Generic unlabeled global rigidity

Let \(\mathbf{p}\) be a configuration of \(n\) points in \(\mathbb{R}^d\) for some \(n\) and some \(d \ge 2\). Each pair of points has a Euclidean length in the configuration. Given some
graph \(G\) on \(n\) vertices, we measure the point-pair lengths corresponding to the edges of \(G\).

In this paper, we study the question of when a generic \(\mathbf{p}\) in \(d\) dimensions will be uniquely determined (up to an unknowable Euclidean transformation) from a given set of point-pair lengths together with knowledge of \(d\) and \(n\). In this setting the lengths are given simply as a set of real numbers; they are not labeled with the combinatorial data that describes which point-pair gave rise to which length, nor is data about \(G\) given.

We show, perhaps surprisingly, that in terms of generic uniqueness, labels have no effect. A generic configuration is determined by an unlabeled set of point-pair lengths(together with \(d\) and \(n\)) iff it is determined by the labeled edge lengths.