Studies of Multilevel Networks via Fault-Tolerant Metric Dimensions
Imtiaz Ali, Muhammad Javaid, Samuel Asefa Fufa · IEEE Access · 2022
A subsetTof the vertex set of a networkGis called a resolving set forGif each pair of vertices ofGhave distinct representations with respect toT. A resolving setBʹ among all the resolving sets of a networkGis called a fault-tolerant resolving set ifBʹ\{t} is as well a resolving set for each vertextϵBʹ. A fault-tolerant resolving setBʹ of a networkGwhich 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 byr-level anti-web wheelAWW(n,r),r-level HelmH(n,r) andr-level anti-web gearAWJ(2n,r) networks in the form of different algebraic expressions consisting of the integral numbersnandr. At the end we discussed a simple method for finding the fault-tolerant metric dimensions and fault-tolerant resolving sets of ar-level wheel related network. We also discussed the importance of these networks in navigation.