Graphs and Combinatorics

, Volume 23, Supplement 1, pp 117–134 | Cite as

Enumerating Non-crossing Minimally Rigid Frameworks

  • David Avis
  • Naoki Katoh
  • Makoto Ohsaki
  • Ileana Streinu
  • Shin-ichi Tanigawa


In this paper, we present an algorithm for enumerating without repetitions all the non-crossing generically minimally rigid bar-and-joint frameworks (simply called non-crossing Laman frameworks) on a given generic set of n points. Our algorithm is based on the reverse search paradigm of Avis and Fukuda. It generates each output graph in O(n4) time and O(n) space, or, with a slightly different implementation, in O(n3) time and O(n2) space. In particular, we obtain that the set of all non-crossing Laman frameworks on a given point set is connected by flips which remove an edge and then restore the Laman property with the addition of a non-crossing edge.


Geometric enumeration Rigidity Non-crossing minimally rigid frameworks Minimally rigid graph 


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Copyright information

© Springer-Verlag Tokyo 2007

Authors and Affiliations

  • David Avis
    • 1
  • Naoki Katoh
    • 2
  • Makoto Ohsaki
    • 2
  • Ileana Streinu
    • 3
  • Shin-ichi Tanigawa
    • 2
  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada
  2. 2.Department of Architecture and Architectural EngineeringKyoto UniversityKyotoJapan
  3. 3.Department of Computer ScienceSmith CollegeNorthamptonUSA

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