Fault-Tolerant Metric Dimension of Interconnection Networks

Abstract

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.

Keywords

• Graph theory
• metric dimension
• fault-tolerant metric dimension
• NP-complete problems
• interconnection networks