The rigidity of infinite graphs

TL;DR

This paper develops a rigidity theory for infinite graphs in various norms, generalizing finite graph results and exploring rigidity properties in non-Euclidean spaces with applications to polyhedra.

Contribution

It extends classical rigidity characterizations to countably infinite graphs and non-Euclidean norms, providing new insights into their rigidity and flexibility properties.

Findings

Countably infinite graphs can be constructed as direct limits of finite rigid graphs.

In dimensions greater than two, rigidity does not imply sequential rigidity.

For two dimensions, rigidity and sequential rigidity are equivalent under the generalized conditions.

Abstract

A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in R^d with respect to the classical l^p norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and Henneberg combinatorial characterisations of generic infinitesimal rigidity for finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation of the rigidity of generic finite body-bar frameworks in d-dimensional Euclidean space is generalised to the non-Euclidean l^p 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 of an inclusion tower of finite graphs for which the inclusions satisfy a relative rigidity property. For d>2 a countable graph which is rigid for generic placements in R^d may fail the stronger property of sequential rigidity, while for d=2 the equivalence with sequential rigidity is obtained from the generalised Laman characterisations. Applications are given to the flexibility of non-Euclidean convex polyhedra and to the infinitesimal and continuous rigidity of compact infinitely-faceted simplicial polytopes.