Partition Dimension of Some Classes of Trees

Arimbawa, Komang; Baskoro, Edy Tri

doi:10.1016/j.procs.2015.12.077

Cited by 4 publications

(4 citation statements)

References 4 publications

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“…Lemma 2 (see [3]). Let C(p; δ) be a homogeneous caterpillar with any integers p, δ ≥ 1. en, pd C(p; δ) � 4 if and only if (δ � 3 and p ≥ 4) or (δ � 4 and p ≤ 4).…”

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

confidence: 99%

“…ey have promising uses in data reduction, modeling of interactions, computational chemistry, and ordering of graphs [1]. e partition dimension for various classes of trees such as stars, caterpillars, and homogeneous firecrackers have been computed; however, the values of partition dimensions for most kind of trees are still to be solved completely [2,3]. Among different parameters of graph theory, partition dimension of the graph is a unique and important parameter and has applications in network discovery and verification [4], mastermind games [5], and image processing [6].…”

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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“…Lemma 2 (see [3]). Let C(p; δ) be a homogeneous caterpillar with any integers p, δ ≥ 1. en, pd C(p; δ) � 4 if and only if (δ � 3 and p ≥ 4) or (δ � 4 and p ≤ 4).…”

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 previous studies, the partition dimension that has been studied is "Dimensions of Partitions of Several Classes of Trees". 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.…”

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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“…in [13] analyzed the partition dimension of cartesian product graphs, Arimbawa et.al. in [14] also studied the partition dimension of some classes of trees, as well as Juan et. al.…”

Section: Introductionmentioning

confidence: 99%

On the partition dimension of comb product of path and complete graph

Darmaji¹,

Alfarisi²

2017

AIP Conference Proceedings

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Abstract. For a vertex v of a connected graph G(V, E) with vertex set V(G), edge set E(G) and S ⊆ V(G). Given an ordered partition Π = {S 1 , S 2 , S 3 , ..., S k } of the vertex set V of G, the representation of a vertex v ∈ V with respect to Π is the vector is a resolving partition if different vertices of G have distinct representations, i.e., for every pair of vertices u, v ∈ V(G), r(u|Π) r(v|Π). The minimum k of Π resolving partition is a partition dimension of G, denoted by pd(G). Finding the partition dimension of G is classified to be a NP-Hard problem. In this paper, we will show that the partition dimension of comb product of path and complete graph. The results show that comb product of complete grapph K m and path P n namely pd(K m P n ) = m where m ≥ 3 and n ≥ 2 and pd(P n K m ) = m where m ≥ 3, n ≥ 2 and m ≥ n.

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Partition Dimension of Some Classes of Trees

Cited by 4 publications

References 4 publications

On Fault-Tolerant Partition Dimension of Homogeneous Caterpillar Graphs

On Fault-Tolerant Partition Dimension of Homogeneous Caterpillar Graphs

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

On the partition dimension of comb product of path and complete graph

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