Classes of Wecken maps of surfaces with boundary

Let F: M → M be a self-map of a hyperbolic surface with boundary. If F is both simple and W-characteristic, we present a formula for the Nielsen number N(F) which depends only on the induced map F# of the fundamental group. We list several distinct classes of maps which are simple, and use the fact that if M is the pants surface, i.e., the disc with two holes, we can find a map Fʹ in one of these classes such that N(F) = N(Fʹ) and M F[F] = M F[Fʹ] where M F[F] denotes the minimum number of fixed points of any map homotopic to F. Using Kellyʹs formulas for the minimum number for self-maps of the pants surface, we determined which maps satisfy the Wecken property N(F) = M F[F].