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Tay characterized the multigraphs which can be realized as infinitesimally rigid d-dimensional body-and-bar frameworks. Subsequently, Tay and Whiteley 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 longstanding conjecture of Tay and Whiteley 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 two spanning trees which use each edge of G at most twice, then its square can be realized as an infinitesimally rigid 3-dimensional bar-and-joint framework.
Bibtex entry:
AUTHOR | = | {Jackson, Bill and Jord{\'a}n, Tibor}, |
TITLE | = | {The generic rank of body-bar-and-hinge frameworks}, |
NOTE | = | {{\tt egres.elte.hu}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2007}, |
NUMBER | = | {TR-2007-06} |