Kitson, D. and C. Power, S.
(2018)
*The rigidity of infinite graphs.*
Discrete and Computational Geometry, 60 (3).
pp. 531-557.
ISSN 0179-5376

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## Abstract

A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in the normed spaces $(\bR^d,\|\cdot \|_q)$, for $d\geq 2$ and $1 <q < \infty$. Generalisations are obtained for the Laman combinatorial characterisation of generic infinitesimal rigidity for finite graphs in $(\bR^2,\|\cdot \|_2)$. Also Tay's multi-graph characterisation of generic infinitesimal rigidity for finite body-bar frameworks in $(\bR^d,\|\cdot\|_2)$ is generalised to the non-Euclidean norms and to countably infinite graphs. For all dimensions and norms it is shown that a generically rigid countable simple graph is the direct limit $G= \varinjlim G_k$ of an inclusion tower of finite graphs $G_1 \subseteq G_2 \subseteq \dots$ for which the inclusions satisfy a relative rigidity property. For $d\geq 3$ a countable graph which is rigid for generic placements in $\bR^d$ may fail the stronger property of sequential rigidity, while for $d=2$ the properties are equivalent.