  In this paper, we develop the basic theory of the shadowing property for
local homeomorphisms of metric locally compact spaces, with a focus on
applications to edge shift spaces connected with C*-algebra theory. For the
local homeomorphism (the Deaconu-Renault system) associated with a direct
graph, we completely characterize the shadowing property in terms of conditions
on sets of paths. Using these results, we single out classes of graphs for
which the associated system presents the shadowing property, fully characterize
the shadowing property for systems associated with certain graphs, and show
that the system associated with the rose of infinite petals presents the
shadowing property and that the Renewal shift system does not present the
shadowing property.
