Fault Tolerant Metric Dimension of Arithmetic Graphs

Abstract

For a graph G, two vertices x, y ∈ G are said to be resolved by a vertex s ∈ G if d(x|s), d(y|s), where d(x|s) denotes the distance between x and s. The minimum cardinality of such a resolving set R in G is called the metric dimension. A resolving set R is said to be fault-tolerant if, for every p ∈ R, the set R − p preserves the property of being a resolving set. The fault-tolerant metric dimension of G is the minimal possible order of a fault-tolerant resolving set. The concept of metric dimension has wide applications in areas where connection, distance, and network connectivity are critical. This includes understanding the structure and dynamics of complex networks, as well as addressing problems in robotic network design, navigation, optimization, and facility placement. By utilizing the concept of metric dimension, robots can optimize their methods for localization and navigation using a limited number of reference points. As a result, various applications in robotics, such as collaborative robotics, autonomous navigation, and environment mapping, have become more precise, efficient, and resilient. The arithmetic graph Al is defined as the graph where the vertex set is the set of all divisors of a composite number l, where l = pγ11 pη22 · · · pαnn and the pi’s are distinct primes with pi ≥ 2. Two distinct divisors x and y of l are said to have the same parity if they share the same prime factors (e.g., x = p1p2 and y = p21p32 have the same parity). Furthermore, two distinct vertices x, y ∈ Al are adjacent if and only if they have different parity and gcd(x, y) = pi (greatest common divisor) for some i ∈ {1, 2, . . ., t}. This article focuses on the investigation of the arithmetic graph of a composite number l, referred to throughout as Al. In this study, we compute the fault-tolerant resolving set and the fault-tolerant metric dimension of the arithmetic graph Al, where l is a composite number. © 2024 the Author(s), licensee Combinatorial Press.