A Henneberg-based algorithm for generating tree-decomposable minimally rigid graphs

In this work we describe an algorithm to generate tree-decomposable minimally rigid graphs on a given set of vertices V. The main idea is based on the well-known fact that all minimally rigid graphs, also known as Laman graphs, can be generated via Henneberg sequences. Given that not each minimally rigid graph is tree-decomposable, we identify a set of conditions on the way Henneberg steps are applied so that the resulting graph is tree-decomposable. We show that the worst case running time of the algorithm is O ( | V | 3 ) .

[1]  Christoph M. Hoffmann,et al.  Decomposition Plans for Geometric Constraint Systems, Part I: Performance Measures for CAD , 2001, J. Symb. Comput..

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

[3]  Sebastià Vila-Marta,et al.  Transforming an under-constrained geometric constraint problem into a well-constrained one , 2003, SM '03.

[4]  Jan Willem Klop,et al.  Term Rewriting Systems: From Church-Rosser to Knuth-Bendix and Beyond , 1990, ICALP.

[5]  Alfred V. Aho,et al.  On Finding Lowest Common Ancestors in Trees , 1976, SIAM J. Comput..

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

[7]  C. Hoffmann,et al.  A Brief on Constraint Solving , 2005 .

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

[9]  Michael A. Bender,et al.  The LCA Problem Revisited , 2000, LATIN.

[10]  András Frank,et al.  An Extension of a Theorem of Henneberg and Laman , 2001, IPCO.

[11]  Christoph M. Hoffmann,et al.  Decomposition Plans for Geometric Constraint Problems, Part II: New Algorithms , 2001, J. Symb. Comput..

[12]  John C. Owen,et al.  The Nonsolvability by Radicals of Generic 3-connected Planar Graphs , 2002, Automated Deduction in Geometry.

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

[14]  W. Whiteley,et al.  Generating Isostatic Frameworks , 1985 .

[15]  Robert Joan-Arinyo,et al.  A correct rule-based geometric constraint solver , 1997, Comput. Graph..

[16]  John E. Savage,et al.  Models of computation - exploring the power of computing , 1998 .

[17]  Ileana Streinu,et al.  The Number of Embeddings of Minimally Rigid Graphs , 2004, Discret. Comput. Geom..

[18]  Brigitte Servatius,et al.  Rigidity, global rigidity, and graph decomposition , 2010, Eur. J. Comb..

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

[20]  Sebastià Vila-Marta,et al.  On the domain of constructive geometric constraint solving techniques , 2001, Proceedings Spring Conference on Computer Graphics.

[21]  Gilles Trombettoni,et al.  Decomposition of Geometric Constraint Systems: a Survey , 2006, Int. J. Comput. Geom. Appl..

[22]  H. Garcia,et al.  Geometric constraint solving in a dynamic geometry framework. , 2013 .

[23]  J. C. Owen,et al.  The non-solvability by radicals of generic 3-connected planar Laman graphs , 2006 .

[24]  Christoph M. Hoffmann,et al.  A graph-constructive approach to solving systems of geometric constraints , 1997, TOGS.