  We show that a $C^*$-algebra $\mathfrak{A}$ which is stably isomorphic to a
unital graph $C^*$-algebra, is isomorphic to a graph $C^*$-algebra if and only
if it admits an approximate unit of projections. As a consequence, a hereditary
$C^*$-subalgebra of a unital real rank zero graph $C^*$-algebra is isomorphic
to a graph $C^*$-algebra. Furthermore, if a $C^*$-algebra $\mathfrak{A}$ admits
an approximate unit of projections, then its minimal unitization is isomorphic
to a graph $C^*$-algebra if and only if $\mathfrak{A}$ is stably isomorphic to
a unital graph $C^*$-algebra.
