  The Hanna Neumann conjecture states that if F is a free group, then for all
nontrivial finitely generated subgroups H,K <= F, rank(H intersect K) - 1 <=
[rank(H)-1] [rank(K)-1]. Where most papers to date have considered a direct
graph theoretic interpretation of the conjecture, here we consider the use of
monomorphisms. We illustrate the effectiveness of this approach with two
results. First, we show that for any finitely generated groups H,K <= F either
the pair H,K or the pair H^{-}, K satisfy the Hanna Neumann conjecture--Here
{-} denotes the automorphism which sends each generator of F to its inverse.
Next, using particular monomorphisms from F to F_2, we obtain that if the Hanna
Neumann conjecture is false then there is a counterexample H,K < F_2 having the
additional property that all the branch vertices in the foldings of H and K are
of degree 3, and all degree 3 vertices have the same local structure or
``type''.
