  Let $F$ be a finite-rank free group and $H$ be a finite-rank subgroup of $F$.
We discuss proofs of two algorithms that sandwich $H$ between an upper-layer
free-product factor of $F$ that contains $H$ and a lower-layer free-product
factor of $F$ that is contained in $H$.
  Richard Stong showed that the unique smallest-possible upper layer, denoted
$\operatorname{Cl}(H)$, is visible in the output of the polynomial-time
cut-vertex algorithm of J. H. C. Whitehead. Stong's proof used bi-infinite
paths in a Cayley tree and sub-surfaces of a three-manifold. We give a variant
of his proof that uses edge-cuts of the Cayley tree induced by edge-cuts of a
Bass-Serre tree.
  A. Clifford and R. Z. Goldstein gave an exponential-time algorithm that
determines whether or not the trivial subgroup is the only possible lower
layer. Their proof used Whitehead's three-manifold techniques. We give a
variant of their proof that uses Whitehead's cut-vertex results, and thereby
obtain a somewhat simpler algorithm that yields a lower layer of
maximum-possible rank.
