A graph 
is fully reconstructible in 
if the graph is determined from its 
-dimensional measurement variety. This is the algebraic variety
of squared edge-length lists obtained from placements of 
in 
. If 
is globally rigid in 
on 
vertices, then 
is fully reconstructible in 
(Garamvölgyi et al. 2021). The full reconstructibility
problem has been solved for 
and 
(Bernstein and Gortler 2022).
This is not reconstructibility in the sense of the graph reconstruction conjecture. A reconstructible graph is determined by its graph deck of vertex-deleted subgraphs, not by edge-length data.
For 
a graph on three or more vertices with no isolated
vertices, 
is fully reconstructible in 
iff it is 3-connected (Garamvölgyi
et al. 2021, Bernstein and Gortler 2022).
For 
a graph on four or more vertices, 
is fully reconstructible in 
iff 
is generically globally
rigid in two dimensions (Garamvölgyi et al. 2021, Bernstein and Gortler
2022).
The disposition of full reconstructibility of graphs in 
is not fully determined, although some necessary and some
sufficient conditions are known. Bernstein and Gortler (2022) showed that the complete bipartite graph 
is fully reconstructible in 
.
is Fully Reconstructible in 
." Submitted to Disc. Math., 2022.Garamvölgyi,
D.; Gortler, S. J.; and Jordán, J. "Globally Rigid Graphs Are Fully
Reconstructible." 10 May 2021.