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arXiv · 2007.15978

Which graphs are rigid in $\ell_p^d$?

Sean Dewar·Derek Kitson·Anthony Nixon
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Abstract

We present three results which support the conjecture that a graph is minimally rigid in $d$-dimensional $\ell_p$-space, where $p\in (1,\infty)$ and $p\not=2$, if and only if it is $(d,d)$-tight. Firstly, we introduce a graph bracing operation which preserves independence in the generic rigidity matroid when passing from $\ell_p^d$ to $\ell_p^{d+1}$. We then prove that every $(d,d)$-sparse graph with minimum degree at most $d+1$ and maximum degree at most $d+2$ is independent in $\ell_p^d$. Finally, we prove that every triangulation of the projective plane is minimally rigid in $\ell_p^3$. A catalogue of rigidity preserving graph moves is also provided for the more general class of strictly convex and smooth normed spaces and we show that every triangulation of the sphere is independent for 3-dimensional spaces in this class.

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Sean Dewar, Derek Kitson, Anthony Nixon. 2020-07-31. Which graphs are rigid in $\ell_p^d$?. https://doi.org/10.1007/s10898-021-01008-z

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