  In this paper, we extend the theory of minimal limit key polynomials of
valuations on the polynomial ring $\kx$. We use the theory of cuts on ordered
abelian groups to show that the previous results on bounded sets of key
polynomials of rank-one valuations, extend to vertically bounded sets of key
polynomials of valuations of an arbitrary rank. We discuss as well properties
of minimal limit key polynomials in the vertically unbounded case.
