  Consider a group G and a family $\mathcal{A}$ of subgroups of G. We say that
vertex finiteness holds for splittings of G over $\mathcal{A}$ if, up to
isomorphism, there are only finitely many possibilities for vertex stabilizers
of minimal G-trees with edge stabilizers in $\mathcal{A}$.
  We show vertex finiteness when G is a toral relatively hyperbolic group and
$\mathcal{A}$ is the family of abelian subgroups.
  We also show vertex finiteness when G is hyperbolic relative to virtually
polycyclic subgroups and $\mathcal{A}$ is the family of virtually cyclic
subgroups; if moreover G is one-ended, there are only finitely many minimal
G-trees with virtually cyclic edge stabilizers, up to automorphisms of G.
