TR-2007-06

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

Bill Jackson, Tibor Jordán

Published in:
European J. Combinatorics 31 (2010) 574-588.



Abstract

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:

@techreport{egres-07-06,
AUTHOR = {Jackson, Bill and Jord{\'a}n, Tibor},
TITLE = {The generic rank of body-bar-and-hinge frameworks},
NOTE= {{\tt www.cs.elte.hu/egres}},
INSTITUTION = {Egerv{\'a}ry Research Group, Budapest},
YEAR = {2007},
NUMBER = {TR-2007-06}
}


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