  We use Gersten's generalization of Whitehead's algorithm to determine whether
a given finitely generated subgroup of a free group $F$ is elliptic in an
elementary cyclic splitting of $F$. We provide a similar result for all
elementary cyclic splittings of a free group of rank two and partial results
towards an algorithm for ellipticity in elementary cyclic splittings of $F$ as
an HNN-extension.
