Reconstructing a graph from the boundary distance matrix

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 b(G) of G is the set of all of its boundary vertices. The boundary distance matrix D_G of a graph G = ([n], E) is the square matrix of order k, with k being the order of b(G), such that for every i,j ¿ b(G), [D_G]ij = dG(i,j). Given a square matrix B of order k, we prove under which conditions B is the distance matrix 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 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 (respectively, a unicyclic graph) from its boundary distance matrix is given, whose time complexity in the worst case is O(n) (respectively, O(n^2)).