The Roman domination number of some special classes of graphs - convex polytopes

Kartelj, Aleksandar; Grbić, Milana; Matić, Dragan; Filipović, Vladimir

doi:10.2298/aadm171211019k

Cited by 7 publications

(3 citation statements)

References 23 publications

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“…In the same paper polynomial-time algorithm is proposed on AT-free graphs. Specific type of graph, known as convex polytopes, as shown in [50], have tighter bounds for Roman domination number, that it is the general case.…”

Section: Roman Domination Problemmentioning

confidence: 99%

Topological Variable Neighborhood Search

Filipovic,

Kartelj

2024

Preprint

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The design of the novel metaheuristic method, called Topological Variable Neighborhood Search, is presented and its theoretical properties are elaborated.The proposed metaheuristic method is implemented, applied to several well-known NP-hard problems on graphs (Metric Dimension Problem, Roman Domination Problem and Maximum Betweeness Problem) and compared with the relevant optimization methods for these problems. The obtained experimental results clearly show that the proposed method achieves a significant improvement for the investigated problems and outperforms other methods that participated in the comparison. In particular, it was shown that the Topological Variable Neighborhood Search is consistently better than the classical Variable Neighborhood Search.

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Section: Roman Domination Problemmentioning

confidence: 99%

Topological Variable Neighborhood Search

Filipovic,

Kartelj

2024

Preprint

Self Cite

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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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“…The binary locating domination number (respectively locating domination number) is studied by Simić et al, 15 and the same problem was further studied for other classes of convex polytopes by Raza et al 16 The open locating domination number in graphs of convex polytopes was studied by Savića et al 17 and Raza 18 . Kartelj et al 19 studied the problem of roman domination number of some classes of convex polytopes. In the next section, we study the problem of locating-TD in some graphs of convex polytopes.…”

Section: Convex Polytopesmentioning

confidence: 99%

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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The Roman domination number of some special classes of graphs - convex polytopes

Cited by 7 publications

References 23 publications

Topological Variable Neighborhood Search

Topological Variable Neighborhood Search

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

Computing locating-total domination number in some rotationally symmetric graphs

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