A distance compatible set labeling (dcsl) of a connected graph G is an injective set assignment  X being a non-empty ground set such that the corresponding induced function    given by  satisfies    for every pair of distinct vertices  where  denotes the path distance between u and v and  is a constant, not necessarily an integer, depending on the pair of vertices u, v chosen. A dcsl f of G is k-uniform if all the constants of proportionality with respect to f are equal to k, and if G admits such a dcsl, then G is called a k-uniform dcsl graph. Let  be a family of subsets of a set X. A tight path between two distinct sets P and Q (or from P to Q) in  is a sequence  in  such that  and  for    The family  is well-graded family (or wg-family), if there is a tight path between any two of its distinct sets. In this paper, we establish the relationship between 1-uniform dcsl graphs and wg-family of sets.

dcsl graphs, 1-uniform dcsl graphs, wg-family of sets.