Global Rigidity of Periodic Graphs under Fixed-lattice Representations

Viktória Kaszanitzky, Bernd Schulze, Shin-ichi Tanigawa

Published in:
Journal of Combinatorial Theory, Series B, Volume 146, 2021, Pages 176-218. DOI link


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}},
INSTITUTION = {Egerv{\'a}ry Research Group, Budapest},
YEAR = {2016},
NUMBER = {TR-2016-21}

Last modification: 22.11.2022. Please email your comments to Tamás Király!