Two forbidden induced subgraphs and well-quasi-ordering

It is known that a class of graphs defined by a single forbidden induced subgraph G is well-quasi-ordered by the induced subgraph relation if and only if G is an induced subgraph of P 4 . However, very little is known about well-quasi-ordered classes of graphs defined by more than one forbidden induced subgraph. We conjecture that for any natural number k , there are finitely many minimal classes of graphs defined by k forbidden induced subgraphs which are not well-quasi-ordered by the induced subgraph relation and prove the conjecture for k = 2 . We explicitly reveal many of the minimal classes defined by two forbidden induced subgraphs which are not well-quasi-ordered and many of those which are well-quasi-ordered by the induced subgraph relation.