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 G is a dominating set S in G with the property that the subgraph induced by the open neighborhood of the set S has no isolated vertex. The neighborhood total domination number, denoted by ^γnt(G), is the minimum cardinality of a NTD-set of G. Every total dominating set is a NTD-set, implying that γ(G)≤^γnt(G)≤^γt(G), where γ(G) and ^γt(G) denote the domination and total domination numbers of G, respectively. We show that if G is a connected graph on n≥3 vertices, then ^γnt(G)≤(n+1)/2 and we characterize the graphs achieving equality in this bound.