Rank connectivity and pivot-minors of graphs

The cut-rank of a set X in graph G is the rank X×(V(G)?X) submatrix adjacency matrix over binary field. A split partition vertex into two sets (X,Y) such that less than 2 and both Y have at least vertices. prime (with respect to decomposition) if it connected has no splits. k+?-rank-connected for every vertices with k, |X| or |V(G)?X| k+?. We prove 3+2-rank-connected 10 3+3-rank-connected pivot-minor H |V(H)|=|V(G)|?1. As corollary, we show excluded class graphs rank-width most k (3.5?6k?1)/5 k?2. also pivot-minors 16