On Sharp Bounds of Fault-Tolerant Partition Dimension of Convex Polytopes

Graph theory is a fundamental and powerful tool for designing and modeling networks. It plays a vital role in diverse real-world systems, including social, computer, biological, ecological, and neural networks. Convex polytopes are convex sets of elements contained in the Euclidean space Rn. They arise in numerous domains such as linear programming, finance, computer science, electrical engineering, bioinformatics, and chemistry. From the graph-theoretic perspective, some sharp upper bounds for the partition dimension of convex polytopes have been established recently. Inspired by this, in this paper, we aim to investigate the fault-tolerant partition dimension (FTPD), which is an extension of the partition dimension. Here, we compute bounds for the FTPD of certain convex polytopes. These results contribute to a deeper understanding of the robustness of network structures modeled by convex polytopes and may support further applications in fault-tolerant system design.