Abstract
Rigidity of bar-and-joint frameworks has been studied for centuries. If the exact positions of the joints of such a framework are known, the rank of the so called rigidity matrix determines whether the framework is rigid1. If the underlying graph is given only, the rigidity of the framework cannot always be determined: if certain conditions (depending on the dimension of the space) are not satisfied then the framework cannot be rigid, no matter what the actual positions of the joints are, otherwise rigidity can be realized by some (in fact, almost all) positions of the joints. These graph theoretic conditions can be checked in polynomial time for the 1- and 2-dimensional frameworks, while the complexity questions are mainly open for higher dimensions. For surveys of such results the reader is referred to [6, 13, 16].
Supported by Grant No. OTKA T67651 of the Hungarian National Science Fund and the National Office for Research and Technology. The useful remarks of the referee are gratefully acknowledged.
For brevity, we use the word “rigid” instead of the more precise “infinitesimally rigid”, see the definitions in the next section.
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© 2008 János Bolyai Mathematical Society and Springer-Verlag
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Recski, A. (2008). Combinatorial Conditions for the Rigidity of Tensegrity Frameworks. In: Győri, E., Katona, G.O.H., Lovász, L., Sági, G. (eds) Horizons of Combinatorics. Bolyai Society Mathematical Studies, vol 17. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77200-2_8
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