The number of realizations of a Laman graph

Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Using ideas from algebraic and tropical geometry, we provide a recursion formula for the number of complex solutions of such systems.

[1]  G. Mikhalkin.  Enumerative tropical algebraic geometry , 2003 .

[2]  W. F. Meyer,et al.  Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen , 2022 .

[3]  Josef Schicho,et al.  The number of realizations of a Laman graph , 2017, Electron. Notes Discret. Math..

[4]  Sergey Bereg,et al.  Certifying and constructing minimally rigid graphs in the plane , 2005, SCG.

[5]  J. Faugère,et al.  Combinatorial classes of parallel manipulators , 1995 .

[6]  Leo Liberti,et al.  On the number of realizations of certain Henneberg graphs arising in protein conformation , 2014, Discret. Appl. Math..

[7]  A. Recski.  A Network Theory Approach to the Rigidity of Skeletal Structures.Part III. An Electric Model of Planar Frameworks , 1984 .

[8]  G. Mikhalkin.  Enumerative tropical algebraic geometry in R^2 , 2003, math/0312530.

[9]  Sam Payne,et al.  Fibers of tropicalization , 2007, 0705.1732.

[10]  Dinesh Manocha,et al.  SOLVING SYSTEMS OF POLYNOMIAL EQUATIONS , 2002 .

[11]  Eric Katz.  A Tropical Toolkit , 2006 .

[12]  B. Hendrickson,et al.  Regular ArticleAn Algorithm for Two-Dimensional Rigidity Percolation: The Pebble Game , 1997 .

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

[14]  Steven J. Gortler,et al.  Characterizing generic global rigidity , 2007, Ad Hoc Networks.

[15]  Orit E. Raz,et al.  Configurations of Lines in Space and Combinatorial Rigidity , 2016, Discret. Comput. Geom..

[16]  N. Sloane.  The on-line encyclopedia of integer sequences , 2018, Notices of the American Mathematical Society.

[17]  H. Pollaczek-Geiringer.  Über die Gliederung ebener Fachwerke , 1927 .

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

[19]  Bill Jackson,et al.  The number of equivalent realisations of a rigid graph , 2012, 1204.1228.

[20]  Ioannis Z. Emiris,et al.  Mixed Volume and Distance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs , 2013, Distance Geometry.

[21]  Helen Murray Roberts,et al.  Elements of mathematics , 1956 .

[22]  B. Hendrickson,et al.  An Algorithm for Two-Dimensional Rigidity Percolation , 1997 .

[23]  J. M. Selig.  Geometric Fundamentals of Robotics , 2004, Monographs in Computer Science.

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

[25]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[26]  Bernard Mourrain,et al.  Computer Algebra Methods for Studying and Computing Molecular Conformations , 1999, Algorithmica.

[27]  Thorsten Theobald,et al.  Mixed volume techniques for embeddings of Laman graphs , 2008, Comput. Geom..

[28]  P. Dietmaier,et al.  The Stewart-Gough Platform of General Geometry can have 40 Real Postures , 1998 .

[29]  András Recski,et al.  A network theory approach to the rigidity of skeletal structures part II. Laman's theorem and topological formulae , 1984, Discret. Appl. Math..

[30]  Ileana Streinu,et al.  Frameworks with crystallographic symmetry , 2011, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[31]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[32]  Jean-Charles Faugère,et al.  FGb: A Library for Computing Gröbner Bases , 2010, ICMS.

[33]  L. Lovász,et al.  On Generic Rigidity in the Plane , 1982 .

[34]  Ileana Streinu,et al.  The Number of Embeddings of Minimally Rigid Graphs , 2002, SCG '02.