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 one assumes the vertex locations are unrelated, then the uniqueness question can be approached from a purely graph theoretic angle that ignores edge lengths. This paper identifies three necessary graph theoretic conditions for a graph to have a unique realization in any dimension. Efficient sequential and NC algorithms are presented for each condition, although these algorithms have very different flavors in different dimensions.

[1]  Joseph Cheriyan,et al.  On determining vertex connectivity , 1990 .

[2]  J. Gilbert.  Predicting Structure in Sparse Matrix Computations , 1994 .

[3]  B. Roth,et al.  The rigidity of graphs , 1978 .

[4]  E. Bolker,et al.  When is a bipartite graph a rigid framework , 1980 .

[5]  Ben Roth.  Questions on the Rigidity of Structures , 1980 .

[6]  Robert E. Tarjan,et al.  Dividing a Graph into Triconnected Components , 1973, SIAM J. Comput..

[7]  W. Whiteley,et al.  Generating Isostatic Frameworks , 1985 .

[8]  L. Lovász,et al.  On Generic Rigidity in the Plane , 1982 .

[9]  杉原 厚吉,et al.  On Redundant Bracing in Plane Skeletal Structures , 1980 .

[10]  H. Gluck.  Almost all simply connected closed surfaces are rigid , 1975 .

[11]  Harold N. Gabow,et al.  Forests, frames, and games: algorithms for matroid sums and applications , 1988, STOC '88.

[12]  Samir Khuller,et al.  Efficient Parallel Algorithms for Testing k-Connectivity and Finding Disjoint s-t Paths in Graphs , 1991, SIAM J. Comput..

[13]  Oscar H. Ibarra,et al.  A Note on the Parallel Complexity of Computing the Rank of Order n Matrices , 1980, Inf. Process. Lett..

[14]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[15]  John R. Gilbert,et al.  Predicting fill for sparse orthogonal factorization , 1986, JACM.

[16]  László Lovász,et al.  A physical interpretation of graph connectivity, and its algorithmic applications , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[17]  B. Hendrickson.  The Molecular Problem: Determining Conformation from Pairwise Distances , 1990 .

[18]  L. Dale,et al.  FORESTS , 1994, Restoration & Management Notes.

[19]  Gary L. Miller,et al.  A new graph triconnectivity algorithm and its parallelization , 1992, Comb..

[20]  Gordon M. Crippen,et al.  Distance Geometry and Molecular Conformation , 1988 .

[21]  Kurt Mehlhorn,et al.  A Probabilistic Algorithm for Vertex Connectivity of Graphs , 1982, Inf. Process. Lett..

[22]  A. Sard,et al.  The measure of the critical values of differentiable maps , 1942 .

[23]  A. George,et al.  Solution of sparse linear least squares problems using givens rotations , 1980 .

[24]  G. Laman.  On graphs and rigidity of plane skeletal structures , 1970 .

[25]  Hiroshi Imai,et al.  On combinatorial structures of line drawings of polyhedra , 1985, Discret. Appl. Math..

[26]  W. Whiteley.  INFINITESIMAL MOTIONS OF A BIPARTITE FRAMEWORK , 1984 .

[27]  Arkady Kanevsky,et al.  Improved algorithms for graph four-connectivity , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).