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 ) . The minimum cardinality of such a resolving set R in G is called its metric dimension. A resolving set R is said to be fault-tolerant, if for every p ∈ R , R − p preserves the property of being a resolving set. A fault-tolerant metric dimension of G is a minimal possible order fault-tolerant resolving set. A wide variety of situations, in which connection, distance, and connectivity are important aspects, call for the utilisation of metric dimension. The structure and dynamics of complex networks, as well as difficulties connected to robotics network design, navigation, optimisation, and facility positioning, are easier to comprehend as a result of this. As a result of the relevant concept of metric dimension, robots are able to optimise their methods of localization and navigation by making use of a limited number of reference locations. As a consequence of this, numerous applications of robotics, including collaborative robotics, autonomous navigation, and environment mapping, have become more precise, efficient, and resilient. The arithmetic graph A l is defined as the graph with its vertex set as the set of all divisors of l , where l is a composite number and l = p γ 1 1 p η 2 2 , … , p n n , where p n ≥ 2 and the p i ’s are distinct primes. Two distinct divisors x , y of l are said to have the same parity if they have the same prime factors (i.e., x = p 1 p 2 and y = p 2 1 p 3 2 have the same parity). Further, two distinct vertices x , y ∈ A l are adjacent if and only if they have different parity and gcd ( x , y ) = p i (greatest common divisor) for some i ∈ { 1 , 2 , … , t } . This article is dedicated to the investigation of the arithmetic graph of a composite number l , which will be referred to throughout the text as A l . In this study, we compute the fault-tolerant resolving set and the fault-tolerant metric dimension of the arithmetic graph A l , where l is a composite number.