Minimally rigid tensegrity frameworks

A \(d\)-dimensional tensegrity framework \((T,p)\) is an edge-labeled geometric graph in \({\mathbb R}^d\), which consists of a graph \(T=(V,B\cup C\cup S)\) and a map \(p:V\to {\mathbb R}^d\). The labels determine whether an edge \(uv\) of \(T\) corresponds to a fixed length bar in \((T,p)\), or a cable which cannot increase in length, or a strut which cannot decrease in length. We consider minimally infinitesimally rigid \(d\)-dimensional tensegrity frameworks and provide tight upper bounds on the number of its edges, in terms of the number of vertices and the dimension \(d\). We obtain stronger upper bounds in the case when there are no bars and the framework is in generic position. The proofs use methods from convex geometry and matroid theory. A special case of our results confirms a conjecture of Whiteley from 1987. We also give an affirmative answer to a conjecture concerning the number of edges of a graph whose three-dimensional rigidity matroid is minimally connected.