Reducible handle additions to weakly reducible Heegaard splittings

Let V∪SW be an irreducible Heegaard splitting of M and F a component of ∂−V. A simple closed curve J in F is called reducible if MJ=VJ⋃SW is reducible, where MJ is obtained by attaching a 2-handle to M along J. In this paper, we give a sufficient condition for the diameter of the set of reducible curves in F to be bounded in C(F) for the keen weakly reducible Heegaard splittings. Moreover, an upper bound of the diameter of the set of the reducible curves in F is given under some circumstance for the weakly reducible Heegaard splittings.