On the Number of Embeddings of Minimally Rigid Graphs

Rigid frameworks in some Euclidian space are embedded graphs having a unique local realization (up to Euclidian 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 ${{2n-4}\choose {n-2}} \approx 4^n$. We also exhibit several families which realize lower bounds of the order of $2^n$, $2.21^n$ and $2.88^n$. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley-Menger variety $CM^{2,n}(C)\subset P_{{{n}\choose {2}}-1}(C)$ 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 $2n-4$ hyperplanes yields at most $deg(CM^{2,n})$ zero-dimensional components, and one finds this degree to be $D^{2,n}={1/2}{{2n-4}\choose {n-2}}$. 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 $2 D^{3,n}= {\frac{2^{n-3}}{n-2}}{{n-6}\choose{n-3}}$ for the number of spatial embeddings with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to rigid motions.