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arXiv · 2605.17544

On Generic Linearly Constrained Frameworks

Zakir Deniz·Hakan Guler·Anthony Nixon
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Abstract

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.

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Zakir Deniz, Hakan Guler, Anthony Nixon. 2026-05-17. On Generic Linearly Constrained Frameworks. https://arxiv.org/abs/2605.17544

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