Year: 2025
Type: article
Abstract: A d -dimensional tensegrity framework ( T , p ) is an edge-labeled geometric graph in $$\mathbb {R}^d$$ R d , which consists of a graph $$T=(V,B\cup C\cup S)$$ T = ( V , B ∪ C ∪ S ) and a map $$p:V\rightarrow \mathbb {R}^d$$ p : V → 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.
Authors: Adam D. W. Clay, Tibor Jordán, Sára Hanna Tóth
Institutions: Purdue University West Lafayette, Eötvös Loránd University
Language: English
Subfield: Civil and Structural Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
Awards: G4183452178, G660829768, G721023497
OpenAlex ID: W4415079388