Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Trees with Large Neighborhood Total Domination Number   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.

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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: comp-gas
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The categories of this paper are: astro-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th