## Abstract

Let

**p**be a configuration of n points in ℝ*for some n and some*^{d}*d*≥ 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**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 distance, 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 distances (together with*d*and*n*) if and only if it is determined by the labeled distances.Original language | English |
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Article number | e21 |

Pages (from-to) | 1-34 |

Number of pages | 34 |

Journal | Forum of Mathematics, Sigma |

Volume | 7 |

DOIs | |

Publication status | Published - 30 Jul 2019 |