### Graphs with Large Disjunctive Total Domination Number

*Michael Anthony Henning, Viroshan Naicker*

#### Abstract

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^{d}_{t}(G), is the minimum cardinality of such a set. We observe that γ^{d}_{t}(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 13;45] that if δ≥2 and n ≥11, then γ_{t}(G) ≤4n/7. Further [J. Graph Theory 46 (2004), 207 13;210] if δ≥3, then γ_{t}(G) ≤n/2. We prove that if δ≥2 and n ≥8, then γ^{d}_{t}(G) ≤n/2 and we characterize the extremal graphs.Full Text: PDF PostScript