Metric dimensions of bicyclic graphs with potential applications in Supply Chain Logistics

Metric dimensions and metric basis are graph invariants studied for their use in locating and indexing nodes in a graph. It was recently established that for bicyclic graph of type-III ($\Theta $-graphs), the metric dimension is $3$ only, when all paths have equal lengths, or when one of the outside path has a length $2$ more than the other two paths. In this article, we refute this claim and show that the case where the middle path is $2$ vertices more than the other two paths, also has metric dimension $3$. We also determine the metric dimension for other values of $p,q,r$ which were omitted in the recent research due to the constraint $p \leq q \leq r$. We also propose a graph-based technique to transform an agricultural supply chain logistics problem into a mathematical model, by using metric basis and metric dimensions. We provide a theoretical groundwork which can be used to model and solve these problems using machine learning algorithms.