Mixed Volume and Distance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs

  • Ioannis Z. Emiris
  • Elias P. Tsigaridas
  • Antonios Varvitsiotis


A graph G is called generically minimally rigid in d if, for any choice of sufficiently generic edge lengths, it can be embedded in d in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining tight bounds on the number of such embeddings, as a function of the number of vertices. The study of rigid graphs is motivated by numerous applications, mostly in robotics, bioinformatics, sensor networks, and architecture. We capture embeddability by polynomial systems with suitable structure so that their mixed volume, which bounds the number of common roots, yields interesting upper bounds on the number of embeddings. We explore different polynomial formulations so as to reduce the corresponding mixed volume, namely by introducing new variables that remove certain spurious roots and by applying the theory of distance geometry. We focus on \({\mathbb{R}}^{2}\) and \({\mathbb{R}}^{3}\), where Laman graphs and 1-skeleta (or edge graphs) of convex simplicial polyhedra, respectively, admit inductive Henneberg constructions. Our implementation yields upper bounds for \(n \leq 10\) in \({\mathbb{R}}^{2}\) and \({\mathbb{R}}^{3}\), which reduce the existing gaps and lead to tight bounds for \(n \leq 7\) in both \({\mathbb{R}}^{2}\) and \({\mathbb{R}}^{3}\); in particular, we describe the recent settlement of the case of Laman graphs with seven vertices. Our approach also yields a new upper bound for Laman graphs with eight vertices, which is conjectured to be tight. We also establish the first lower bound in \({\mathbb{R}}^{3}\) of about 2. 52 n , where n denotes the number of vertices.


Rigid graph Laman graph Euclidean embedding Henneberg construction Polynomial system Mixed volume Cayley–Menger matrix Cyclohexane caterpillar. 



I.Z. Emiris is partially supported by FP7 contract PITN-GA-2008-214584 SAGA: Shapes, Algebra, and Geometry. Part of this work was done while he was on sabbatical at team Salsa of INRIA Rocquencourt. E. Tsigaridas is partially supported by an individual postdoctoral grant from the Danish Agency for Science, Technology and Innovation, and also acknowledges support from the Danish National Research Foundation and the National Science Foundation of China (under grant 61061130540) for the Sino-Danish Center for the Theory of Interactive Computation, within which part of this work was performed and from the EXACTA grant of the National Science Foundation of China (NSFC 60911130369) and the French National Research Agency (ANR-09-BLAN-0371-01). E. Tsigaridas performed part of this work while he was with the Aarhus University, Denmark. A. Varvitsiotis started work on this project as a graduate student at the University of Athens.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Ioannis Z. Emiris
    • 1
  • Elias P. Tsigaridas
    • 2
  • Antonios Varvitsiotis
    • 3
  1. 1.University of AthensAthensGreece
  2. 2.INRIA Paris-RocquencourtParisFrance
  3. 3.Centrum Wiskunde & Informatica (CWI)AmsterdamThe Netherlands

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