Computing locating-total domination number in some rotationally symmetric graphs

Raza, Hassan; Iqbal, Naveed; Khan, Hassan; Botmart, Thongchai

doi:10.1177/00368504211053417

Cited by 3 publications

(1 citation statement)

References 19 publications

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“…The prism graph 𝐷 n can be constructed by taking the Cartesian product of the cycle 𝐶 n and the path 𝑃 2 [13], resulting in a 3-regular graph depicted in Figure 3. According to the research conducted by Raza et al [14] and Kartelj et al [15], this graph belongs to the category of Archimedean convex polytopes. Furthermore, it should be emphasized that the prism graph can be considered identical to the Petersen graph, denoted as 𝑃(𝑛, 1).…”

Section: The Prism Graph 𝑫 𝒏mentioning

confidence: 99%

Dominance Parameters in Prism Graphs: A Comparative Study of Minimum Dominating Sets

Rajagopal,

Narayan

2024

MMEP

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Let G=(V, E) be a graph. A dominating set S of graph G is defined as a set of vertices such that every vertex in V\S is adjacent to at least one vertex in S. The domination number of graph G, denoted as γ(G), corresponds to the size of the smallest dominating set within G. In other words, γ(G) represents the number of vertices required in the minimum dominating set to cover all other vertices in the graph G. In the graph G, our objective is to position a protector at each vertex within a subset S of V, ensuring that S forms a dominating set, effectively covering all other vertices in G. Moreover, in the event that a protector positioned at vertex 𝑣 needs to move along an edge to protect an unguarded vertex u, the arrangement of protectors should maintain the property of forming a dominating set for the graph. In other words, the movement of protectors should maintain the property of domination within the graph, ensuring efficient coverage and defense across the network. The bare minimum of security guards is necessary to protect all vertices in the graphs. In this article, we find the bounds for domination, independent domination number (IDN), connected domination number(CDN), total domination number(TDN), and the secure domination number(SDN) denoted byγ(A n ), γ i (A n ), γ c (A n ), γ t and γ s (A n ) respectively for the antiprism graph, where A n denoted the 4 -regular graph with girth 3. We further establish that the TDN is greater than or equal to the SDN of the antiprism graph for 𝑛 ≥ 3.

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Section: The Prism Graph 𝑫 𝒏mentioning

confidence: 99%

Dominance Parameters in Prism Graphs: A Comparative Study of Minimum Dominating Sets

Rajagopal,

Narayan

2024

MMEP

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Computing locating-total domination number in some rotationally symmetric graphs

et al. 2021

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Let [Formula: see text] be a connected graph. A locating-total dominating set in a graph G is a total dominating set S of a G, for every pair of vertices [Formula: see text], such that [Formula: see text]. The minimum cardinality of a locating-total dominating set is called locating-total domination number and represented as [Formula: see text]. In this paper, locating-total domination number is determined for some cycle-related graphs. Furthermore, some well-known graphs of convex polytopes from the literature are also considered for the locating-total domination number.

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Finding the domination number of triangular belt networks

Almotairi,

Alharbi,

Alzaid

et al. 2024

Math. models, eng.

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Domination in graphs has been an extensively researched branch of graph theory. A set S⊆V is said to be the dominating set of graph G if for every v∈V-S there exists a vertex u∈S such that uv∈E. The minimum cardinality of vertices among the dominating set of G is called the domination number of G denoted by γ(G). The main object of this paper is to study the domination number of some networks such as triangular belt networks TBn, alternate triangular belt networks ATBn and restricted square of bistar networks Bn,n.

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Computing locating-total domination number in some rotationally symmetric graphs

Cited by 3 publications

References 19 publications

Dominance Parameters in Prism Graphs: A Comparative Study of Minimum Dominating Sets

Dominance Parameters in Prism Graphs: A Comparative Study of Minimum Dominating Sets

Computing locating-total domination number in some rotationally symmetric graphs

Finding the domination number of triangular belt networks

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