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In [8] 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 $\mathbb{R}^d$. Jackson and Jord\'{a}n [9] confirmed that these conditions are also sufficient in $\mathbb{R}^2$, giving a combinatorial characterization of graphs whose generic realizations in $\mathbb{R}^d$ 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 $\mathbb{R}^2$ in particular implies toroidal and cylindrical counterparts of the theorem by Jackson and Jord\'{a}n.
Bibtex entry:
| AUTHOR | = | {Kaszanitzky, Vikt{\'o}ria and Schulze, Bernd and Tanigawa, Shin-ichi}, |
| TITLE | = | {Global Rigidity of Periodic Graphs under Fixed-lattice Representations}, |
| NOTE | = | {{\tt egres.elte.hu}}, |
| INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
| YEAR | = | {2016}, |
| NUMBER | = | {TR-2016-21} |