The notion of a boundary graph property is a relaxation of that of a
minimal property. Several fundamental results in graph theory have been obtained in
terms of identifying minimal properties. For instance, Robertson and Seymour showed that
there is a unique minimal minor-closed property with unbounded tree-width (the planar
graphs), while Balogh, Bollobás and Weinreich identified nine minimal hereditary
properties of labeled graphs with the factorial speed of growth. However, there are
situations where the notion of minimal property is not applicable. A typical example of this type
is given by graphs of large girth. It is known that for each particular value of k, the
graphs of girth at least k are of unbounded tree-width and their speed of growth is
superfactorial, while the limit property of this sequence (i.e., the acyclic graphs) has bounded
tree-width and its speed of growth is factorial. To overcome this difficulty, the notion of
boundary properties of graphs has been recently introduced. In the present talk, we use this
notion in order to identify some classes of graphs which are well-quasi-ordered with
respect to the induced subgraph relation.