标题： 具有大邻域总数支配数的树 显示英文标题

标题： Trees with Large Neighborhood Total Domination Number

摘要： 在本文中，我们继续研究图中的邻域全支配，该研究最初由Arumugam和Sivagnanam [Opuscula Math. 31 (2011), 519--531]进行。图$G$中的邻域全支配集，简称为NTD集，是在$G$中的一个支配集$S$，其特性是该集合$S$的开邻域所诱导的子图没有孤立顶点。邻域全支配数，记为$\gnt(G)$，是$G$的NTD集的最小基数。 每个全支配集都是NTD集，这意味着$\gamma(G) \le \gnt(G) \le \gt(G)$，其中$\gamma(G)$和$\gt(G)$分别表示$G$的支配数和全支配数。Arumugam和Sivagnanam提出了一个问题是表征连通图$G$的阶数$n \ge 3$，以达到最大的邻域全支配数，即$\gnt(G) = \lceil n/2 \rceil$。 这一问题的部分解由Henning和Rad[离散应用数学161 (2013), 2460--2466]提出，他们证明当$n$为奇数时，$5$-圈和细分星形是唯一达到该界限的图。在本文中，我们表征了当$n$为偶数时达到该界限的极值树。作为这种树表征的结果，当$n$为偶数时，可以通过注意到这样的图的每个生成树都属于我们的极值树家族，从而得到达到该界限的连通图的表征。 显示英文摘要

摘要： 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 posed the problem of characterizing the connected graphs $G$ of order $n \ge 3$ achieving the largest possible neighborhood total domination number, namely $\gnt(G) = \lceil n/2 \rceil$. A partial solution to this problem was presented by Henning and Rad [Discrete Applied Mathematics 161 (2013), 2460--2466] who showed that $5$-cycles and subdivided stars are the only such graphs achieving equality in the bound when $n$ is odd. In this paper, we characterize the extremal trees achieving equality in the bound when $n$ is even. As a consequence of this tree characterization, a characterization of the connected graphs achieving equality in the bound when $n$ is even can be obtained noting that every spanning tree of such a graph belongs to our family of extremal trees.