Algebraic methods for counting Euclidean embeddings of rigid graphs

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 ℝ2 and ℝ3, where Laman graphs and 1-skeleta of convex simplicial polyhedra, respectively, admit inductive Henneberg constructions. We establish the first general lower bound in ℝ3 of about 2.52n , where n denotes the number of vertices. Moreover, our implementation yields upper bounds for n ≤ 10 in ℝ2 and ℝ3, which reduce the existing gaps, and tight bounds up to n = 7 in ℝ3. © 2010 Springer-Verlag.