We revisit the problem of deciding whether a finitely generated subgroup H is a free factor of a given free group F. Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of H and exponential in the rank of F. We show that the latter dependency can be made exponential in the rank difference rank(F) - rank(H), which often makes a significant change.

We study one-relator free products in which the relator has free-product length $4$. We find conditions for such presentations to have a Freiheitssatz and classify all non-aspherical presentations under certain conditions.

AMS 2000 Mathematics subject classification: Primary 20F05; 20F06; 20F10