  The paper systematically classifies rings based on the dominant metric
dimensions (Ddim) of their associated CZDG, establishing consequential bounds
for the Ddim of these compressed zero-divisor graphs. The authors investigate
the interplay between the ring-theoretic properties of a ring ( R ) and
associated CZDG. An undirected graph consisting of vertex set ( Z(R_E)\{[0]}\
=\ R_E\{[0],[1]}), where ( R_E=\{[x]:\ x\in R\} ) and ([x]=\{y\in R:\
\text{ann}(x)=\text{ann}(y)\} ) is called a compressed zero-divisor graph,
denoted by ( \Gamma_E(R) ). An edge is formed between two vertices ([x]) and
([y]) of ( Z(R_E) ) if and only if ([x][y]=[xy]=[0]), that is, iff ( xy=0 ).
For a ring ( R ), graph ( G ) is said to be realizable as ( \Gamma_E(R) ) if (
G ) is isomorphic to ( \Gamma_E(R) ). Moreover, an exploration into the Ddim of
realizable graphs for rings is conducted, complemented by illustrative examples
reinforcing the presented results. A recent discussion within the paper
elucidates the nuanced relationship between Ddim, diameter, and girth within
the domain of compressed zero-divisor graphs. This research offers a
comprehensive and insightful analysis at the intersection of algebraic
structures and graph theory, providing valuable contributions to the current
mathematical discourse.
