Reconstructing a pseudotree from the distance matrix of its boundary

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 ∂(G) of G is the set of all of its boundary vertices. The distance matrix D̂_G of the boundary of a graph G is the square matrix of order κ, being κ the order of ∂(G), such that for every i,j∈∂(G), [D̂_G]_ij=d_G(i,j). Given a square matrix B̂ of order κ, we prove under which conditions B̂ is the distance matrix D̂_T of the set of leaves of a tree T, which is precisely its boundary. We show that if G is either a tree or a unicyclic graph with girth g≥ 5 vertices, then G is uniquely determined by the distance matrix D̂_G of the boundary of G and we also conjecture that this statement holds for every connected graph. Moreover, two algorithms for reconstructing a tree and a unicyclic graph from the distance matrix of their boundaries are given, whose time complexities in the worst case are, respectively, O(κ n) and O(n^2).