Algebraic Methods for Counting Euclidean Embeddings of Rigid Graphs

  • Ioannis Z. Emiris
  • Elias P. Tsigaridas
  • Antonios E. Varvitsiotis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5849)

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 \({\mathbb R}^3\).

Keywords

Rigid graph Euclidean embedding Henneberg construction polynomial system root bound cyclohexane caterpillar 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Ioannis Z. Emiris
    • 1
  • Elias P. Tsigaridas
    • 2
  • Antonios E. Varvitsiotis
    • 3
  1. 1.University of AthensAthensGreece
  2. 2.INRIA MéditerranéeSophia-AntipolisFrance
  3. 3.CWIAmsterdamThe Netherlands

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