The Partition Dimension of Some Families of Trees

Fredlina, Ketut Queena; Baskoro, Edy Tri

doi:10.1016/j.procs.2015.12.076

Cited by 12 publications

(5 citation statements)

References 4 publications

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“…Lemma 1 (see [18]). Let C(p; δ) be a homogeneous caterpillar with p, δ ≥ 1. en, pd C(p; δ) � 3 if and only if (δ � 1 and p ≥ 3) or (δ � 2 and p ≥ 2) or (δ � 3 and p ≤ 3).…”

Section: Fault-tolerant Partition Dimension Of the Homogeneous Caterp...mentioning

confidence: 99%

“…In [7], Chartrand et al characterised the graphs with partition dimension 2 or n. e partition dimensions for various classes of connected graphs have been obtained. For instance, some families of trees were discussed by Fredlina et al [18] and Rodŕiguez et al [19]. Amrullah studied the partition dimension problem for a subdivision of a homogeneous firecracker in [2].…”

Section: Introduction and Basic Terminologiesmentioning

confidence: 99%

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On Fault-Tolerant Partition Dimension of Homogeneous Caterpillar Graphs

Azhar

Zafar

Kashif

2021

Mathematical Problems in Engineering

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Metric-related parameters in graph theory have several applications in robotics, navigation, and chemical strata. An important such parameter is the partition dimension of graphs that plays an important role in engineering, computer science, and chemistry. In the context of chemical and pharmaceutical engineering, these parameters are used for unique representation of chemical compounds and their structural analysis. The structure of benzenoid hydrocarbon molecules is represented in the form of caterpillar trees and studied for various attributes including UV absorption spectrum, molecular susceptibility, anisotropy, and heat of atomization. Several classes of trees have been studied for partition dimension; however, in this regard, the advanced variant, the fault-tolerant partition dimension, remains to be explored. In this paper, we computed fault-tolerant partition dimension for homogeneous caterpillars C p ; 1 , C p ; 2 , and C p ; 3 for p ≥ 5 , p ≥ 3 , and p ≥ 4 , respectively, and it is found to be constant. Further numerical examples and an application are furnished to elaborate the accuracy and significance of the work.

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Section: Fault-tolerant Partition Dimension Of the Homogeneous Caterp...mentioning

confidence: 99%

Section: Introduction and Basic Terminologiesmentioning

confidence: 99%

On Fault-Tolerant Partition Dimension of Homogeneous Caterpillar Graphs

Azhar

Zafar

Kashif

2021

Mathematical Problems in Engineering

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“…In 2014, the partition dimension of unicyclic graphs was discussed in [12]. e partition dimension of some families of trees was studied by Fredlina and Baskaro [13] in 2015. In 2020, Estrado-Moreno [14] presented the concept of κ−partition dimension of the network.…”

Section: Background and Related Workmentioning

confidence: 99%

On the Fault Tolerant Partition Resolvability of Toeplitz Networks

Nadeem

Kashif

Aljaedi

et al. 2022

Mathematical Problems in Engineering

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In any interconnection network, fault tolerance is the most desirable property to achieve reliability. Toeplitz networks are used as interconnection networks due their smaller diameter, symmetry, simpler routing, high connectivity, and reliability. The partition dimension of a network is presented as an extension of metric dimension of networks. Its applications can be seen in several areas including robot navigation, network designing, image processing, and chemistry. In this article, the fault tolerant partition dimension, pd 2 T n 1 , t , of Toeplitz networks, is shown to be bounded below by 4 for t ≥ 2 , n ≥ 4 , whereas it is bounded above by 5 for t = 3 , n ≥ 14 . Further, it is shown that the exact value of pd 2 T n 1 , t equals 4 for t = 2 , n ≥ 4 ; t = 3 , n ∈ 5,6 , … , 13 ; and t ≥ 4 , n ∈ t + 2 , t + 3 , t + 4 .

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“…This results in the partition dimensions of several tree classes including a maximum of 17 possible representations of 𝑣 𝑖,1 , thus the maximum limit is 17 (Arimbawa & Baskoro, 2015). In addition, it was also investigated by Fredlina & Baskoro (2015) with the title "Dimensions of Partitions in Several Tree Graph Families" for example 𝐶(𝑚; 𝑛) is a homogeneous caterpillar with 𝑚, 𝑛 ≥ 1. Then, 𝑝𝑑(𝐶(𝑚; 𝑛)) = 3 if and only if (𝑛 = 1 and 𝑚 ≥ 3) or (𝑛 = 2 and 𝑚 ≥ 2) or (𝑛 = 3 and 𝑚 ≤ 3).…”

Section: A Introductionmentioning

confidence: 99%

Local Partition Dimension of Grid Graph and Its Application to the Coordinates of Potential Disaster Areas in Jember Regency

Saifudin,

Umilasari,

Rizal

2023

JTAM

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Partition dimension was introduced as a part of interesting topic in graph theory. It was focus to observe about distance. The local partition dimension is an expansion of the partition dimension by adding certain conditions to the representation of the vertices 𝑣 to which is Π denoted by 𝑟(𝑣| Π). One of the conditions that must be met for 𝑟(𝑣| Π) is discussed. The minimum value of k so that there is a local distinguishing partition from V (G) is the local partition dimension of G or it can can be said that the distance of each neighbor is different. The local partition dimension of a graph G is denoted 〖pd〗_l (G). In this study, we used an axiomatic deductive methods and pattern recognition. In order to construct the discriminating set on the metric dimension (dim) and the discriminating partition on the partition dimension (pd), the pattern detection method looks for patterns in which the coordinate values are minimum and different. By all observations, the local partition dimensions of P_n×P_m Grid graph has two condition about the results of resolving partition. The Result of local partition dimension of a Grid graph 〖〖pd〗_l (P〗_n×P_m)=2, where n≥2 dan m≥2. In addition, it will decide how to convert the coordinates of areas in the Jember district that are prone to flooding and landslides into metric dimensions. It was about Coordinates of Flood and Landslide Disaster Locations in Jember Regency. The number of local and minimal partition sets generated for flood-prone areas in Jember Regency is 〖pd〗_l (G_Jember)=3.

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The Partition Dimension of Some Families of Trees

Cited by 12 publications

References 4 publications

On Fault-Tolerant Partition Dimension of Homogeneous Caterpillar Graphs

On Fault-Tolerant Partition Dimension of Homogeneous Caterpillar Graphs

On the Fault Tolerant Partition Resolvability of Toeplitz Networks

Local Partition Dimension of Grid Graph and Its Application to the Coordinates of Potential Disaster Areas in Jember Regency

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