Vertices contained in all or in no minimum k-dominating sets of a tree ∗

Let k be a positive integer and G = (V,E) be a simple graph. A subset S ⊆ V is dominating in G, if for each vertex v ∈ V \ S, N(v) ∩ S 6= ∅. In 1985, Fink and Jacobson gave a generalization of the concept of dominating sets in graphs. A subset S of V is kdominating in G, if every vertex of V \ S is adjacent to at least k vertices in S. In this paper, we characterize vertices that are in all or in no minimum k-dominating sets in a tree for k ≥ 2.