I think there are two problems: - the arcs of \alpha and \beta that you used can be on the boundary of the same disk (not the disk you removed, I'm saying in the remaining part of the surface) -if a pair of curves decompose the surface in a union of disks you cannot say that the corresponding pair realizing minimal intersection do the same

Prof. Majer: suppose that the surface minus the filling curves is a connected disk (it happens for the torus, for example). I can't figure out how to do the connected sum in a suitable way.