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, γt(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, γtd(G)$\gamma _t^d (G)$ , is the minimum cardinality of such a set. We observe that γtd(G)≤γt(G)$\gamma _t^d (G) \le \gamma _t (G)$ . 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 ℓ leaves and s support vertices, then 2(n−ℓ+3)/5≤γtd(T)≤(n+s−1)/2$2(n - \ell + 3)/5 \le \gamma _t^d (T) \le (n + s - 1)/2$ and we characterize the families of trees which attain these bounds. For every tree T, we show have γt(T)/γtd(T)<2$\gamma _t (T)/\gamma _t^d (T) < 2$ and this bound is asymptotically tight