The set of prime extensions of a graph: the finite and the infinite case

Let H be a graph then a graph H ′ is a prime extension of H if H ′ is prime (in the sense of modular decomposition), it contains an induced subgraph isomorphic to H and is minimal with respect to set inclusion and primality. An open problem concerning the set of prime extension Ext(H) of H is the following: find the necessary and sufficient conditions establishing the finiteness of Ext(H). We solve the above problem by characterizing all classes of graphs whose set of prime exensions is finite. We give also a simple way for generating an infinite number of extensions for each graph belonging to any other class of graphs.