  In this article we study a partial ordering on knots in the 3-sphere where
K_1 is greater than or equal to K_2 if there is an epimorphism from the knot
group of K_1 onto the knot group of K_2 which preserves peripheral structure.
If K_1 is a 2-bridge knot and K_1 > K_2, then it is known that K_2 must also be
2-bridge. Furthermore, Ohtsuki, Riley, and Sakuma give a construction which,
for a given 2-bridge knot K_{p/q}, produces infinitely 2-bridge knots K_{p'/q'}
with K_{p'/q'}>K_{p/q}. After characterizing all 2-bridge knots with 4 or less
distinct boundary slopes, we use this to prove that in any such pair, K_{p'/q'}
is either a torus knot or has 5 or more distinct boundary slopes. We also prove
that 2-bridge knots with exactly 3 distinct boundary slopes are minimal with
respect to the partial ordering. This result provides some evidence for the
conjecture that all pairs of 2-bridge knots with K_{p'/q'}>K_{p/q} arise from
the Ohtsuki-Riley-Sakuma construction.
