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 , , and for , , and , 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.

Graph invariants provide an amazing tool to analyze the abstract structures of networks. The interaction and interconnection between devices, sensors, and service providers have opened the door for an eruption of mobile over the web applications. Structure of web sites containing number of pages can be represented using graph, where web pages are considered to be the vertices, and an edge is a link between two pages. Figuring resolving partition of the graph is an intriguing inquest in graph theory as it has many applications such as sensor design, compound classification in chemistry, robotic navigation, and Internet network. The partition dimension is a graph parameter akin to the concept of metric dimension, and fault-tolerant partition dimension is an advancement in the line of research of partition dimension of the graph. In this paper, we compute fault-tolerant partition dimension of alternate triangular cycle, mirror graph, and tortoise graphs.

In this paper, we study the fault-tolerant metric dimension in graph theory, an important measure against failures in unique vertex identification. The metric dimension of a graph is the smallest number of vertices required to uniquely identify every other vertex based on their distances from these chosen vertices. Building on existing work, we explore fault tolerance by considering the minimal number of vertices needed to ensure that all other vertices remain uniquely identifiable even if a specified number of these vertices fails. We compute the fault-tolerant metric dimension of various chemical graphs, namely fullerenes, benzene, and polyphenyl graphs.

The concept of metric-related parameters permeates all of graph theory and plays an important role in diverse networks, such as social networks, computer networks, biological networks and neural networks. The graph parameters include an incredible tool for analyzing the abstract structures of networks. An important metric-related parameter is the partition dimension of a graph holding auspicious applications in telecommunication, robot navigation and geographical routing protocols. A fault-tolerant resolving partition is a preference for the concept of a partition dimension. A system is fault-tolerant if failure of any single unit in the originally used chain is replaced by another chain of units not containing the faulty unit. Due to the optimal fault tolerance, cycle-related graphs have applications in network analysis, periodic scheduling and surface reconstruction. In this paper, it is shown that the partition dimension (PD) and fault-tolerant partition dimension (FTPD) of cycle-related graphs, including kayak paddle and flower graphs, are constant and free from the order of these graphs. More explicitly, the FTPD of kayak paddle and flower graphs is four, whereas the PD of flower graphs is three. Finally, an application of these parameters in a scenario of installing water reservoirs in a locality has also been furnished in order to verify our findings.

The distance‐based graph parameters are being used extensively in the fields of chemistry, computer science, scheduling and in many operations’ management problems. The unique identification of vertices plays a key role in the computation of these parameters. The local versions of these parameters deal with adjacent vertices and, in this manner, provide more applicability. In this manuscript, the exact values of local metric dimension and local partition dimension of tadpole and necklace graphs have been computed. The article also discusses the application of a local metric basis in the optimal allocation of fire stations in a region.

Fault tolerant metric basis assigns a unique code in terms of distances to each node of the graph which works in the presence of faults. The distance based parameters are widely used in different fields like robot navigation, interconnection networks, sensor deployments, image processing and chemistry. The convex polytopes are stable and destroy resistant which make them attractive choice for interconnection networks and anti-tracking networks. The current article, computes the fault tolerant metric dimension (FTMD) of convex polytopes $B_n, C_n, E_n$ and $U_n$. The applications of FTMD have also been considered in the manuscript.