I'm currently working my way through An Efficient Algorithm for the Stable Roommates Problem by Robert Irving (Journal of Algorithms, 6:577–595, 1984).On page 7 the paper starts with describing the algorithm for phase 2. The only thing that phase two does is reducing all-or-nothing cycles.  The paper seems to assume that after phase 1, the proposal and filter phase, the filtered preference lists can be divided into three categories:One or more of the filtered preference lists is empty. Therefore no stable roommate assignment exists.Each filtered preference list contains only one element. Therefore a stable roommate assignment exists.The filtered preference list contains an all-or-nothing cycle (wikipedia calls this rotations).But why can't you have filtered preference lists that contain more than one element but does not contain an all-or-nothing cycle.
Properties of the filtered preference list (phase 1) in the Stable Roommates problem
algorithms;matching