Fault-Tolerant Strong Metric Dimension of Rooted Product Graphs

Abstract

The concept of fault tolerance in graph theory is critical in designing robust networks, ensuring that essential graph properties are preserved despite failures of vertices or edges. In this paper, we investigate the fault-tolerant strong metric dimension of rooted product graphs, a class of graphs derived by attaching multiple copies of a rooted graph to each vertex of a base graph. We extend the notion of a strong metric dimension by considering scenarios where the strong resolving set remains effective even after a specific number of vertices have failed. We provide exact values for specific families of rooted product graphs and demonstrate how the fault-tolerant property varies with the structure of the root and base graphs.