Bounds on neighborhood total domination in graphs

In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [S. Arumugam, C. Sivagnanam, Neighborhood total domination in graphs, Opuscula Math. 31 (2011) 519–531]. A neighborhood total dominating set, abbreviated NTD-set, in a graph GG is a dominating set SS in GG with the property that the subgraph induced by the open neighborhood of the set SS has no isolated vertex. The neighborhood total domination number, denoted by γnt(G), is the minimum cardinality of a NTD-set of GG. Every total dominating set is a NTD-set, implying that γ(G)≤γnt(G)≤γt(G), where γ(G)γ(G) and γt(G)γt(G) denote the domination and total domination numbers of GG, respectively. We show that if GG is a connected graph on n≥3n≥3 vertices, then γnt(G)≤(n+1)/2 and we characterize the graphs achieving equality in this bound.