Connected rigidity matroids and unique realizations of graphs
Journal of Combinatorial Theory Series B
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The graph embedding 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 embedding. Both these problems are NP-hard, but the proofs rely upon special combinations of edge lengths. If we assume the edge lengths are unrelated then the uniqueness question can be approached from a purely graph theoretic framework that ignores edge lenghts. This paper identifies three necessary graph theoretic conditions for a graph to have a unique embedding in any dimension. Efficient sequential and NC algorithms are presented for each condition, although these algorithms have very different flavors in different dimensions.