The generic rank of body–bar-and-hinge frameworks

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.