Sufficient Conditions for the Global Rigidity of Periodic Graphs

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 $$d>2$$ 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 $${\mathbb {R}}^d$$ 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 $${\mathbb {R}}^{2d}$$ R 2 d 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.

Graph Reconstruction from Unlabeled Edge Lengths

A d-dimensional framework is a pair (G, p), where $$G=(V,E)$$ G = ( V , E ) is a graph and p is a map from V to $$\mathbb {R}^d$$ R d . The length of an edge $$uv\in E$$ u v ∈ E in (G, p) is the distance between p(u) and p(v). The framework is said to be globally rigid in $$\mathbb {R}^d$$ R d if every other d-dimensional framework (G, q), in which the corresponding edge lengths are the same, is congruent to (G, p). In a recent paper Gortler, Theran, and Thurston proved that if every generic framework (G, p) in $$\mathbb {R}^d$$ R d is globally rigid for some graph G on $$n\ge d+2$$ n ≥ d + 2 vertices (where $$d\ge 2$$ d ≥ 2 ), then already the set of (unlabeled) edge lengths of a generic framework (G, p), together with n, determine the framework up to congruence. In this paper we investigate the corresponding unlabeled reconstruction problem in the case when the above generic global rigidity property does not hold for the graph. We provide families of graphs G for which the set of (unlabeled) edge lengths of any generic framework (G, p) in d-space, along with the number of vertices, uniquely determine the graph, up to isomorphism. We call these graphs weakly reconstructible. We also introduce the concept of strong reconstructibility; in this case the labeling of the edges is also determined by the set of edge lengths of any generic framework. For $$d=1,2$$ d = 1 , 2 we give a partial characterization of weak reconstructibility as well as a complete characterization of strong reconstructibility of graphs. In particular, in the low-dimensional cases we describe the family of weakly reconstructible graphs that are rigid but not redundantly rigid.

Global Rigidity of 2D Linearly Constrained Frameworks

A linearly constrained framework in $\mathbb{R}^d$ is a point configuration together with a system of constraints that fixes the distances between some pairs of points and additionally restricts some of the points to lie in given affine subspaces. It is globally rigid if the configuration is uniquely defined by the constraint system. We show that a generic linearly constrained framework in $\mathbb{R}^2$ is globally rigid if and only if it is redundantly rigid and “balanced”. For unbalanced generic frameworks, we determine the precise number of solutions to the constraint system whenever the rigidity matroid of the framework is connected. We obtain a stress matrix sufficient condition and a Hendrickson type necessary condition for a generic linearly constrained framework to be globally rigid in $\mathbb{R}^d$.