Sufficient conditions for the global rigidity of periodic graphs

TL;DR

This paper extends the concept of vertex-redundant rigidity implying global rigidity from finite graphs to periodic graphs, providing new conditions and proofs for their global rigidity in arbitrary dimensions.

Contribution

It generalizes Tanigawa's result to periodic graphs, introduces new proof techniques, and offers a necessary and sufficient condition for global rigidity of periodic body-bar frameworks.

Findings

Vertex-redundant rigidity implies global rigidity for periodic graphs.

Established conditions for global rigidity of periodic body-bar frameworks.

Extended known finite graph results to infinite periodic frameworks.

Abstract

Tanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic graphs under fixed lattice representations. A periodic graph is vertex-redundantly rigid if the deletion of a single vertex orbit under the periodicity results in a periodically rigid graph. Our proof is similar to the one of Tanigawa, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property. This issue is resolved via a slight modification of a recent result of Kaszanitzy, Schulze and Tanigawa (2016). Secondly, while the rigidity of finite graphs in $\mathbb{R}^d$ on at most $d$ vertices obviously implies their global rigidity, it is non-trivial to prove a similar result for periodic graphs. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous movement between two equivalent $d$-dimensional realisations of a single graph in $\mathbb{R}^{2d}$ to periodic frameworks. As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic body-bar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly, Jordan and Whiteley (2013) regarding the global rigidity of generic finite body-bar frameworks.