Double outer-independent domination number of graphs

Martínez, Abel Cabrera

doi:10.2989/16073606.2020.1834001

Cited by 3 publications

(3 citation statements)

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“…A double dominating set D ⊆ V (G) is a total outer-independent dominating set of G if V (G)\D is an independent set. The total outer-independent domination number of G, denoted by γ t,oi (G), is the minimum cardinality among all total outer-independent dominating sets of G. We define a γ t,oi (G)-set as a total outer-independent dominating set of cardinality γ t,oi (G) (see [5,6,8,24]).…”

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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“…The study of this domination parameter was initiated in [7]. Some recent and excellent results on this concept can be found, for example, in [6,8,9].…”

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confidence: 99%

New Bounds for Three Outer-Independent Domination-Related Parameters in Cactus Graphs

Cabrera-Martínez,

Rueda-Vázquez,

Segarra

2024

Axioms

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Let G be a nontrivial connected graph. For a set D⊆V(G), we define D¯=V(G)∖D. The set D is a total outer-independent dominating set of G if |N(v)∩D|≥1 for every vertex v∈V(G) and D¯ is an independent set of G. Moreover, D is a double outer-independent dominating set of G if |N[v]∩D|≥2 for every vertex v∈V(G) and D¯ is an independent set of G. In addition, D is a 2-outer-independent dominating set of G if |N(v)∩D|≥2 for every vertex v∈D¯ and D¯ is an independent set of G. The total, double or 2-outer-independent domination number of G, denoted by γtoi(G), γ×2oi(G) or γ2oi(G), is the minimum cardinality among all total, double or 2-outer-independent dominating sets of G, respectively. In this paper, we first show that for any cactus graph G of order n(G)≥4 with k(G) cycles, γ2oi(G)≤n(G)+l(G)2+k(G), γtoi(G)≤2n(G)−l(G)+s(G)3+k(G) and γ×2oi(G)≤2n(G)+l(G)+s(G)3+k(G), where l(G) and s(G) represent the number of leaves and the number of support vertices of G, respectively. These previous bounds extend three known results given for trees. In addition, we characterize the trees T with γ×2oi(T)=γtoi(T). Moreover, we show that γ2oi(T)≥n(T)+l(T)−s(T)+12 for any tree T with n(T)≥3. Finally, we give a constructive characterization of the trees T that satisfy the equality above.

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“…A basic problem in the study of product graphs consists of finding closed formulas or sharp bounds for specific invariants of the product of two graphs and expressing these in terms of parameters of the graphs involved in the product. In this sense, for recent results on rooted product graphs, we cite the following works [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. As we can expect, the products of graphs are not alien to applications in other fields.…”

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confidence: 99%

Total Domination in Rooted Product Graphs

Martínez¹,

Rodríguez-Velázquez²

2020

Symmetry

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During the last few decades, domination theory has been one of the most active areas of research within graph theory. Currently, there are more than 4400 published papers on domination and related parameters. In the case of total domination, there are over 580 published papers, and 50 of them concern the case of product graphs. However, none of these papers discusses the case of rooted product graphs. Precisely, the present paper covers this gap in the theory. Our goal is to provide closed formulas for the total domination number of rooted product graphs. In particular, we show that there are four possible expressions for the total domination number of a rooted product graph, and we characterize the graphs reaching these expressions.

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Double outer-independent domination number of graphs

Cited by 3 publications

References 8 publications

New Bounds on the Double Total Domination Number of Graphs

New Bounds on the Double Total Domination Number of Graphs

New Bounds for Three Outer-Independent Domination-Related Parameters in Cactus Graphs

Total Domination in Rooted Product Graphs

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