The algebraic geometry of motions of bar-and-body frameworks
SIAM Journal on Algebraic and Discrete Methods
The union of matroids and the rigidity of frameworks
SIAM Journal on Discrete Mathematics
Pin-Collinear Body-and-Pin Frameworks and the Molecular Conjecture
Discrete & Computational Geometry
Periodic body-and-bar frameworks
Proceedings of the twenty-eighth annual symposium on Computational geometry
Generic global rigidity of body-bar frameworks
Journal of Combinatorial Theory Series B
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Tay [T.S. Tay, Rigidity of multi-graphs I Linking Bodies in n-space, J. Combin. Theory B 26 (1984) 95-112] characterized the multigraphs which can be realized as infinitesimally rigid d-dimensional body-and-bar frameworks. Subsequently, Tay [T.S. Tay, Linking (n-2)-dimensional panels in n-space II: (n-2,2)-frameworks and body and hinge structures, Graphs Combin. 5 (1989) 245-273] and Whiteley [W. Whiteley, The union of matroids and the rigidity of frameworks, SIAM J. Discrete Math. 1 (2) (1988) 237-255] independently characterized the multigraphs which can be realized as infinitesimally rigid d-dimensional body-and-hinge frameworks. We adapt Whiteley's proof technique to characterize the multigraphs which can be realized as infinitesimally rigid d-dimensional body-bar-and-hinge frameworks. More importantly, we obtain a sufficient condition for a multigraph to be realized as an infinitesimally rigid d-dimensional body-and-hinge framework in which all hinges lie in the same hyperplane. This result is related to a long-standing conjecture of Tay and Whiteley [T.S. Tay, W. Whiteley, Recent advances in the generic rigidity of structures, Structural Topology 9 (1984) 31-38] which would characterize when a multigraph can be realized as an infinitesimally rigid d-dimensional body-and-hinge framework in which all the hinges incident to each body lie in a common hyperplane. As a corollary we deduce that if a graph G has three spanning trees which use each edge of G at most twice, then its square can be realized as an infinitesimally rigid three-dimensional bar-and-joint framework.