Fault-tolerant basis and fault-tolerant edge basis of three classes of French windmill graphs

A resolving set is a subset of vertices that uniquely identifies every vertex based on distances. A fault-tolerant resolving set maintains this condition under any single-vertex removal, and the minimum size of such a set is the fault-tolerant metric dimension. Similarly, the edge metric dimension is defined as the minimum number of vertices required to distinguish all edge pairs, and its fault-tolerant variant ensures resiliency against node failures. Interconnection networks, which model parallel architectures, are naturally represented by graphs where vertices correspond to processing nodes and edges denote communication links. Among these, the windmill graph is a well-studied topology formed by joining several copies of a complete graph at a shared central vertex. In this work, we determine the exact values of the fault-tolerant metric dimension and fault-tolerant edge metric dimension for the French windmill graph and its subcases, providing insights into the structural robustness of such networks.