Metric dimension of line graphs of optimal fault-tolerant token ring networks

Interconnection networks are commonly modeled as graphs, where vertices represent processors or devices and edges represent communication links. In such networks, locating or identifying components using a small set of reference points is fundamental for routing, fault diagnosis, monitoring, and navigation. A standard graph-theoretic measure of this capability is the metric dimension (or locating number), defined as the minimum cardinality of a resolving set whose distance vectors uniquely distinguish all vertices. Since determining the metric dimension is NP-hard in general, exact values for structured network families are of both theoretical and practical interest. In this paper, we studied the metric dimension of the line graph of an optimal $2$-fault-tolerant token ring network. The underlying network ${\mathbb{T}}_m^2$ augments a simple ring with additional links to ensure robust connectivity under up to two link or node failures, while the line graph $L({\mathbb{T}}_m^2)$ represents the network at the link level. A lemma was established to prove the lower bound of the metric dimension via contradiction, while the upper bound was determined by explicitly constructing resolving sets. The analysis was conducted case by case according to the congruence of the network order modulo $4$, which simplified verification of all representation vectors. Our results showed that the fault-tolerant links increase the metric dimension compared with ordinary token rings, highlighting the influence of additional links on network distinguishability and providing insights for the design of robust interconnection networks.