The dominant strong metric dimension of graphs

Graphs are useful for analyzing the structure models in computer science, operations research, and sociology. Also, different types of graph products have several applications in modeling, including those found in network analysis, communication protocols, organizational structures, diverse fields, network analysis and even chemistry. In this article, we introduce the concept of dominant strong metric dimension of graphs (DSMD for short) as a generalization of strong metric dimension and also dominant metric dimension. We determine the DSMD for several families of graphs that include for instance [Formula: see text], the fan graph [Formula: see text], the wheel graph [Formula: see text] and the helm graph [Formula: see text]. Additionally, we study this invariant under the join product, corona product of two graphs and the generalized join graphs. Moreover, this article investigates the concept of dominant strong metric dimensions for some algebraic graphs associated with rings, which is the comaximal graph. In fact, by using the results on DSMD of the generalized join graphs, the dominant strong metric dimension of the comaximal graph of the ring of integers modulo n is investigated. Through this exploration, besides considering various graph products, we aim to provide a comprehensive framework for analyzing commutative rings and their associated graphs, thereby advancing both theoretical knowledge and practical applications in diverse domains.