Studies of Multilevel Networks via Fault-Tolerant Metric Dimensions

A subset
$T$
of the vertex set of a network
$G$
is called a resolving set for
$G$
if each pair of vertices of
$G$
have distinct representations with respect to
$T$
. A resolving set
$B^{\prime} $
among all the resolving sets of a network
$G$
is called a fault-tolerant resolving set if
$B^{\prime} \backslash \left \{{t}\right \}$
is as well a resolving set for each vertex
$t \in ~B^{\prime} $
. A fault-tolerant resolving set
$B^{\prime} $
of a network
$G$
which contains minimum number of vertices is called a fault-tolerant metric basis. The cardinality of a fault-tolerant metric basis is called fault-tolerant metric dimension. This concept is widely used to find the integral solution of the problems existing in different disciplines of computer science and chemistry such as linear optimization problems, robot navigation, operation research problems, sensor networking, classification of chemical compounds, drug discoveries, source localization, embedding biological sequence data, detecting network motifs, comparing the interconnected networks and image processing. In this paper, we compute the fault-tolerant metric dimensions of three wheel related networks called by
$r$
-level anti-web wheel
$AWW_{(n, r)}$
,
$r$
-level Helm
$H_{(n, r)}$
and
$r$
-level anti-web gear
$AWJ_{(2n, r)}$
networks in the form of different algebraic expressions consisting of the integral numbers
$n$
and
$r$
. At the end we discussed a simple method for finding the fault-tolerant metric dimensions and fault-tolerant resolving sets of a
$r$
-level wheel related network. We also discussed the importance of these networks in navigation.