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arXiv · 2409.12199

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

Tauseef Asif·Ghulam Haidar·Faisal Yousafzai·Murad Ul Islam Khan·Qaisar Khan·Rakea Fatima
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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.

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Tauseef Asif, Ghulam Haidar, Faisal Yousafzai, Murad Ul Islam Khan, Qaisar Khan, Rakea Fatima. 2024-09-09. Fault Tolerant Metric Dimensions of Leafless Cacti Graphs with Application in Supply Chain Management. https://doi.org/10.1016/j.tcs.2026.116025

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