Bounds on weak and strong total domination in graphs

Akhbari, Mohammad Hadi; Rad, Nader Jafari

doi:10.5614/ejgta.2016.4.1.10

Cited by 6 publications

(5 citation statements)

References 7 publications

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“…A vertex v ∈ V(G) strongly dominates a vertex u ∈ V(G) in a graph G if uv ∈ E(G) and deg(u) ≥ deg(v). A dominating set S ⊆ V(G) in which every vertex u ∈ V − S is strongly dominated by some vertex v ∈ S is said to be a strong dominating set of the graph G, and the minimum cardinality of a strong dominating set is the strong domination number γ s (G) of the graph G (see [202]). A total dominating set S ⊆ V(G) in which every vertex u ∈ V − S is strongly dominated by some vertex, v ∈ S is said to be a total strong dominating set of a graph G, and the minimum cardinality of total strong dominating set of G is called the total strong dominating number of the graph, denoted by γ ts (refer to [202]).…”

Section: Unitary Addition Cayley Graphmentioning

confidence: 99%

Graphs Defined on Rings: A Review

Madhumitha,

Naduvath

2023

Mathematics

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The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two mathematical fields that focus on creating and analysing structures. There are numerous studies linking algebraic structures and graphs, which began with the introduction of Cayley graphs of groups. Several algebraic graphs have been defined on rings, a fast-growing area in the literature. In this article, we systematically review the literature on some variants of Cayley graphs that are defined on rings and highlight the properties and characteristics of such graphs, to showcase the research in this area.

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Section: Unitary Addition Cayley Graphmentioning

confidence: 99%

Graphs Defined on Rings: A Review

Madhumitha,

Naduvath

2023

Mathematics

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“…Stewart [15], and ReVelle and Rosing [14] defined and discussed the concept of Roman domination. Many papers were published on the Roman domination and its several variants, see, for examples, [2,9,10].…”

Section: Wwwejgtaorgmentioning

confidence: 99%

Total Roman domination for proper interval graphs

Poureidi

2020

EJGTA

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A function f : V → {0, 1, 2} is a total Roman dominating function (TRDF) on a graph G = (V, E) if for every vertex v ∈ V with f (v) = 0 there is a vertex u adjacent to v with f (u) = 2 and for every vertex v ∈ V with f (v) > 0 there exists a vertex u ∈ N G (v) with f (u) > 0. The weight of a total Roman dominating function f on G is equal to f (V ) = v∈V f (v). The minimum weight of a total Roman dominating function on G is called the total Roman domination number of G. In this paper, we give an algorithm to compute the total Roman domination number of a given proper interval graph G = (V, E) in O(|V |) time.

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“…The minimum cardinality of strong dominating set is called strong domination number and denote by γ st (G). Afterward, the strong domination number was introduced in [13] and in [17], and for more references about the strong dominating set we can see at [1,5,17,18].…”

Section: Introductionmentioning

confidence: 99%

On the resolving strong domination number of some wheel related graphs

Humaizah

Dafik

Kristiana

et al. 2022

J. Phys.: Conf. Ser.

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This study aims to analyse the resolving strong dominating set. This concept combinations of two notions, they are metric dimension and strong domination set. By a resolving strong domination set, we mean a set D s ⊂ V(G) which satisfies the definition of strong dominating set as well as resolving set. The resolving strong domination number of graph G, denoted by γrst (G), is the minimum cardinality of resolving strong dominating set of G. In this paper, we determine the resolving strong domination number of some wheel related graphs, namely helm graph Hn , gear graph Gn , and flower graph Fln . Through this paper, we will use the notations γst (G) and dim(G) which show the strong domination and dimension numbers.

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Bounds on weak and strong total domination in graphs

Cited by 6 publications

References 7 publications

Graphs Defined on Rings: A Review

Graphs Defined on Rings: A Review

Total Roman domination for proper interval graphs

On the resolving strong domination number of some wheel related graphs

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