Geodesic
Bounds On The Disjunctive Total Domination Number Of A Tree
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 1, pp. 153-171

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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). 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 ≤γ_t^d (T) ≤ (n+s−1)//2 and we characterize the families of trees which attain these bounds. For every tree T, we show have γ_t(T) // γ_t^d (T) lt;2 and this bound is asymptotically tight.
Keywords: total domination, disjunctive total domination, trees 
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Henning, Michael A.; Naicker, Viroshan. Bounds On The Disjunctive Total Domination Number Of A Tree. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 1, pp. 153-171. http://geodesic.mathdoc.fr/item/DMGT_2016_36_1_a11/