions for paradoxical mobility, in Proceedings of the 12th World Congress on Mechanism and Machine Science, 2007.
W. Wunderlich, On deformable nine-bar linkages with six triple joints, Indag. Math. (Proceedings), 79 (1976), pp. 257--262, https://doi.org/10.1016/1385-7258(76)90052-4.

Information & Authors


Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 325 - 361
ISSN (online): 1095-7146


Submitted: 24 September 2019
Accepted: 16 November 2020
Published online: 11 March 2021


  1. graph
  2. sphere
  3. Dixon
  4. flexibility
  5. moduli space

MSC codes

  1. 05C99
  2. 53A17
  3. 70B99
  4. 51F99



Funding Information

Austrian Science Fund https://doi.org/10.13039/501100002428 : W1214-N15
H2020 Marie Skłodowska-Curie Actions https://doi.org/10.13039/100010665 : 675789
Erwin Schrödinger International Institute for Mathematics and Physics https://doi.org/10.13039/501100003066 : J425

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