A Unified Framework for Fault‐Tolerant Metric Dimension of Layered Sequential Join Graphs With Applications to Hierarchical Networks

ABSTRACT Fault‐tolerant monitoring and reliable node identification are essential requirements in modern hierarchical communication systems such as IoT–Fog–Cloud architectures and distributed sensing networks. The fault‐tolerant metric dimension (FTMD) provides an effective graph‐theoretic framework for resilient localization and monitoring in such systems. In this paper, we introduce the layered sequential join graph, a structured graph construction obtained by joining graph components sequentially so that complete adjacency occurs only between consecutive layers. Unlike classical join constructions, which collapse inter‐component distances and destroy hierarchical distance information, the framework preserves a natural layer‐indexed metric structure across the network. Using the framework of the join fault‐tolerant metric dimension (JFTMD), we establish an exact additive decomposition for the FTMD of graphs. Specifically, for where are connected component graphs, we prove that where denotes the JFTMD of the component graph . The obtained decomposition eliminates the correction terms that typically arise in classical join constructions and shows that fault‐tolerant monitoring tasks can be performed independently within individual layers. Finally, we illustrate the applicability of the proposed framework to hierarchical IoT–Fog–Cloud systems, where scalable and resilient monitor placement can be achieved through independent layer‐wise analysis.