Fault Tolerant Metric Dimensions of Leafless Cacti Graphs with Application in Supply Chain Management

TL;DR

This paper determines the fault tolerant metric dimensions of leafless cacti graphs and applies these findings to supply chain management scenarios.

Contribution

It introduces the concept of fault tolerant metric dimension for leafless cacti graphs and computes these values in terms of inner and outer cycles.

Findings

Fault tolerant metric dimension of type-I and II bicyclic graphs is always 4.

Derived formulas for fault tolerant metric dimensions of leafless cacti graphs.

Demonstrated application in supply chain management scenarios.

Abstract

A resolving set for a simple graph $G$ is a subset of vertex set of $G$ such that it distinguishes all vertices of $G$ using the shortest distance from this subset. This subset is a metric basis if it is the smallest set with this property. A resolving set is a fault tolerant resolving set if the removal of any vertex from the subset still leaves it a resolving set. The smallest set satisfying this property is the fault tolerant metric basis, and the cardinality of this set is termed as fault tolerant metric dimension of $G$, denoted by $\beta'(G)$. In this article, we determine the fault tolerant metric dimension of bicyclic graphs of type-I and II and show that it is always $4$ for both types of graphs. We then use these results to form our basis to consider leafless cacti graphs, and calculate their fault tolerant metric dimensions in terms of \textit{inner cycles} and \textit{outer cycles}. We then consider a detailed real world example of supply and distribution center management, and discuss the application of fault tolerant metric dimension in such a scenario. We also briefly discuss some other scenarios where leafless cacti graphs can be used to model real world problems.