Graphs having distance-n domination number half their order Original Research Article

For any positive integer n and any graph G a set D of vertices of G is a distance-n dominating set, if every vertex v∈V(G)−D has exactly distance n to at least one vertex in D. The distance-n domination number γ=n(G) is the smallest number of vertices in any distance-n dominating set. If G is a graph of order p and each vertex in G has distance n to at least one vertex in G, then the distance-n domination number has the upper bound p/2 as Oreʹs upper bound on the classical domination number. In this paper, a characterization is given for graphs having distance-n domination number equal to half their order, when the diameter is greater or equal 2n−1. With this result we confirm a conjecture of Boland, Haynes, and Lawson.