  Previously, Reynolds showed that any irreducible nonsurjective endomorphism
can be represented by an irreducible immersion on a finite graph. We give a new
proof of this and also show a partial converse holds when the immersion has
connected Whitehead graphs with no cut vertices. The next result is a
characterization of finitely generated subgroups of the free group that are
invariant under an irreducible nonsurjective endomorphism. Consequently,
irreducible nonsurjective endomorphisms are fully irreducible. The
characterization and Reynolds' theorem imply that the mapping torus of an
irreducible nonsurjective endomorphism is word-hyperbolic.
