  Let X be a compact 2-manifold with nonempty boundary dX and let f: (X, dX)
--> (X, dX) be a boundary-preserving map. Denote by MF_d[f] the minimum number
of fixed point among all boundary-preserving maps that are homotopic through
boundary-preserving maps to f. The relative Nielsen number N_d(f) is the sum of
the number of essential fixed point classes of the restriction f-bar : dX -->
dX and the number of essential fixed point classes of f that do not contain
essential fixed point classes of f-bar. We prove that if X is the Moebius band
with one (open) disc removed, then MF_d[f] - N_d(f) < 2 for all maps f : (X,
dX) --> (X, dX). This result is the final step in the boundary-Wecken
classification of surfaces, which is as follows. If X is the disc, annulus or
Moebius band, then X is boundary-Wecken, that is, MF_d[f] = N_d(f) for all
boundary-preserving maps. If X is the disc with two discs removed or the
Moebius band with one disc removed, then X is not boundary-Wecken, but MF_d[f]
- N_d(f) < 2. All other surfaces are totally non-boundary-Wecken, that is,
given an integer k > 0, there is a map $f_k : (X, dX) --> (X, dX) such that
MF_d[f_k] - N_d(f_k) >= k.
