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Corpus ID: 220919904

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

@article{Dewar2020WhichGA,
title={Which graphs are rigid in \$\ell_p^d\$?},
author={S. Dewar and D. Kitson and Anthony Nixon},
journal={arXiv: Metric Geometry},
year={2020}
}

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… CONTINUE READING