Abstract
The study of (minimally) rigid graphs is motivated by numerous applications, mostly in robotics and bioinformatics. A major open problem concerns the number of embeddings of such graphs, up to rigid motions, in Euclidean space. We capture embeddability by polynomial systems with suitable structure, so that their mixed volume, which bounds the number of common roots, to yield interesting upper bounds on the number of embeddings. We focus on \({\mathbb R}^2\) and \({\mathbb R}^3\), where Laman graphs and 1-skeleta of convex simplicial polyhedra, respectively, admit inductive Henneberg constructions. We establish the first general lower bound in \({\mathbb R}^3\) of about 2.52n, where n denotes the number of vertices. Moreover, our implementation yields upper bounds for n ≤ 10 in \({\mathbb R}^2\) and \({\mathbb R}^3\), which reduce the existing gaps, and tight bounds up to n = 7 in \({\mathbb R}^3\).
Chapter PDF
Similar content being viewed by others
Keywords
References
Bernstein, D.N.: The number of roots of a system of equations. Fun. Anal. Pril. 9, 1–4 (1975)
Borcea, C., Streinu, I.: The number of embeddings of minimally rigid graphs. Discrete Comp. Geometry 31(2), 287–303 (2004)
Bowen, R., Fisk, S.: Generation of triangulations of the sphere. Math. of Computation 21(98), 250–252 (1967)
Emiris, I.Z., Mourrain, B.: Computer algebra methods for studying and computing molecular conformations. Algorithmica, Special Issue 25, 372–402 (1999)
Emiris, I.Z., Varvitsiotis, A.: Counting the number of embeddings of minimally rigid graphs. In: Proc. Europ. Workshop Comput. Geometry, Brussels (2009)
Gluck, H.: Almost all simply connected closed surfaces are rigid. Lect. Notes in Math. 438, 225–240 (1975)
Steffens, R., Theobald, T.: Mixed volume techniques for embeddings of Laman graphs. In: Proc. Europ. Workshop Comput. Geometry, Nancy, France, pp. 25–28 (2008); Final version accepted in Comp. Geom: Theory & Appl., Special Issue
Walter, D., Husty, M.: On a 9-bar linkage, its possible configurations and conditions for paradoxical mobility. In: IFToMM Congress, Besançon, France (2007)
Wunderlich, W.: Gefärlice Annahmen der Trilateration und bewegliche Fachwerke I. Zeitschrift für Angewandte Mathematik und Mechanik 57, 297–304 (1977)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Emiris, I.Z., Tsigaridas, E.P., Varvitsiotis, A.E. (2010). Algebraic Methods for Counting Euclidean Embeddings of Rigid Graphs. In: Eppstein, D., Gansner, E.R. (eds) Graph Drawing. GD 2009. Lecture Notes in Computer Science, vol 5849. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11805-0_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-11805-0_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11804-3
Online ISBN: 978-3-642-11805-0
eBook Packages: Computer ScienceComputer Science (R0)