A minmax relationship between embeddable and rigid graphs Original Research Article

Weighted graphs that can be embedded in Euclidean space in such a way as to preserve the weight of an edge as the distance between its two end points are of interest in a variety of applications. The concept of elastic embeddability, introduced in [1], is designed to deal with distances subject to error. Elastic graphs are related to, but distinct from, generically rigid graphs known in structural engineering. Whereas rigidity is defined via the possible motions of vertices that leave edge lengths invariant, elasticity deals with the behavior of embeddings as edge lengths are perturbed. Although these two classes are nearly disjoint, we prove that they meet at a common boundary: a graph is maximal with respect to elastic embedding if and only if it is minimal with respect to infinitesimal rigidity.