The recent preprint [Determining Generic Point Configurations From Unlabeled Path or Loop Lengths](https://arxiv.org/abs/1709.03936) by Gkioulekas, Gortler, Theran, and Zickler, treats a generalization of this question to higher dimensions, and to the situation where the list of distances provided may not correspond to the lengths of single edges, but may also be the lengths of arbitrary paths and loops on the graph.