Algorithms for the d-Dimensional Rigidity Matroid of Sparse Graphs

  • Sergey Bereg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3742)


Let \({\mathcal R}_{d}(G)\) be the d-dimensional rigidity matroid for a graph G=(V,E). Combinatorial characterization of generically rigid graphs is known only for the plane d=2 [11]. Recently Jackson and Jordán [5] derived a min-max formula which determines the rank function in \({\mathcal R}_{d}(G)\) when G is sparse, i.e. has maximum degree at most d + 2 and minimum degree at most d + 1.

We present three efficient algorithms for sparse graphs G that

(i) detect if E is independent in the rigidity matroid for G, and

(ii) construct G using vertex insertions preserving if G is isostatic, and

(iii) compute the rank of \({\mathcal R}_{d}(G)\).

The algorithms have linear running time assuming that the dimension d is fixed.


Linear Time Connected Graph Rank Function Sparse Graph Rigidity Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Connelly, R., Demaine, E.D., Rote, G.: Straightening polygonal arcs and convexifying polygonal cycles. In: Proc. 41th Annu. Sympos. on Found. of Computer Science, pp. 432–442 (2000)Google Scholar
  2. 2.
    Eren, T., Anderson, B.D., Whiteley, W., Morse, A.S., Belhumeur, P.N.: Information structures to control formation splitting and merging. In: Proc. of the American Control Conference (2004) (to appear)Google Scholar
  3. 3.
    Eren, T., Whiteley, W., Morse, A.S., Belhumeur, P.N., Anderson, B.D.: Sensor and network topologies of formations with direction, bearing and angle information between agents. In: Proc. of the 42nd IEEE Conference on Decision and Control, pp. 3064–3069 (2003)Google Scholar
  4. 4.
    Graver, J., Servatius, B., Servatius, H.: Combinatorial Rigidity. Graduate Studies in Mathematics, Amer. Math. Soc., Providence 2 (1993)Google Scholar
  5. 5.
    Jackson, B., Jordán, T.: The d-dimensional rigidity matroid of sparse graphs. Technical Report TR-2003-06, EGRES Technical Report Series (2003)Google Scholar
  6. 6.
    Jacobs, D., Rader, A.J., Kuhn, L., Thorpe, M.: Protein flexibility predictions using graph theory. Proteins 44, 150–165 (2001)CrossRefGoogle Scholar
  7. 7.
    Laman, G.: On graphs and rigidity of plane skeletal structures. J. Engineering Math. 4, 331–340 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Streinu, I.: A combinatorial approach to planar non-colliding robot arm motion planning. Proc. 41st Annu. IEEE Sympos. Found. Comput. Sci., 443–453 (2000)Google Scholar
  9. 9.
    Whiteley, W.: Some matroids from discrete applied geometry. In: Bonin, J.E., Oxley, J.G., Servatius, B. (eds.) Contemp. Mathematics, Seattle, WA, vol. 197, pp. 171–311. Amer. Math. Soc., Providence (1997)Google Scholar
  10. 10.
    Whiteley, W.: Rigidity of molecular structures: generic and geometric analysis. In: Thorpe, M.F., Duxbury, P.M. (eds.) Rigidity Theory and Applications, pp. 21–46. Kluwer, Dordrecht (1999)Google Scholar
  11. 11.
    Whiteley, W.: Rigidity and scene analysis. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, ch. 60, pp. 1327–1354. CRC Press LLC, Boca Raton (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Sergey Bereg
    • 1
  1. 1.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA

Personalised recommendations