  A $d$-dimensional framework is a pair $(G,p)$, where $G=(V,E)$ is a graph and
$p$ is a map from $V$ to $\mathbb{R}^d$. The length of an edge $uv\in E$ in
$(G,p)$ is the distance between $p(u)$ and $p(v)$. The framework is said to be
globally rigid in $\mathbb{R}^d$ if the graph $G$ and its edge lengths uniquely
determine $(G,p)$, up to congruence. A graph $G$ is called globally rigid in
$\mathbb{R}^d$ if every $d$-dimensional generic framework $(G,p)$ is globally
rigid.
  In this paper, we consider the problem of reconstructing a graph from the set
of edge lengths arising from a generic framework. Roughly speaking, a graph $G$
is strongly reconstructible in $\mathbb{C}^d$ if the set of (unlabeled) edge
lengths of any generic framework $(G,p)$ in $d$-space, along with the number of
vertices of $G$, uniquely determine both $G$ and the association between the
edges of $G$ and the set of edge lengths. It is known that if $G$ is globally
rigid in $\mathbb{R}^d$ on at least $d+2$ vertices, then it is strongly
reconstructible in $\mathbb{C}^d$. We strengthen this result and show that
under the same conditions, $G$ is in fact fully reconstructible in
$\mathbb{C}^d$, which means that the set of edge lengths alone is sufficient to
uniquely reconstruct $G$, without any constraint on the number of vertices
(although still under the assumption that the edge lengths come from a generic
realization).
  As a key step in our proof, we also prove that if $G$ is globally rigid in
$\mathbb{R}^d$ on at least $d+2$ vertices, then the $d$-dimensional generic
rigidity matroid of $G$ is connected. Finally, we provide new families of fully
reconstructible graphs and use them to answer some questions regarding
unlabeled reconstructibility posed in recent papers.
