On free-group algorithms that sandwich a subgroup between free-product factors

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.