  Additively Separable Hedonic Game (ASHG) are coalition-formation games where
we are given a graph whose vertices represent $n$ selfish agents and the weight
of each edge $uv$ denotes how much agent $u$ gains (or loses) when she is
placed in the same coalition as agent $v$. We revisit the computational
complexity of the well-known notion of core stability of ASHGs, where the goal
is to construct a partition of the agents into coalitions such that no group of
agents would prefer to diverge from the given partition and form a new
(blocking) coalition. Since both finding a core stable partition and verifying
that a given partition is core stable are intractable problems
($\Sigma_2^p$-complete and coNP-complete respectively) we study their
complexity from the point of view of structural parameterized complexity, using
standard graph-theoretic parameters, such as treewidth.
