(untitled) (Communications in Mathematical Research, 2019)
Let V be a nontrivial compression body which is not simple. An essential simple closed curve J in _+V is called weakly disk-busting if _+V-J has only one compressing disk up to isotopy. In this paper, we give an upper bound of the diameter of the image of boundaries of essential disks in V under any projection determined by a weakly disk-busting curve. Moreover, we give a sufficient condition for the handle additions to be boundary irreducible.