Hereditary C∗-subalgebras of graph C∗-algebras

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.