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.