Optimally Bracing Grid Frameworks with Holes

  • Yoshihiko Ito
  • Yuki KobayashiEmail author
  • Yuya Higashikawa
  • Naoki Katoh
  • Sheung-Hung Poon
  • Maria Saumell
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8881)


We consider the bracing problem of a square grid framework possibly with holes and present an efficient algorithm for making the framework infinitesimally rigid by augmenting it with the minimum number of diagonal braces. This number of braces matches the lower bound given by Gáspár, Radics and Recski [2]. Our contribution extends the famous result on bracing the rectangular grid framework by Bolker and Crapo [1].


Combinatorial rigidity Bar-joint framework Square grid framework Bracing 


  1. 1.
    Bolker, E.D., Crapo, H.: Bracing Rectangular Frameworks. I. SIAM J. Appl. Math. 36(3), 473–490 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Gáspár, Z., Radics, N., Recski, A.: Rigidity of Square Grids with Holes. Comput. Assist. Mech. Eng. Sci. 6(3–4), 329–335 (1999)zbMATHGoogle Scholar
  3. 3.
    Henneberg, L.: Die Graphische Statik der Starren System, Leipzig 1911, Johnson Reprint 1968Google Scholar
  4. 4.
    Maxwell, J.C.: On the Calculation of the Equilibrium and Stiffness of Frames. Phil. Mag. 27, 294–299 (1864)Google Scholar
  5. 5.
    Radics, N., Recski, A.: Applications of Combinatorics to Statics - Rigidity of Grids. Discrete Appl. Math. 123, 473–485 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bar-Yehuda, R., Ben-Hanoch, E.: A Linear-Time Algorithm for Covering Simple Polygons with Similar Rectangles. Int. J. Comput. Geom. Appl. 6(1), 79–102 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Whiteley, W.: Some Matroids from Discrete Applied Geometry. AMS Contemp. Math. 197, 171–313 (1996)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Yoshihiko Ito
    • 1
  • Yuki Kobayashi
    • 1
    Email author
  • Yuya Higashikawa
    • 1
  • Naoki Katoh
    • 1
  • Sheung-Hung Poon
    • 2
  • Maria Saumell
    • 3
  1. 1.Department of Architecture and Architectural EngineeringKyoto UniversityKyotoJapan
  2. 2.Department of Computer ScienceNational Tsing Hua UniversityHsinchuTaiwan
  3. 3.Department of Mathematics and European Centre of Excellence NTISUniversity of West BohemiaPilsenCzech Republic

Personalised recommendations