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 e
nd points are of interest in a variety of applications. The concept of
elastic embeddability, introduced in [I], is designed to deal with di
stances subject to error. Elastic graphs are related to, but distinct
from, generically rigid graphs known in structural engineering. Wherea
s rigidity is defined via the possible motions of vertices that leave
edge lengths invariant, elasticity deals with the behavior of embeddin
gs as edge lengths are perturbed. Although these two classes are nearl
y disjoint, we prove that they meet at a common boundary: a graph is m
aximal with respect to elastic embedding if and only if it is minimal
with respect to infinitesimal rigidity.