Reconstruction of Tetrahedra from Sets of Edge Lengths

The problem of reconstructing tetrahedra from given sets of edge lengths is studied. This is a special case of the problem of determining, up to isometry, the position of a complete graph in $\mathbb R^3$ from the set of all pairwise distances between its vertices without knowing their distribution over the edges of the graph. This problem arises in the physics of molecular clusters. Traditionally, the problem of minimizing the potential energy of a molecular cluster is reduced to a computationally complex global optimization problem. However, analyzing the solution thus obtained requires the knowledge of whether the congruence of multiedge constructions is preserved under rearrangements of edge lengths.