Ruth Haas, David Orden, Günter Rote, Francisco Santos, Brigitte Servatius, Herman Servatius, Diane Souvaine, Ileana Streinu, and Walter Whiteley:

Planar minimally rigid graphs and pseudo-triangulations

  1. in: Proceedings of the Nineteenth Annual Symposium on Computational Geometry, San Diego, June 8–10, 2003. Association for Computing Machinery, 2003, pp. 154–163. doi:10.1145/777792.777817
  2. Computational Geometry, Theory and Applications 31 (2005), 31–61. doi:10.1016/j.comgeo.2004.07.003, arXiv:math.CO/0307347. (This is an extended version of parts of the results of 1.)


Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent to an angle larger than pi). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces. These constraints are described by combinatorial pseudo-triangulations, first defined and studied in this paper. Also of interest are our two proof techniques, one based on Henneberg inductive constructions from combinatorial rigidity theory, the other on a generalization of Tutte's barycentric embeddings to directed graphs.
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Last update: April 5, 2005.