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 partial derivative ( G ) of G is the set of all of its boundary vertices. The boundary distance matrix D G of a graph G = ([n], n ] , E ) is the square matrix of order kappa , with kappa being the order of partial derivative ( G ), such that for every i, j is an element of partial derivative ( G ), [ D G ] ij = dG(i, G ( i, j ). Given a square matrix B of order kappa , 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 D G of G and we also uniquely determined by the boundary distance matrix 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 )).