Graph rigidity via Euclidean distance matrices

Linear Algebra and Its Applications

Volumn 310, Issue 1-3, 2000, Pages 149-165

  Alfakih, Abdo Y ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

Author keywords

05C50; 15A57; 52A25; 52C25; 94C15; Convex sets; Euclidean distance matrices; Normal cones; Rigidity; Semidefinite programming; Weighted graphs

Indexed keywords

EID: 0034395202     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(00)00066-5     Document Type: Article

References (24)

1. A.Y. Alfakih, A. Khandani, H. Wolkowicz, Solving Euclidean distance matrix completion problems via semidefinite programming, Comput. Optim. Appl. 12 (1999) 13-30.
2. A.Y. Alfakih, H. Wolkowicz, On the embeddability of weighted graphs in Euclidean spaces, Technical Report, CORR 98-12, Department of Combinatorics and Optimization, University of Waterloo, 1998.
3. Alizadeh F. Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5:1995;13-51.
4. Asimow L., Roth B. The rigidity of graphs. Trans. Amer. Math. soc. 245:1978;279-289.
5. Auslander L., Mackenzie R.E. Introduction to Differentiable Manifolds. 1977;Dover, New York.
6. R. Connelly, Rigidity, in: P.M. Gruber, J.M. Wills (Eds.), Handbook of Convex Geometry, 1993, pp. 223-271.
7. Crippen G.M., Havel T.F. Distance Geometry and Molecular Conformation. 1988;Wiley, New York.
8. Critchley F. On certain linear mappings between inner-product and squared distance matrices. Linear Algebra Appl. 105:1988;91-107.
9. Gower J.C. Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra Appl. 67:1985;81-97.
10. Hayden T.L., Wells J., Liu W.-M., Tarazaga P. The cone of distance matrices. Linear Algebra Appl. 144:1991;153-169.
11. Hendrickson B. Conditions for unique graph realizations. SIAM J. Comput. 21:1992;65-84.
12. Johnson C.R., Tarazaga P. Connections between the real positive semidefinite and distance matrix completion problems. Linear Algebra Appl. 223/224:1995;375-391.
13. Laman G. On graphs and rigidity of plane skeletal structures. J. Engrg. Math. 4:1970;331-340.
14. Laurent M. Cuts, matrix completions and graph rigidity. Math. Programming Ser. B. 79:1997;255-283.
15. Lovász L., Yemini Y. On generic rigidity in the plane. SIAM J. Algebraic Discrete Methods. 3:1982;91-98.
16. J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematical Studies, vol. 61, Princeton University Press, Princeton, NJ, 1968.
17. G. Pataki, Cone-LP's and semidefinite programs: Geometry and a simplex-type method, in: W.H. Cunningham, S. McCormick, M. Queyranne (Eds.), Integer Programming and Combinatorial Optimization: Proceedings of the Fifth International IPCO Conference, Vancouver, 3-5 June 1996, pp. 162-174.
18. Ramana M., Goldman A.J. Some geometric results in semi-definite programming. J. Global Optim. 7:1995;33-50.
19. Roth B. Rigid and flexible frameworks. Amer. Math. Monthly. 88:1981;6-21.
20. J.B. Saxe, Embeddability of weighted graphs in k-space is strongly NP-hard, in: Proceedings of the 17th Allerton Conference in Communications, Control, and Computing, 1979, pp. 480-489.
21. Schoenberg I.J. Remarks to Maurice Frechet's article: Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert. Ann. Math. 36:1935;724-732.
22. Vandenberghe L., Boyd S. Semidefinite programming. SIAM Rev. 38:1996;49-95.
23. Whiteley W. Matroids and rigid structures. White N. Matroid Applications, Encyclopedia of Mathematics and its Applications 40. 1992;1-53.
24. Wolkowicz H. Some applications of optimization in matrix theory. Linear Algebra Appl. 40:1981;101-118.