Conditions for Unique Graph Realizations

The graph realization problem is that of computing the relative locations of a set of vertices placed in Euclidean space, relying only upon some set of inter-vertex distance measurements. This paper is concerned with the closely related problem of determining whether or not a graph has a unique realization. Both these problems are NP-hard, but the proofs rely upon special combinations of edge lengths. If we assume the vertex locations are unrelated then the uniqueness question can be approached from a purely graph theoretic angle that ignores edge lengths. This paper identiies three necessary graph theoretic conditions for a graph to have a unique realization in any dimension. EEcient sequential and NC algorithms are presented for each condition, although these algorithms have very diierent avors in diierent dimensions.