  An edge-weighted, vertex-capacitated graph G is called stable if the value of
a maximum-weight capacity-matching equals the value of a maximum-weight
fractional capacity-matching. Stable graphs play a key role in characterizing
the existence of stable solutions for popular combinatorial games that involve
the structure of matchings in graphs, such as network bargaining games and
cooperative matching games.
  The vertex-stabilizer problem asks to compute a minimum number of players to
block (i.e., vertices of G to remove) in order to ensure stability for such
games. The problem has been shown to be solvable in polynomial-time, for
unit-capacity graphs. This stays true also if we impose the restriction that
the set of players to block must not intersect with a given specified maximum
matching of G.
  In this work, we investigate these algorithmic problems in the more general
setting of arbitrary capacities. We show that the vertex-stabilizer problem
with the additional restriction of avoiding a given maximum matching remains
polynomial-time solvable. Differently, without this restriction, the
vertex-stabilizer problem becomes NP-hard and even hard to approximate, in
contrast to the unit-capacity case.
  Finally, in unit-capacity graphs there is an equivalence between the
stability of a graph, existence of a stable solution for network bargaining
games, and existence of a stable solution for cooperative matching games. We
show that this equivalence does not extend to the capacitated case.
