We show that a C*-algebra 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 A admits an approximate unit of projections, then its minimal unitization is isomorphic to a graph C*-algebra if and only if A is stably isomorphic to a unital graph C*-algebra.