Combinatorial Conditions for the Rigidity of Tensegrity Frameworks

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].