Fault-Tolerant Metric Dimension of Interconnection Networks
Sakander Hayat, Asad Khan, Muhammad Yasir Hayat Malik, Muhammad Imran, Muhammad Kamran Siddiqui · IEEE Access · 2020
A fixed interconnection parallel architecture is characterized by a graph, with vertices corresponding to processing nodes and edges representing communication links. An ordered set$R$of nodes in a graph$G$is said to be a resolving set of$G$if every node in$G$is uniquely determined by its vector of distances to the nodes in$R$. Each node in$R$can be thought of as the site for a sonar or loran station, and each node location must be uniquely determined by its distances to the sites in$R$. A fault-tolerant resolving set$R$for which the failure of any single station at node location$v$in$R$leaves us with a set that still is a resolving set. The metric dimension (resp. fault-tolerant metric dimension) is the minimum cardinality of a resolving set (resp. fault-tolerant resolving set). In this article, we study the metric and fault-tolerant dimension of certain families of interconnection networks. In particular, we focus on the fault-tolerant metric dimension of the butterfly, the Benes and a family of honeycomb derived networks called the silicate networks. Our main results assert that three aforementioned families of interconnection have an unbounded fault-tolerant resolvability structures. We achieve that by determining certain maximal and minimal results on their fault-tolerant metric dimension.