Investigating the Metric and Fault-Tolerant Dimensions in Para-Line Network Topologies

The resolving set (RS) and metric dimension (MD) are critical concepts used in various fields such as computer networks, robot navigation, chemical structures, communication networks, transportation, and electric circuits. In the context of a security system, RS has been employed as a sensor or node to detect intruders within indoor environments. However, a single point of failure in the RS chain can render the security system unable to detect an intruder. To address this issue, the fault-tolerant metric dimension (FTMD) was introduced, which proposes a fault-tolerant, self-stable system capable of identifying intruders even if a sensor malfunctions. A fault-tolerant resolving set (FTRS) is an RS that remains functional even after removing an element, and the FTMD represents its minimal cardinality. In this study, we focused on the MD for the para-line prism networks as the primary contribution. Additionally, we delved into determining the FTMD for para-line networks, with specific emphasis on para-line prism networks. Our findings suggest that the MD and FTMD for this network family remain constant. Furthermore, we observed that the FTMD of the para-line prism network exceeds its MD by exactly one.