  Consider a network with $N$ nodes in $d$-dimensional Euclidean space, and $M$
subsets of these nodes $P_1,\cdots,P_M$. Assume that the nodes in a given $P_i$
are observed in a local coordinate system. The registration problem is to
compute the coordinates of the $N$ nodes in a global coordinate system, given
the information about $P_1,\cdots,P_M$ and the corresponding local coordinates.
The network is said to be uniquely registrable if the global coordinates can be
computed uniquely (modulo Euclidean transforms). We formulate a necessary and
sufficient condition for a network to be uniquely registrable in terms of
rigidity of the body graph of the network. A particularly simple
characterization of unique registrability is obtained for planar networks.
Further, we show that $k$-vertex-connectivity of the body graph is equivalent
to quasi $k$-connectivity of the bipartite correspondence graph of the network.
Along with results from rigidity theory, this helps us resolve a recent
conjecture due to Sanyal et al. (IEEE TSP, 2017) that quasi $3$-connectivity of
the correspondence graph is both necessary and sufficient for unique
registrability in two dimensions. We present counterexamples demonstrating that
while quasi $(d+1)$-connectivity is necessary for unique registrability in any
dimension, it fails to be sufficient in three and higher dimensions.
