Tanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global
rigidity in arbitrary dimension. We extend this result to periodic graphs under fixed
lattice representations. A periodic graph is vertex-redundantly rigid if the deletion of a single
vertex orbit under the periodicity results in a periodically rigid graph. 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. This issue is
resolved via a slight modification of a recent result of Kaszanitzy, Schulze and Tanigawa (2016).
Secondly, while the rigidity of finite graphs in $\mathbb{R}^d$ on at most $d$ vertices
obviously implies their global rigidity, it is non-trivial to prove a similar result for
periodic graphs. This is accomplished by extending a result of Bezdek and Connelly (2002)
on the existence of a continuous movement between two equivalent $d$-dimensional realisations
of a single graph in $\mathbb{R}^{2d}$ 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, Jordán and Whiteley (2013)
regarding the global rigidity of generic finite body-bar frameworks.
Bibtex entry:
@techreport{egres-18-01,
AUTHOR | = | {Kaszanitzky, Vikt{\'o}ria and Kir{\'a}ly, Csaba and Schulze, Bernd}, |
TITLE | = | {Sufficient conditions for the global rigidity of periodic graphs}, |
NOTE | = | {{\tt www.cs.elte.hu/egres}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2018}, |
NUMBER | = | {TR-2018-01} |