On homometric sets in graphs

For a vertex set S ⊆ V ( G ) in a graph G, the distance multiset is the multiset of pairwise distances of vertices of S in G, denoted by D ( S ) . Two vertex sets S , T ⊆ V ( G ) are called homometric if D ( S ) = D ( T ) . For a graph G, the largest integer h, such that there are two disjoint homometric sets of order h in G, is denoted by h ( G ) . We investigate h ( n ) = min h ( G ) over all graphs on n vertices, as well as over special classes of graphs such as trees.