Lower Bounds on the Distance Domination Number of a Graph

For an integer k ≥ 1, a (distance) k-dominating set of a connected graph G is a set S of vertices of G such that every vertex of V (G) \ S is at distance at most k from some vertex of S. The k-domination number, γk(G), of G is the minimum cardinality of a k-dominating set of G. In this talk, we establish lower bounds on the k-domination number of a graph in terms of its diameter, radius and girth. We prove that for connected graphs G and H, γk(G×H) ≥ γk(G) + γk(H)− 1, where G×H denotes the direct product of G and H. ∗Research supported in part by the South African National Research Foundation and the University of Johannesburg