An upper bound on Euclidean embeddings of rigid graphs with 8 vertices

A graph is called (generically) rigid in Rd if, for any choice of sufficiently generic edge lengths, it can be embedded in Rd in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining the maximum number of planar Euclidean embeddings of minimally rigid graphs with 8 vertices, because this is the smallest unknown case in the plane. Until now, the best known upper bound was 128. Our result is an upper bound of 116 (notice that the best known lower bound is 112), and we show it is achieved by exactly two such graphs. We conjecture that the bound of 116 is tight. 1 Definitions Given a graph G = (V,E) with |V | = n and a collection of edge lengths dij ∈ R, a Euclidean embedding of G in R is a mapping of its vertices to a set of points p1, . . . , pn, such that dij = ‖pi−pj‖, for all {i, j} ∈ E. We call a graph (generically) rigid in R iff, for generic edge lengths, it can be embedded in R in a finite number of ways, modulo rigid transformations (translations and rotations). A graph is minimally rigid in R iff it is no longer rigid once any edge is removed. The study of rigid graphs is motivated by numerous applications, mostly in robotics, mechanism and linkage theory, structural bioinformatics, and architecture. The main goal is to determine the maximum number of distinct planar Euclidean embeddings of minimally rigid graphs, up to rigid transformations, as a function of the number of vertices. We shall focus on the planar case. 1 ar X iv :1 20 4. 65 27 v1 [ cs .C G ] 2 9 A pr 2 01 2