Fault-Tolerant Metric Dimension of Interconnection Networks

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.