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| Autori principali: | , , |
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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2605.17544 |
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| _version_ | 1866916020996800512 |
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| author | Deniz, Zakir Guler, Hakan Nixon, Anthony |
| author_facet | Deniz, Zakir Guler, Hakan Nixon, Anthony |
| contents | A linearly constrained framework in $\mathbb{R}^d$ is a bar-joint framework where, in addition, vertices with loops are constrained to lie in given affine subspaces. In the generic case, when each vertex is incident to sufficiently many loops, a characterisation of rigidity was obtained by Jackson, Nixon and Tanigawa for all $d\geq 3$. By extending this to characterise the rank function of the linearly constrained rigidity matroid (under the same loop hypothesis), sufficient conditions for a looped simple graph to be (globally) rigid in $\mathbb{R}^d$ are obtained. In the 2-dimensional case generic rigidity was characterised by Streinu and Theran, and we obtain a sharper sufficient condition in this case. A key technique is the application of the discharging method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17544 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Generic Linearly Constrained Frameworks Deniz, Zakir Guler, Hakan Nixon, Anthony Combinatorics 52C25 A linearly constrained framework in $\mathbb{R}^d$ is a bar-joint framework where, in addition, vertices with loops are constrained to lie in given affine subspaces. In the generic case, when each vertex is incident to sufficiently many loops, a characterisation of rigidity was obtained by Jackson, Nixon and Tanigawa for all $d\geq 3$. By extending this to characterise the rank function of the linearly constrained rigidity matroid (under the same loop hypothesis), sufficient conditions for a looped simple graph to be (globally) rigid in $\mathbb{R}^d$ are obtained. In the 2-dimensional case generic rigidity was characterised by Streinu and Theran, and we obtain a sharper sufficient condition in this case. A key technique is the application of the discharging method. |
| title | On Generic Linearly Constrained Frameworks |
| topic | Combinatorics 52C25 |
| url | https://arxiv.org/abs/2605.17544 |