On exact n-step domination Original Research Article

We generalize to n steps the notion of exact 2-step domination introduced by Chartrand et al. (Math. Bohem. 120 (1995) 125–134) and suggest a related minimization problem for which we find a lower bound. A graph G is an exact n-step domination graph if there is some set of vertices in G such that each vertex in the graph is distance n from exactly one vertex in the set. We prove that such subsets have order at least ⌊log2 n⌋+2 and limit how much better a bound is possible. We also prove a related conjecture of Alavi et al. (Graph Theory, Combinatorics, and Applications, vol. 1, Wiley, New York, 1991, pp. 1–8) that if each vertex in a connected graph G has exactly one vertex distance n from it then the diameter is n unless G is a path consisting of 2n vertices.