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.