The characterization of rigid graphs in ${\mathbb{R}}^d$ for $d\ge 3$ is a major open problem in rigidity theory. The same holds for globally rigid graphs. In this paper our goal is to give necessary and/or sufficient conditions for the (global) rigidity of the square $G^2$ (and more generally, the power $G^k$) of a graph G in ${\mathbb{R}}^d$, for some values of $k,d$. Our work is motivated by some results and conjectures of M. Cheung and W. Whiteley from 2008, the Molecular Theorem of N. Katoh and S. Tanigawa from 2011, which settled the case of rigidity for $k=2,d=3$, and the potential applications in molecular conformation and sensor network localization.

We first consider the case when $k=d$ and characterize those graphs G for which $G^d$ is globally rigid in ${\mathbb{R}}^d$, for all $d\ge 1$, and then focus on the case when $k=d-1$. We provide a new, direct proof for the 3-dimensional bar-and-joint version of the Molecular Theorem ($d=3$) and a necessary condition for the rigidity of $G^{d-1}$ in ${\mathbb{R}}^d$, for all $d\ge 3$. We conjecture that this condition is also sufficient.

The global rigidity of square graphs in ${\mathbb{R}}^3$ is still an open problem. We formulate a Molecular Global Rigidity Conjecture, which proposes a combinatorial characterization of globally rigid square graphs in terms of vertex partitions and edge count conditions. We prove that the condition is necessary. For the general case we give a best possible connectivity based sufficient condition by showing that if G is 3-edge-connected then $G^{d-1}$ is globally rigid in ${\mathbb{R}}^d$, for all $d\ge 3$.

Our results imply affirmative answers to the conjectures of M. Cheung and W. Whiteley in two special cases.

刚性图的表征${\mathbb{R}}^d$为了$d\ge 3$是刚性理论中的一个主要开放问题。这同样适用于全局刚性图。在本文中，我们的目标是为正方形的（全局）刚性提供必要和/或充分条件$G^2$（更一般地说，权力$G^{ķ}$)中的图G${\mathbb{R}}^d$, 对于某些值$ķ,d$. 我们的工作受到 M. Cheung 和 W. Whiteley 2008 年的一些结果和猜想以及 2011 年 N. Katoh 和 S. Tanigawa 的分子定理的推动，这解决了$ķ=2,d=3$，以及在分子构象和传感器网络定位方面的潜在应用。

我们首先考虑以下情况$ķ=d$并刻画那些图G$G^d$是全局刚性的${\mathbb{R}}^d$， 对所有人$d\ge 1$，然后关注情况$ķ=d-1$. 我们为分子定理 ($d=3$) 和刚性的必要条件$G^{d-1}$在${\mathbb{R}}^d$， 对所有人$d\ge 3$. 我们推测这个条件也是充分的。

方图的全局刚性${\mathbb{R}}^3$仍然是一个悬而未决的问题。我们制定了一个分子全局刚性猜想，该猜想根据顶点划分和边数条件提出了全局刚性方形图的组合表征。我们证明条件是必要的。对于一般情况，我们通过证明如果G是 3-edge-connected 则给出基于最佳可能连通性的充分条件$G^{d-1}$是全局刚性的${\mathbb{R}}^d$， 对所有人$d\ge 3$.

我们的结果暗示了在两个特殊情况下对 M. Cheung 和 W. Whiteley 猜想的肯定回答。