Bounds on a generalized domination parameter

Abstract If n is an integer, n ≥ 2 and u and v are vertices of a graph G, then u and v are said to be Kn-adjacent vertices of G if there is a subgraph of G, isomorphic to Kn , containing u and v. For n ≥ 2, a Kn- dominating set of G is a set D of vertices such that every vertex of G belongs to D or is Kn-adjacent to a vertex of D. The Kn-domination number γKn (G) of G is the minimum cardinality among the Kn-dominating sets of vertices of G. It is shown that, for n e {3,4}, if G is a graph of order p with no Kn-isolated vertex, then γKn (G) ≤ p/n. We establish that this is a best possible upper bound. It is shown that the result is not true for n ≥ 5.