Discrete & Computational Geometry
, Volume 31, Issue 2, pp 287303
First online:
The Number of Embeddings of Minimally Rigid Graphs
 Ciprian BorceaAffiliated withDepartment of Mathematics, Rider University, Lawrenceville, NJ 08648 Email author
 , Ileana StreinuAffiliated withDepartment of Computer Science, Smith College, Northampton, MA 01063 Email author
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Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with $n$ vertices. We show that, modulo planar rigid motions, this number is at most ${{2n4}\choose {n2}} \approx 4^n$. We also exhibit several families which realize lower bounds of the order of $2^n$, $2.21^n$ and $2.28^n$. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) CayleyMenger variety ${\it CM}^{2,n}(C)\subset P_{{{n}\choose {2}}1}(C)$ over the complex numbers $C$. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with $2n4$ hyperplanes yields at most $deg({\it CM}^{2,n})$ zerodimensional components, and one finds this degree to be $D^{2,n}=\frac{1}{2}{{2n4}\choose {n2}}$. The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences. The same approach works in higher dimensions. In particular, we show that it leads to an upper bound of $2 D^{3,n}= {({2^{n3}}/({n2}})){{2n6}\choose{n3}}$ for the number of spatial embeddings with generic edge lengths of the $1$skeleton of a simplicial polyhedron, up to rigid motions. Our technique can also be adapted to the nonEuclidean case.
 Title
 The Number of Embeddings of Minimally Rigid Graphs
 Journal

Discrete & Computational Geometry
Volume 31, Issue 2 , pp 287303
 Cover Date
 200402
 DOI
 10.1007/s0045400329020
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 SpringerVerlag
 Additional Links
 Industry Sectors
 Authors

 Ciprian Borcea ^{(1)}
 Ileana Streinu ^{(2)}
 Author Affiliations

 1. Department of Mathematics, Rider University, Lawrenceville, NJ 08648, USA
 2. Department of Computer Science, Smith College, Northampton, MA 01063, USA