Minimally rigid tensegrity frameworks
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912067083042816 |
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| author | Clay, Adam D. W. Jordán, Tibor Tóth, Sára Hanna |
| author_facet | Clay, Adam D. W. Jordán, Tibor Tóth, Sára Hanna |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_07452 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimally rigid tensegrity frameworks Clay, Adam D. W. Jordán, Tibor Tóth, Sára Hanna Combinatorics 52C25 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. |
| title | Minimally rigid tensegrity frameworks |
| topic | Combinatorics 52C25 |
| url | https://arxiv.org/abs/2410.07452 |