  We continue and extend previous work on the parameterized complexity analysis
of the NP-hard Stable Roommates with Ties and Incomplete Lists problem, thereby
strengthening earlier results both on the side of parameterized hardness as
well as on the side of fixed-parameter tractability. Other than for its famous
sister problem Stable Marriage which focuses on a bipartite scenario, Stable
Roommates with Incomplete Lists allows for arbitrary acceptability graphs whose
edges specify the possible matchings of each two agents (agents are represented
by graph vertices). Herein, incomplete lists and ties reflect the fact that in
realistic application scenarios the agents cannot bring all other agents into a
linear order. Among our main contributions is to show that it is W[1]-hard to
compute a maximum-cardinality stable matching for acceptability graphs of
bounded treedepth, bounded tree-cut width, and bounded disjoint paths modulator
number (these are each time the respective parameters). However, if we `only'
ask for perfect stable matchings or the mere existence of a stable matching,
then we obtain fixed-parameter tractability with respect to tree-cut width but
not with respect to treedepth. On the positive side, we also provide
fixed-parameter tractability results for the parameter feedback edge set
number.
