Planar minimally rigid graphs and pseudo-triangulations

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than p). 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---to the best of our knowledge---the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.

[1]  Ileana Streinu,et al.  A combinatorial approach to planar non-colliding robot arm motion planning , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[2]  W. Whiteley.  Rigidity of Molecular Structures: Generic and Geometric Analysis , 2002 .

[3]  Michel Pocchiola,et al.  Pseudo-triangulations: theory and applications , 1996, SCG '96.

[4]  Ileana Streinu,et al.  Combinatorial Roadmaps in Configuration Spaces of Simple Planar Polygons , 2001, Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science.

[5]  Brigitte Servatius,et al.  The Structure of Locally Finite Two-Connected Graphs , 1995, Electron. J. Comb..

[6]  Tibor Jordán,et al.  A proof of Connelly's conjecture on 3-connected circuits of the rigidity matroid , 2003, J. Comb. Theory, Ser. B.

[7]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[8]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[9]  Walter Whiteley,et al.  Rigidity and scene analysis , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[10]  Michel Pocchiola,et al.  Tutte's barycenter method applied to isotopies , 2003, Comput. Geom..

[11]  W. T. Tutte.  Connectivity in graphs , 1966 .

[12]  Ileana Streinu,et al.  Stretchability of star-like pseudo-visibility graphs , 1999, SCG '99.

[13]  Bettina Speckmann,et al.  Tight degree bounds for pseudo-triangulations of points , 2001, CCCG.

[14]  Günter Rote,et al.  Counting triangulations and pseudo-triangulations of wheels , 2001, CCCG.

[15]  Bettina Speckmann,et al.  The Zigzag Path of a Pseudo-Triangulation , 2003, WADS.

[16]  D. Jacobs,et al.  Protein flexibility predictions using graph theory , 2001, Proteins.

[17]  Michel Pocchiola,et al.  Topologically sweeping visibility complexes via pseudotriangulations , 1996, Discret. Comput. Geom..

[18]  Leonidas J. Guibas,et al.  Deformable Free-Space Tilings for Kinetic Collision Detection† , 2002, Int. J. Robotics Res..

[19]  A. Stephen Morse,et al.  Sensor and Network Topologies of Formations with distance-direction-angle constraints , 2003 .

[20]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[21]  W. T. Tutte.  Convex Representations of Graphs , 1960 .

[22]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[23]  B. Aronov,et al.  Discrete and computational geometry : the Goodman-Pollack Festschrift , 2003 .

[24]  G. Rothe.  Two solvable cases of the Traveling Salesman Problem , 1988 .

[25]  H. Whitney.  2-Isomorphic Graphs , 1933 .

[26]  Jack Snoeyink,et al.  Counting and enumerating pseudo-triangulations with the greedy flip algorithm ∗ , 2005 .

[27]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[28]  Bettina Speckmann,et al.  On the Number of Pseudo-Triangulations of Certain Point Sets , 2003, CCCG.

[29]  C. Gotsman,et al.  How to morph tilings injectively , 1999 .

[30]  Sergey Bereg,et al.  Enumerating pseudo-triangulations in the plane , 2005, Comput. Geom..

[31]  Lebrecht Henneberg,et al.  Die graphische Statik der Starren Systeme , 1911 .

[32]  János Pach,et al.  Monotone drawings of planar graphs , 2002, J. Graph Theory.

[33]  Stefan Felsner,et al.  Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes , 2001, Order.

[34]  Gert Vegter,et al.  In handbook of discrete and computational geometry , 1997 .

[35]  Ioannis G. Tollis,et al.  Graph Drawing , 1994, Lecture Notes in Computer Science.

[36]  Marek Chrobak,et al.  Convex drawings of graphs in two and three dimensions (preliminary version) , 1996, SCG '96.

[37]  Günter Rote,et al.  Blowing Up Polygonal Linkages , 2003 .

[38]  Günter Rote,et al.  Planar minimally rigid graphs and pseudo-triangulations , 2005, Comput. Geom..

[39]  G. Laman.  On graphs and rigidity of plane skeletal structures , 1970 .

[40]  Francisco Santos,et al.  The Polytope of Non-Crossing Graphs on a Planar Point Set , 2005, Discret. Comput. Geom..

[41]  Francisco Santos,et al.  The polytope of non-crossing graphs on a planar point set , 2004, ISSAC '04.

[42]  William T. Tutte,et al.  A theory of 3-connected graphs , 1961 .

[43]  Bettina Speckmann,et al.  Allocating Vertex π-Guards in Simple Polygons via Pseudo-Triangulations , 2003, SODA '03.

[44]  Kokichi Sugihara,et al.  Detection of structural inconsistency in systems of equations with degrees of freedom and its applications , 1985, Discret. Appl. Math..

[45]  Joseph O'Rourke,et al.  Handbook of Discrete and Computational Geometry, Second Edition , 1997 .

[46]  János Pach,et al.  How to draw a planar graph on a grid , 1990, Comb..

[47]  M. Thorpe,et al.  Rigidity theory and applications , 2002 .

[48]  Walter Schnyder,et al.  Embedding planar graphs on the grid , 1990, SODA '90.

[49]  Robert E. Tarjan,et al.  Efficient Planarity Testing , 1974, JACM.

[50]  F. Frances Yao,et al.  Computational Geometry , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[51]  Franz Aurenhammer,et al.  Convexity minimizes pseudo-triangulations , 2002, CCCG.

[52]  Joseph O'Rourke,et al.  Vertex-edge pseudo-visibility graphs: characterization and recognition , 1997, SCG '97.

[53]  Robert E. Tarjan,et al.  Applications of a planar separator theorem , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[54]  D. Rose,et al.  Generalized nested dissection , 1977 .

[55]  Walter Whiteley,et al.  Plane Self Stresses and projected Polyhedra I: The Basic Pattem , 1993 .

[56]  Hiroshi Imai,et al.  On combinatorial structures of line drawings of polyhedra , 1985, Discret. Appl. Math..

[57]  A. Stephen Morse,et al.  Operations on Rigid Formations of Autonomous Agents , 2003, Commun. Inf. Syst..

[58]  Günter Rote,et al.  Non-Crossing Frameworks with Non-Crossing Reciprocals , 2004, Discret. Comput. Geom..

[59]  Günter Rote,et al.  Straightening polygonal arcs and convexifying polygonal cycles , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[60]  Francisco Santos,et al.  Expansive Motions and the Polytope of Pointed Pseudo-Triangulations , 2002 .

[61]  Walter Whiteley,et al.  Some matroids from discrete applied geometry , 1996 .

[62]  Jürgen Richter-Gebert.  Realization Spaces of Polytopes , 1996 .