Some new results on the k-tuple domination number of graphs

Martínez, Abel Cabrera

doi:10.1051/ro/2022159

Cited by 3 publications

(2 citation statements)

References 12 publications

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“…As in 2020, Ekinci G.B., Bujtas C. characterized bipartite graphs satisfying the equality for k ≥ 3 and presented a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily if k = 3 (10) . In 2022 (11) , author studied relationships between the k-tuple domination number and parameters of graph and different dominations. Also, (12) studied the total k-domination number of Cartesian product of two complete graphs and obtained some lower and upper bounds for the total k-domination number of Cartesian product of two complete graphs.…”

Section: Related Workmentioning

confidence: 99%

Lict k-Domination in Graphs

Gade,

Vyavahare

2024

IJST

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Objectives: In this study, we find lict -domination number of various types of graphs. Methods: Let be any graph, a set is said to be an -dominating set of lict graph if every vertex is dominated by at least vertices in , that is . The Lict -domination number is the minimum cardinality of -dominating set of . Findings: This study is centered on the lict -domination number of the graph and developed its relationship with other different domination parameters. Novelty: This study introduces the concept of “Lict -Domination in Graphs”. It obtains many bounds on in terms of vertices, edges, and other different parameters of . Keywords: Domination number, k-domination number, Lict graph, Lict k-domination number, Total domination number, Independent domination number

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Section: Related Workmentioning

confidence: 99%

Lict k-Domination in Graphs

Gade,

Vyavahare

2024

IJST

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“…F. Harary and T.W. Haynes in [10,11] introduced double domination, which generalises domination in graphs, and more generally, the concept of k-tuple domination, which has been studied in [12,13] too. Let k be a positive integer.…”

Section: Introductionmentioning

confidence: 99%

On Proper 2-Dominating Sets in Graphs

Bednarz,

Pirga

2024

Symmetry

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This paper introduces the concept of proper 2-dominating sets in graphs, providing a comprehensive characterisation of graphs that possess such sets. We give the necessary and sufficient conditions for a graph to have a proper 2-dominating set. Graphs with proper dominating sets can have a symmetric structure. Moreover, we estimate the bounds of the proper 2-domination number in the graphs with respect to the 2-domination and 3-domination numbers. We show that the cardinality of γ2¯-set is greater by one at most than the cardinality of γ2-set.

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Further results on the total Italian domination number of trees

Martínez

Peiró

Rueda-Vázquez

2023

MATH

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<abstract>Let $ f:V(G)\rightarrow \{0, 1, 2\} $ be a function defined from a connected graph $ G $. Let $ W_i = \{x\in V(G): f(x) = i\} $ for every $ i\in \{0, 1, 2\} $. The function $ f $ is called a total Italian dominating function on $ G $ if $ \sum_{v\in N(x)}f(v)\geq 2 $ for every vertex $ x\in W_0 $ and if $ \sum_{v\in N(x)}f(v)\geq 1 $ for every vertex $ x\in W_1\cup W_2 $. The total Italian domination number of $ G $, denoted by $ \gamma_{tI}(G) $, is the minimum weight $ \omega(f) = \sum_{x\in V(G)}f(x) $ among all total Italian dominating functions $ f $ on $ G $. In this paper, we provide new lower and upper bounds on the total Italian domination number of trees. In particular, we show that if $ T $ is a tree of order $ n(T)\geq 2 $, then the following inequality chains are satisfied. (ⅰ) $ 2\gamma(T)\leq \gamma_{tI}(T)\leq n(T)-\gamma(T)+s(T) $, (ⅱ) $ \frac{n(T)+\gamma(T)+s(T)-l(T)+1}{2}\leq \gamma_{tI}(T)\leq \frac{n(T)+\gamma(T)+l(T)}{2}, $ where $ \gamma(T) $, $ s(T) $ and $ l(T) $ represent the classical domination number, the number of support vertices and the number of leaves of $ T $, respectively. The upper bounds are derived from results obtained for the double domination number of a tree.</abstract>

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Some new results on the k-tuple domination number of graphs

Cited by 3 publications

References 12 publications

Lict k-Domination in Graphs

Lict k-Domination in Graphs

On Proper 2-Dominating Sets in Graphs

Further results on the total Italian domination number of trees

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