Rotationally symmetrical plane graphs and their Fault-tolerant metric dimension

Abstract

Consider a robot which is investigating in a space exhibited by a graph (network), and which needs to know its current location. It can grant a sign to find how far it is from each among a lot of fixed places of interest (tourist spots or landmarks). We study the problem of calculating the minimum number of tourist spots required, and where they ought to be set, with the ultimate objective that the robot can generally decide its location. The set of nodes where the places of interest are placed is known as the metric basis of the graph, and the cardinality of tourist spots is known as the location number (or metric dimension) of the graph. Another graph invariant related to resolving set (say $\mathfrak{L}$) is the fault-tolerant resolving set $\mathfrak{L}^{\ast}$, in which the expulsion of a discretionary vertex from $\mathfrak{L}$ keeps up the resolvability. The problem of characterizing the classes of plane graphs with a bounded fault-tolerant metric dimension is of great interest nowadays. In this article, we obtain the fault-tolerant metric dimension of three interminable classes of symmetrical plane graphs, that are found to be constant for each of these three families of the plane graphs. We set lower and upper bounds for the fault-tolerant metric dimension of these three classes of the plane graphs.