Double domination in lexicographic product graphs

Martínez, Abel Cabrera; García, Suitberto Cabrera; Rodríguez-Velázquez, Juan A.

doi:10.1016/j.dam.2020.03.045

Cited by 22 publications

(5 citation statements)

References 21 publications

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“…is the minimum cardinality among all double dominating sets of G. We define a γ ×2 (G)-set as a double dominating set of cardinality γ ×2 (G). For more information about 2-domination and double domination in graphs, we suggest the chapter [11] due to Hansberg and Volkmann and the recent works [7,9]. A double dominating set D ⊆ V (G) is a total outer-independent dominating set of G if V (G)\D is an independent set.…”

Section: Additional Definitions and Previous Resultsmentioning

confidence: 99%

New Bounds on the Double Total Domination Number of Graphs

Martínez

Hernández-Mira

2021

Bull. Malays. Math. Sci. Soc.

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Let G be a graph of minimum degree at least two. A set $$D\subseteq V(G)$$ D ⊆ V ( G ) is said to be a double total dominating set of G if $$|N(v)\cap D|\ge 2$$ | N ( v ) ∩ D | ≥ 2 for every vertex $$v\in V(G)$$ v ∈ V ( G ) . The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.

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Section: Additional Definitions and Previous Resultsmentioning

confidence: 99%

New Bounds on the Double Total Domination Number of Graphs

Martínez

Hernández-Mira

2021

Bull. Malays. Math. Sci. Soc.

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“…For a basic introduction to the lexicographic product of two graphs we suggest the books [11,13]. One of the main problems in the study of G • H consists of finding exact values or tight bounds of specific parameters of these graphs and express them in terms of known invariants of G and H. In particular, we cite the following works on domination theory of lexicographic product graphs: (total) domination [18,21,28], Roman domination [25], weak Roman domination [27], rainbow domination [26], super domination [8], doubly connected domination [1], secure domination [15], double domination [3] and total Roman domination [6,4].…”

Section: The Case Of Lexicographic Product Graphsmentioning

confidence: 99%

On the 2-Packing Differential of a Graph

2021

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Let G be a graph of order $${\text {n}}(G)$$ n ( G ) and vertex set V(G). Given a set $$S\subseteq V(G)$$ S ⊆ V ( G ) , we define the external neighbourhood of S as the set $$N_e(S)$$ N e ( S ) of all vertices in $$V(G){\setminus } S$$ V ( G ) \ S having at least one neighbour in S. The differential of S is defined to be $$\partial (S)=|N_e(S)|-|S|$$ ∂ ( S ) = | N e ( S ) | - | S | . In this paper, we introduce the study of the 2-packing differential of a graph, which we define as $$\partial _{2p}(G)=\max \{\partial (S):\, S\subseteq V(G) \text { is a }2\text {-packing}\}.$$ ∂ 2 p ( G ) = max { ∂ ( S ) : S ⊆ V ( G ) is a 2 -packing } . We show that the 2-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of 2-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions. Among other results, we obtain a Gallai-type theorem, which states that $$\partial _{2p}(G)+\mu _{_R}(G)={\text {n}}(G)$$ ∂ 2 p ( G ) + μ R ( G ) = n ( G ) , where $$\mu _{_R}(G)$$ μ R ( G ) denotes the unique response Roman domination number of G. As a consequence of the study, we derive several combinatorial results on $$\mu _{_R}(G)$$ μ R ( G ) , and we show that the problem of finding this parameter is NP-hard. In addition, the particular case of lexicographic product graphs is discussed.

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“…This parameter was introduced by Cabrera et al in [4], and independently by Abdollahzadeh Ahangar et al in [1], under the name of total Roman {2}-domination number. The total Italian domination number of lexicographic product graphs was studied in [5].…”

Section: Introductionmentioning

confidence: 99%

From Italian domination in lexicographic product graphs to w-domination in graphs

2022

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In this paper, we show that the Italian domination number of every lexicographic product graph G • H can be expressed in terms of five different domination parameters of G. These parameters can be defined under the following unified approach, which encompasses the definition of several well-known domination parameters and introduces new ones.Let N (v) denote the open neighbourhood of v ∈ V (G), and let w = (w 0 , w 1 , . . . , w l ) be a vector of nonnegative integers such that w 0 ≥ 1. We say that a function f :The weight of f is defined to be ω(f ) = v∈V (G) f (v). The w-domination number of G, denoted by γ w (G), is the minimum weight among all wdominating functions on G.Specifically, we show that γ I (G • H) = γ w (G), where w ∈ {2} × {0, 1, 2} l and l ∈ {2, 3}. The decision on whether the equality holds for specific values of w 0 , . . . , w l will depend on the value of the domination number of H. This paper also provides preliminary results on γ w (G) and raises the challenge of conducting a detailed study of the topic.

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Double domination in lexicographic product graphs

Cited by 22 publications

References 21 publications

New Bounds on the Double Total Domination Number of Graphs

New Bounds on the Double Total Domination Number of Graphs

On the 2-Packing Differential of a Graph

From Italian domination in lexicographic product graphs to w-domination in graphs

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