Planar minimally rigid graphs and pseudo-triangulations

  • Authors:

  • Ruth Haas;David Orden;Günter Rote;Francisco Santos;Brigitte Servatius;Herman Servatius;Diane Souvaine;Ileana Streinu;Walter Whiteley

  • Affiliations:

  • Department of Mathematics, Smith College, Northampton, MA;Departamento de Matemáticas, University of Alcalá de Henares, E-28871 Alcalá de Henares, Spain;Institut für Informatik, Freie Universität Berlin, Takustraße 9, D-14195 Berlin, Germany;Departamento de Matemáticas, Estadistica y Computacion, Universidad de Cantabria, E-39005 Santander, Spain;Mathematics Department, Worcester Polytechnic Institute, Worcester MA;Mathematics Department, Worcester Polytechnic Institute, Worcester MA;Department of Computer Science, Tufts University, Medford, MA;Department of Computer Science, Smith College, Northampton, MA;Department of Mathematics and Statistics, York University, Toronto, Canada

  • Venue:
  • Computational Geometry: Theory and Applications - Special issue on the 19th annual symposium on computational geometry - SoCG 2003
  • Year:
  • 2005

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Abstract

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent to an angle larger than π). 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.