Graph rigidity via Euclidean distance matrices

Abstract

Let G=(V,E,ω) be an incomplete graph with node set V, edge set E, and nonnegative weights ω ij 's on the edges. Let each edge (v i,v j) be viewed as a rigid bar, of length ω ij , which can rotate freely around its end nodes. A realization of a graph G is an assignment of coordinates, in some Euclidean space, to each node of G. In this paper, we consider the problem of determining whether or not a given realization of a graph G is rigid. We show that each realization of G can be epresented as a point in a compact convex set Ω⊂ R m ̄ ; and that a generic realization of G is rigid if and only if its corresponding point is a vertex of Ω, i.e., an extreme point with full-dimensional normal cone.