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Discussiones Mathematicae Graph Theory
Article . 2020 . Peer-reviewed
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Discussiones Mathematicae Graph Theory
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Bounds on the locating-total domination number in trees

Authors: Wang Kun; Ning Wenjie; Lu Mei;

Bounds on the locating-total domination number in trees

Abstract

The authors determine lower and upper bounds on the locating-total domination number of trees of order at least three. The locating-total domination number \(\gamma_t^L(G)\) of a graph \(G\) is the minimum cardinality of a locating-total dominating set, that is, a set of vertices that is both a total dominating set (every vertex in the graph has a neighbour in the set) and a locating set (no two vertices outside the set have the same neighbourhood in the set). Two lower bounds and one upper bound are proved in the paper. The upper bound is expressed in terms of \(n\) and \(d\), the order and the diameter of the tree, respectively. The two lower bounds are expressed in terms of the order, the number \(\ell\) of leaves, the number \(s\) of support vertices (neighbours of leaves), and the number \(s_1\) of strong support vertices (vertices adjacent to at least two leaves) of the tree. Extremal trees achieving the given bounds are characterized. In the abstract and in the introduction of the paper, the authors announce that they also show a lower bound in terms of the diameter. However, as they explain in Section 2, this bound holds for all connected graphs of order at least two and was proved by \textit{M. A. Henning} and \textit{N. J. Rad} [Discrete Appl. Math. 160, No. 13--14, 1986--1993 (2012; Zbl 1246.05119)]. Analyzing the proof of this lower bound, the authors characterize extremal trees achieving this bound, too. The four bounds can be summarized as follows: \[ \frac{d+1}{2}\leq \gamma_t^L(T)\leq n-\frac{d-1}{2} \] and \[ \gamma_t^L(T)\geq \max\left\{\frac{n+\ell-s+1}{2}-\frac{s+s_1}{4},\frac{2(n+1)+3(\ell-s)-s_1}{5}\right\}\,. \] The lower bound \(\gamma_t^L(T)\geq \frac{2(n+1)+3(\ell-s)-s_1}{5}\) refines (and implies) the bound \(\gamma_t^L(T)\geq \frac{2(n+\ell-s+1)}{5}\) proved by \textit{M. Chellali} [Discuss. Math., Graph Theory 28, No. 3, 383--392 (2008; Zbl 1173.05034)]. The lower bound \(\gamma_t^L(T)\geq \frac{n+\ell-s+1}{2}-\frac{s+s_1}{4}\) refines (and implies) the bound \(\gamma_t^L(T)\geq \frac{n+\ell+1}{2}-s\) proved by \textit{X.-G. Chen} and \textit{M. Y. Sohn} [Discrete Appl. Math. 159, No. 8, 769--773 (2011; Zbl 1223.05207)].

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Keywords

Extremal problems in graph theory, locating-total dominating set, total dominating set, tree, Trees, locating-total domination number, Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.), 05c69, QA1-939, Mathematics

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
1
Average
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