Bounds on the disjunctive total domination number of a tree

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, Υ_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, Υ^d_t (G), is the minimum cardinality of such a set. We observe that Υ^d_t (G) ≤ Υ_t(G). A leaf of G is a vertex of degree 1, while a support vertex of G is a vertex adjacent to a leaf. We show that if T is a tree of order n with . leaves and s support vertices, then 2(n-ℓ+3)/5 ≤ Υ^d_t (T) . (n+s.1)/2 and we characterize the families of trees which attain these bounds. For every tree T, we show have Υ_t(T)/Υ^d_t (T) ≤ 2 and this bound is asymptotically tight.