A generically globally rigid graph in 
is a graph for which every (equivalently,
some) generic realization in 
is globally rigid (Garamvölgyi et al. 2021).
A 
-dimensional
framework is a pair 
where 
is a graph with vertex set 
and edge set 
and 
is a map that assigns a point in 
to each vertex of 
.
The length of an edge 
in 
is the Euclidean distance between 
and 
.
Call 
a realization of 
in 
.
The framework is said to be generic if the set of the coordinates of its points is
algebraically independent over 
,
and is said to be globally rigid if every other realization 
of 
in 
in which corresponding edges have the same length is congruent
to 
, i.e., the graph 
and its edge lengths in 
uniquely determine the pairwise distances of all vertices
in 
.
If 
is globally rigid in 
on 
vertices, then 
is fully reconstructible
in 
(Garamvölgyi et al. 2021).
For generic global rigidity in the plane, Cong et al. (2026) gave deterministic polynomial-time algorithms to find a smallest edge set or vertex set whose deletion destroys global rigidity. Assigning positive integer deletion costs makes both optimization problems NP-hard, even when the input is globally rigid.
is Fully Reconstructible in 
." Submitted to Disc. Math., 2022.Cong,
Y.; Xu, C.; and Zhao, Y. "Destroying Rigidity in the Plane." 28 Oct 2026.