The boundary-Wecken classification of surfaces
  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. 
Robert F. Brown, Michael R. Kelly