Optimizing emergency response services in urban areas through the fault-tolerant metric dimension of hexagonal nanosheet.

In this work, we find the fault-tolerant metric dimension of a hexagonal nanosheet. This concept ensures robust identity of vertices inside a graph, even in situations in which a few resolving vertices fail. By extending the applicability of this parameter to city systems, we explore its practical use in optimizing emergency reaction and carrier management in cities. Here, houses are modeled as vertices, roads as edges, and crucial service hubs (e.g., hospitals, heart stations) as resolving points. The fault-tolerant metric dimension presents a scientific technique to identifying minimum, strategic places for carrier hubs such that the town stays uniquely resolvable even at some point of partial disruptions, including natural screw ups or infrastructural failures. We show that the hexagonal nanosheet structure offers precious insights into resilience techniques for city structures, due to its symmetry and fault tolerance. This work bridges the gap between theoretical graph structures and real-world applications, showcasing how the fault-tolerant metric dimension may be harnessed to design resilient, price-powerful city structures that ensure uninterrupted provider shipping under unfavorable situations. Our findings lay the foundation for similarly interdisciplinary packages of graph ideas in city planning and disaster control.