  Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem $OP(F)$,
posed in 1967 and still open, asks for a decomposition of $K_v$ into copies of
$F$. In this paper we show that $OP(F)$ has a solution whenever $F$ has a
sufficiently large cycle which meets a given lower bound and, in addition, has
a single-flip automorphism, which is an involutory automorphism acting as a
reflection on exactly one of the cycles of $F$. Furthermore, we prove analogous
results for the minimum covering version and the maximum packing version of the
problem. We also show a similar result when the edges of $K_v$ have
multiplicity 2, but in this case we do not require that $F$ be single-flip.
  Our approach allows us to explicitly construct solutions to the Oberwolfach
Problem with well-behaved automorphisms, in contrast with some recent
asymptotic results, based on probabilistic methods, which are nonconstructive
and do not provide a lower bound on the order of $F$ that guarantees the
solvability of $OP(F)$.
  Our constructions are based on a doubling construction which applies to
graceful labelings of $2$-regular graphs with a vertex removed. We show that
this class of graphs is graceful as long as the length of the path-component is
sufficiently large. A much better lower bound on the length of the path is
given for an $\alpha$-labeling of such graphs to exist.
