Characterizing rings based on resolvability in associated compressed zero divisor graphs

This paper explores the concept of dominant metric dimensions (D[Formula: see text]dim) of compressed zero-divisor graphs (CZDG) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring [Formula: see text] and associated compressed zero-divisor graph. A undirected graph consisting of a vertex set [Formula: see text], where [Formula: see text] and [Formula: see text] is called a compressed zero-divisor graph, denoted by [Formula: see text]. An edge is formed between two vertices [Formula: see text] and [Formula: see text] of [Formula: see text] if and only if [Formula: see text], that is, iff [Formula: see text]. For a ring [Formula: see text], the graph [Formula: see text] is said to be realizable as [Formula: see text] if [Formula: see text] is isomorphic to [Formula: see text]. We classify the rings based on the D[Formula: see text]dim of their associated CZDG and obtain the bounds for the D[Formula: see text]dim of the compressed zero-divisor graphs. We also study the D[Formula: see text]dim of realizable graphs of rings. Moreover, some examples are provided to support our results. Of late, we have discussed the relationship between D[Formula: see text]dim, diameter, and girth of compressed zero-divisor graphs.