On Generic Linearly Constrained Frameworks
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916020996800512 |
|---|---|
| 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 |