  We consider Exel's interaction $(V,H)$ over a unital $C^*$-algebra $A$, such
that $V(A)$ and $H(A)$ are hereditary subalgebras of $A$. For the associated
crossed product, we obtain a uniqueness theorem, ideal lattice description,
simplicity criterion and a version of Pimsner-Voiculescu exact sequence. These
results cover the case of crossed products by endomorphisms with hereditary
ranges and complementary kernels. As model examples of interactions not coming
from endomorphisms we introduce and study in detail interactions arising from
finite graphs.
  The interaction $(V,H)$ associated to a graph $E$ acts on the core $F_E$ of
the graph algebra $C^*(E)$. By describing a partial homeomorphism of
$\widehat{F}_E$ dual to $(V,H)$ we find Cuntz-Krieger uniqueness theorem,
criteria for gauge-invariance of all ideals and simplicity of $C^*(E)$ as
results concerning reversible noncommutative dynamics. We also provide a new
approach to calculation of $K$-theory of $C^*(E)$ using only an induced partial
automorphism of $K_0(F_E)$ and the six-term exact sequence.
