Birecognition of prime graphs, and minimal prime graphs

Given a graph G, a subset M of V (G) is a module of G if for each v is an element of V (G) set minus M, v is adjacent to all the elements of M or to none of them. For instance, V (G), null and {v} (v is an element of V (G)) are the trivial modules of G. A graph G is prime if |V (G)|>= 4 and all its modules are trivial. Given a prime graph G, consider X (sic) V (G) such that G[X] is prime. Given a graph H such that V (G) = V (H) and G[X] = H[X], G and H are G[X]-similar if for each W (sic) V (G) set minus X, G[X boolean OR W] and H[X boolean OR W] are both prime or not. The graph G is said to be G[X]-birecognizable if every graph, G[X]-similar to G, is prime. We study the graphs G that are not G[X]-birecognizable, where X (sic) V (G) such that G[X] is prime, by using the following notion of a minimal prime graph. Given a prime graph G, consider X (sic) V (G) such that G[X] is prime. Given v,w is an element of V (G) set minus X, G is G[X boolean OR{v,w}]-minimal if for each W (sic) V (G) such that X ?{v,w}subset of W, G[W] is not prime.