Minimally Rigid Tensegrity Frameworks

Abstract A d -dimensional tensegrity framework ( T , p ) is an edge-labeled geometric graph in $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , which consists of a graph $$T=(V,B\cup C\cup S)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>∪</mml:mo> <mml:mi>C</mml:mi> <mml:mo>∪</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and a map $$p:V\rightarrow \mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>:</mml:mo> <mml:mi>V</mml:mi> <mml:mo>→</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> . 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.