On Fault-Tolerant Partition Dimension of Homogeneous Caterpillar Graphs

Kamran Azhar,Agha Kashif,Sohail Zafar,Shaojian Qu

doi:10.1155/2021/7282245

Kamran Azhar, Agha Kashif + Show 2 more

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https://doi.org/10.1155/2021/7282245

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Abstract

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.

Highlights

Introduction and Basic TerminologiesGraph theory is a widely excelling branch of mathematics that is used to model and simplify the solution of daily-life problems
We extend this study by considering homogeneous caterpillars C(p; 1), C(p; 2), and C(p; 3) and show that they have constant fault-tolerant partition dimension
We have computed that F(C(p; δ)) for δ 1, 2, and 3 is between 3 and 5. e obtained results led us to the conclusion that the structures of homogeneous caterpillars C(p; 1), C(p; 2), and C(p; 3) have constant fault-tolerant partition dimension for p ≥ 5, p ≥ 3, and p ≥ 4, respectively

Summary

Introduction and Basic Terminologies

Graph theory is a widely excelling branch of mathematics that is used to model and simplify the solution of daily-life problems. In 2008, Hernando et al [14] initiated the concept of fault-tolerant metric dimension of graphs. E fault-tolerant metric dimension of Ψ is the minimum cardinality of fault-tolerant resolving set μ and is denoted by β′(Ψ). Discussed fault-tolerant metric dimension of circulant graphs in [17]. We define the partition dimension of graph Ψ as min􏼈|Ω|: Ω is resolving partition of Ψ􏼉, and it is denoted by pd(Ψ). E concept of fault-tolerant version of partition dimension of graphs was initiated by Salman et al [28]. Ψ is defined as min􏼈|Ω|: Ω is fault − tolerant resolving partition of Ψ􏼉 and is denoted by F(Ψ). We extend this study by considering homogeneous caterpillars C(p; 1), C(p; 2), and C(p; 3) and show that they have constant fault-tolerant partition dimension.

Fault-Tolerant Partition Dimension of the Homogeneous Caterpillar Graph

Conclusion