  A vertex $v$ of a connected graph $G$ is said to be a boundary vertex of $G$
if for some other vertex $u$ of $G$, no neighbor of $v$ is further away from
$u$ than $v$. The boundary $\partial(G)$ of $G$ is the set of all of its
boundary vertices.
  The boundary distance matrix $\hat{D}_G$ of a graph $G=([n],E)$ is the square
matrix of order $\kappa$, being $\kappa$ the order of $\partial(G)$, such that
for every $i,j\in \partial(G)$, $[\hat{D}_G]_{ij}=d_G(i,j)$.
  Given a square matrix $\hat{B}$ of order $\kappa$, we prove under which
conditions $\hat{B}$ is the distance matrix $\hat{D}_T$ of the set of leaves of
a tree $T$, which is precisely its boundary.
  We show that if $G$ is either a block graph or a unicyclic graph, then $G$ is
uniquely determined by the boundary distance matrix $\hat{D}_{G}$ of $G$ and we
also conjecture that this statement holds for every connected graph $G$,
whenever both the order $n$ and the boundary (and thus also the boundary
distance matrix) of $G$ are prefixed.
  Moreover, an algorithm for reconstructing a 1-block graph (resp., a unicyclic
graph) from its boundary distance matrix is given, whose time complexity in the
worst case is $O(\kappa n)$ (resp., $O(n^2)$).
