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.

Cylindrical framework, Global rigidity, Group-labeled graph, Matroid connectivity, Periodic framework, Toroidal framework
Journal of Combinatorial Theory - Series B
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

Kaszanitzky, V.E, Schulze, B, & Tanigawa, S.-I. (2021). Global rigidity of periodic graphs under fixed-lattice representations. Journal of Combinatorial Theory - Series B, 146, 176–218. doi:10.1016/j.jctb.2020.09.009