On Fault-Tolerant Resolving Sets of Some Families of Ladder Networks

Hua Wang,Muhammad Faisal Nadeem,Adnan Aslam,Muhammad Azeem,Ata Ur-Rehman,Muhammad Javaid

doi:10.1155/2021/9939559

Abstract

In computer networks, vertices represent hosts or servers, and edges represent as the connecting medium between them. In localization, some special vertices (resolving sets) are selected to locate the position of all vertices in a computer network. If an arbitrary vertex stopped working and selected vertices still remain the resolving set, then the chosen set is called as the fault-tolerant resolving set. The least number of vertices in such resolving sets is called the fault-tolerant metric dimension of the network. Because of the variety of applications of the metric dimension in different areas of sciences, many generalizations were proposed, and fault tolerant is one of them. In this paper, we computed the fault-tolerant metric dimension of triangular snake, ladder, Mobius ladder, and hexagonal ladder networks. It is important to observe that, in all these classes of networks, the fault-tolerant metric dimension and metric dimension differ by 1.

Highlights

Let G (V(G), E(G)) be a simple connected graph, where V(G) and E(G) are the set of vertices and edges, respectively.dimf(G) ≥ dim(G) + 1. (1)First time, the idea of metric dimension was studied by Slater [2] in 1975 and later by Harary and Melter [3] in 1976
The idea of metric dimension was studied by Slater [2] in 1975 and later by Harary and Melter [3] in 1976
Hernando et al [5] presented the idea of faulttolerant metric dimension to overcome this kind of problems

Summary

Introduction

Let G (V(G), E(G)) be a simple connected graph, where V(G) and E(G) are the set of vertices and edges, respectively. The study of fault-tolerant resolving sets of different networks is as sundry as the study of the metric dimension is. Computing the fault-tolerant metric dimension is an NP-complete problem. Computing the fault-tolerant metric dimension is considered as one of the interesting but difficult problems in combinatorics. In the following Lemma 1, we calculate the metric dimension dim(TSn) of triangular snake graph TSn. Lemma 1. Let TSn be a triangular snake graph with s ≥ 2. As all the vertices have unique representations regarding the resolving set Q, dim(TSn) ≤ 2. It follows from the above discussion that dimf(TSn) ≤ 4 since every vertex of TSn has a unique representation regarding resolving set Qf. We prove the reverse inequality dimf(TSn) ≥ 4 by the contradiction method. It follows from the above discussion that dimf(TSn) ≠ 3 when s ≥ 4. is implies that dimf(TSn) ≥ 4 when s ≥ 4. is completes the proof

Fault-Tolerant Metric Dimension of Ladder Graphs

Conclusion