  Leighton's graph covering theorem states that a pair of finite graphs with
isomorphic universal covers have a common finite cover. We provide a new proof
of Leighton's theorem that allows generalizations; we prove the corresponding
result for graphs with fins. As a corollary we obtain pattern rigidity for free
groups with line patterns, building on the work of Cashen-Macura and
Hagen-Touikan. To illustrate the potential for future applications, we give a
quasi-isometric rigidity result for a family of cyclic doubles of free groups.
