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)=minh(G) over all graphs on n vertices, as well as over special classes of graphs such as trees.