Graphs with large disjunctive total domination number

Let G be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, γ_t(G). A set S of vertices in G is a disjunctive total dominating set of G if every vertex is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it. The disjunctive total domination number, γ_t^d(G), is the minimum cardinality of such a set. We observe that γ_t^d(G) ≤ γ_t(G). Let G be a connected graph on n vertices with minimum degree δ. It is known [J. Graph Theory 35 (2000), 21-45] that if δ ≥ 2 and n ≥ 11, then γ_t(G) ≤ 4n/7. Further [J. Graph Theory 46 (2004), 207-210] if δ ≥ 3, then γ_t(G) ≤ n/2. We prove that if δ ≥ 2 and n ≥ 8, then γ_t^d(G) ≤ n/2 and we characterize the extremal graphs.