Bounds on the locating-domination number and differentiating-total  domination number in trees

Rad, Nader Jafari; Rahbani, Hadi

doi:10.7151/dmgt.2012

Cited by 4 publications

(2 citation statements)

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“…For an identifiable graph G, we denote by γ ID (G) the smallest size of an identifying code of G. In the context of total dominating identifying codes, by saying a graph is identifiable we also assume implicitly that it admits a total dominating set. For such an identifiable graph G, we denote by γ ID t (G) the smallest size of a total dominating identifying code of G. Total dominating identifying codes have been studied only in a handful of papers, see [8,21,30,31,32,33,35].…”

Section: Introductionmentioning

confidence: 99%

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Bounds and Extremal Graphs for Total Dominating Identifying Codes

Foucaud

Lehtilä

2023

Electron. J. Combin.

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An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. The smallest size of an identifying code of $G$ is denoted $\gamma^{\text{ID}}(G)$. When every vertex of $G$ also has a neighbour in $C$, it is said to be a total dominating identifying code of $G$, and the smallest size of a total dominating identifying code of $G$ is denoted by $\gamma_t^{\text{ID}}(G)$. Extending similar characterizations for identifying codes from the literature, we characterize those graphs $G$ of order $n$ with $\gamma_t^{\text{ID}}(G)=n$ (the only such connected graph is $P_3$) and $\gamma_t^{\text{ID}}(G)=n-1$ (such graphs either satisfy $\gamma^{\text{ID}}(G)=n-1$ or are built from certain such graphs by adding a set of universal vertices, to each of which a private leaf is attached). Then, using bounds from the literature, we remark that any (open and closed) twin-free tree of order $n$ has a total dominating identifying code of size at most $\frac{3n}{4}$. This bound is tight, and we characterize the trees reaching it. Moreover, by a new proof, we show that this upper bound actually holds for the larger class of all twin-free graphs of girth at least 5. The cycle $C_8$ also attains the upper bound. We also provide a generalized bound for all graphs of girth at least 5 (possibly with twins). Finally, we relate $\gamma_t^{\text{ID}}(G)$ to the similar parameter $\gamma^{\text{ID}}(G)$ as well as to the location-domination number of $G$ and its variants, providing bounds that are either tight or almost tight.

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Section: Introductionmentioning

confidence: 99%

“…The computational problem associated with determining γ ID t (G) for an input graph G is NP-hard, and has been studied in [33]. Lower and upper bounds for parameter γ ID t in trees have been proved in [8,21,30,35]. Different graph classes, in particular graph products, were studied in [31,32].…”

Section: Introductionmentioning

confidence: 99%

Bounds and Extremal Graphs for Total Dominating Identifying Codes

Foucaud

Lehtilä

2023

Electron. J. Combin.

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Locating-Domination and Identification

Lobstein

Hudry

Charon

2020

Topics in Domination in Graphs

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Locating-domination and identification are two particular, related, types of domination: a set C of vertices in a graph G = (V, E) is a locating-dominating code if it is dominating and any two vertices of V \ C are dominated by distinct sets of codewords; C is an identifying code if it is dominating and any two vertices of V are dominated by distinct sets of codewords. This chapter presents a survey of the major results on locating-domination and on identification.

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Bounds on the disjunctive domination number of a tree

Zhuang

2022

RAIRO-Oper. Res.

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A set $D$ of vertices in a graph $G$ is a disjunctive dominating set in $G$ if every vertex not in $D$ is adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it in $G$. The disjunctive domination number, $\gamma^{d}_2(G)$, of $G$ is the minimum cardinality of a disjunctive dominating set in $G$. We show that if $T$ is a tree of order $n$ with $l$ leaves and $s$ support vertices, then $\frac{n-l+3}{4}\leq \gamma^{d}_2(T)\leq \frac{n+l+s}{4}$. Moreover, we characterize the families of trees which attain these bounds.

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Bounds on the locating-domination number and differentiating-total domination number in trees

Cited by 4 publications

References 8 publications

Bounds and Extremal Graphs for Total Dominating Identifying Codes

Bounds and Extremal Graphs for Total Dominating Identifying Codes

Locating-Domination and Identification

Bounds on the disjunctive domination number of a tree

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