Tanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic frameworks under fixed lattice representations. That is, we show that if a generic periodic framework is vertex-redundantly rigid, in the sense that the deletion of a single vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. 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 in dimension \(d>2\). We work around this issue by using slight modifications of recent results of Kaszanitzky et al. (2021). Secondly, while the rigidity of finite frameworks 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 frameworks. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion 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 et al. (2013) regarding the global rigidity of generic finite body-bar frameworks.

Tanigawa (2016) 表明，图的顶点冗余刚性意味着其在任意维度上的全局刚性。我们将此结果扩展到固定点阵表示下的周期性框架。也就是说，我们表明，如果通用周期框架是顶点冗余刚性的，在周期性下单个顶点轨道的删除导致周期性刚性框架的意义上，那么它也是周期性全局刚性的。我们的证明类似于 Tanigawa 的证明，但还有一些额外的困难。首先，不知道周期性全局刚度是否是维度\(d>2\) 中的通用属性。我们通过对 Kaszanitzky 等人的最新结果稍加修改来解决这个问题。(2021)。其次，虽然\({\mathbb {R}}^d\)中有限框架的刚性在最多d个顶点上显然意味着它们的全局刚性，证明周期性框架的类似结果并非易事。这是通过将 Bezdek 和 Connelly (2002) 关于\({\mathbb {R}}^{2d}\) 中单个图的两个等效d维实现之间存在连续运动的结果扩展到周期性构架。作为我们结果的应用，我们给出了任意维度的通用周期性体杆框架的全局刚度的充分必要条件。这提供了 Connelly 等人的结果的周期性对应物。(2013) 关于通用有限车身杆框架的整体刚度。