  This paper is a contribution to the study of hereditary classes of finite
graphs. We classify these classes according to the number of prime structures
they contain. We consider such classes that are \emph{minimal prime}: classes
that contain infinitely many primes but every proper hereditary subclass
contains only finitely many primes. We give a complete description of such
classes. In fact, each one of these classes is a well-quasi-ordered (w.q.o) age
and there are uncountably many of them. Eleven of these ages are almost
multichainable; they remain w.q.o when labels in a w.q.o are added, hence have
finitely many bounds. Five ages among them are exhaustible. Among the remaining
ones, only countably many remain w.q.o when one label is added, and these have
finitely many bounds (except for the age of the infinite path and its
complement). The others have infinitely many bounds.
  Except for six examples, members of these ages we characterize are
permutation graphs. In fact, every age which is not among the eleven ones is
the age of a graph associated to a uniformly recurrent word on the integers. A
description of minimal prime classes of posets and bichains is also provided.
  Our results support the conjecture that if a hereditary class of finite
graphs does not remain w.q.o when adding labels from a w.q.o set to these
graphs, then it is not w.q.o if we add just two constants to each of these
graphs
  Our description of minimal prime classes uses a description of minimal prime
graphs \cite{pouzet-zaguia2009} and previous work by Sobrani
\cite{sobranithesis, sobranietat} and the authors \cite{oudrar, pouzettr} on
properties of uniformly recurrent words and the associated graphs. The
completeness of our description is based on classification results of
Chudnovsky, Kim, Oum and Seymour \cite{chudnovsky} and Malliaris and Terry
\cite {malliaris}.
