Given a set of distances (no info regarding what points the distances correspond to) from a complete graph, is the realization of the graph unique? There are $n$ points in $R^2$ (i.e. the 2D real space). We can think of them as a complete graph where edge weights correspond to the distance between points.

Let $D$ be the distance matrix between all pairs of points, such that entry $D_{ij}$ corresponds to the distance between points $i$ and $j$. This matrix is of size $n \times n$ and the upper-triangle contains the distances between unique pairings of points.

Now imagine I put the elements from the upper triangle into a set, such that you no longer know which pairs of points produce the distance values. This is essentially the set of edge weights, with no information regarding the relationship between the weight and the edge the weight belongs to.

If you are given two sets and they are identical, does this imply the graphs corresponding to each set are also essentially the same?
I think the following example with four points answers your question (though it gives multi-sets of distances). See Reconstructing Sets From Interpoint Distances by Skiena, Smith, and Lemke for more information.