General distance domination

For any graph G=(V,E), a subset S⊆V dominates G if all vertices are contained in the closed neighborhood of S, that is N[S]=V. The minimum cardinality over all such S is called the domination number, written γ(G). For any positive integer k, a general k-distance domination function of a graph G is a function f:V→{0,1,…,k} such that every vertex with label 0 is at most distance j−1 away from a vertex with label j, for 2⩽j⩽k. We show some bounds for this function, produce a Vizing-like bound for the simplest case, and conjecture other more general bounds.