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The rigidity of infinite graphs

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E-pub ahead of print
<mark>Journal publication date</mark>11/04/2018
<mark>Journal</mark>Discrete and Computational Geometry
Number of pages26
<mark>State</mark>E-pub ahead of print
Early online date11/04/18
<mark>Original language</mark>English


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.

Bibliographic note

The final publication is available at Springer via http://dx.doi.org/10.1007/s00454-018-9993-0