  We characterize when there exists a diagonal preserving $*$-isomorphism
between two graph $C^*$-algebras in terms of the dynamics of the boundary path
spaces. In particular, we refine the notion of "orbit equivalence" between the
boundary path spaces of the directed graphs $E$ and $F$ and show that this is a
necessary and sufficient condition for the existence of a diagonal preserving
$*$-isomorphism between the graph $C^*$-algebras $C^*(E)$ and $C^*(F)$.
