Unavoidable vertex-minors in large prime graphs

A graph is prime (with respect to the split decomposition) if its vertex set does not admit a partition ( A , B ) (called a split) with | A | , | B | ⩾ 2 such that the set of edges joining A and B induces a complete bipartite graph. ve that for each n , there exists N such that every prime graph on at least N vertices contains a vertex-minor isomorphic to either a cycle of length n or a graph consisting of two disjoint cliques of size n joined by a matching.