  The compressed zero-divisor graph $\Gamma_C(R)$ associated with a commutative
ring $R$ has vertex set equal to the set of equivalence classes $\{ [r] \mid r
\in Z(R), r \neq 0 \}$ where $r \sim s$ whenever $ann(r) = ann(s)$. Distinct
classes $[r],[s]$ are adjacent in $\Gamma_C(R)$ if and only if $xy = 0$ for all
$x \in [r], y \in [s]$. In this paper, we explore the compressed zero-divisor
graph associated with quotient rings of unique factorization domains.
Specifically, we prove several theorems which exhibit a method of constructing
$\Gamma(R)$ for when one quotients out by a principal ideal, and prove
sufficient conditions for when two such compressed graphs are graph-isomorphic.
We show these conditions are not necessary unless one alters the definition of
the compressed graph to admit looped vertices, and conjecture necessary and
sufficient conditions for two compressed graphs with loops to be isomorphic
when considering any quotient ring of a unique factorization domain.
