  We give necessary and sufficient conditions which a graph should satisfy in
order for its associated $C^\ast$-algebra to have a $T_1$ primitive ideal
space. We give a description of which one-point sets in such a primitive ideal
space are open, and use this to prove that any purely infinite graph
$C^\ast$-algebra with a $T_1$ (in particular Hausdorff) primitive ideal space,
is a $c_0$-direct sum of Kirchberg algebras. Moreover, we show that graph
$C^\ast$-algebras with a $T_1$ primitive ideal space canonically may be given
the structure of a $C(\widetilde{\mathbb N})$-algebra, and that isomorphisms of
their $\widetilde{\mathbb N}$-filtered $K$-theory (without coefficients) lift
to $E(\widetilde{\mathbb N})$-equivalences, as defined by Dadarlat and Meyer.
