Second-Order Rigidity and Prestress Stability for TensegrityFrameworks
SIAM Journal on Discrete Mathematics
Handbook of discrete and computational geometry
A proof of Connelly's conjecture on 3-connected circuits of the rigidity matroid
Journal of Combinatorial Theory Series B
Connected rigidity matroids and unique realizations of graphs
Journal of Combinatorial Theory Series B
Note: Strongly rigid tensegrity graphs on the line
Discrete Applied Mathematics
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Tensegrity frameworks are defined on a set of points in R^d and consist of bars, cables, and struts, which provide upper and/or lower bounds for the distance between their endpoints. The graph of the framework, in which edges are labeled as bars, cables, and struts, is called a tensegrity graph. It is said to be rigid in R^d if it has an infinitesimally rigid realization in R^d as a tensegrity framework. The characterization of rigid tensegrity graphs is not known for d=2. A related problem is how to find a rigid labeling of a graph using no bars. Our main result is an efficient combinatorial algorithm for finding a rigid cable-strut labeling of a given graph in the case when d=2. The algorithm is based on a new inductive construction of redundant graphs, i.e. graphs which have a realization as a bar framework in which each bar can be deleted without increasing the degree of freedom. The labeling is constructed recursively by using labeled versions of some well-known operations on bar frameworks.