Abstract
In [9] Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in Rd. Jackson and Jordán [10] confirmed that these conditions are also sufficient in R2, giving a combinatorial characterization of graphs whose generic realizations in R2 are globally rigid. In this paper, we establish analogues of these results for infinite periodic frameworks under fixed lattice representations. Our combinatorial characterization of globally rigid generic periodic frameworks in R2 in particular implies toroidal and cylindrical counterparts of the theorem by Jackson and Jordán.
Highlights
A bar-joint framework in Rd is a pair (G, p), where G = (V, E) is a graph and p : V → Rd is a map
Jackson and Jordan [10] confirmed that these conditions are sufficient in R2, giving a combinatorial characterization of graphs whose generic realizations in R2 are globally rigid
Based on the theory of stress matrices by Connelly [3, 4], Gortler, Healy, and Thurston [8] gave an algebraic characterization of the global rigidity of generic frameworks
Summary
Based on the theory of stress matrices by Connelly [3, 4], Gortler, Healy, and Thurston [8] gave an algebraic characterization of the global rigidity of generic frameworks This in particular implies that all generic realizations of a given graph share the same global rigidity properties in Rd, as in the case of local rigidity. Our main result (Theorem 4.2) is that these necessary conditions are sufficient in R2, giving a first combinatorial characterization of the global rigidity of generic periodic frameworks. The proof of this result is inspired by the work in [11, 26].
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