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Abstract
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In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [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 $\gnt(G)$, is the minimum cardinality of a NTD-set of $G$. Every total dominating set is a NTD-set, implying that $\gamma(G) \le \gnt(G) \le \gt(G)$, where $\gamma(G)$ and $\gt(G)$ denote the domination and total domination numbers of $G$, respectively. Arumugam and Sivagnanam showed that if $G$ is a connected graph on $n$ vertices with maximum degree~$\Delta < n-1$, then $\gnt(G) \le n - \Delta$ and pose the problem of characterizing the graphs $G$ achieving equality in this bound. We provide a complete solution to this problem for triangle-free graphs and a characterization for general graphs involving the packing number.
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