The cut-rank of a set X in a graph G is the rank of the XΓ(V(G)βX) submatrix of the adjacency matrix over the binary field. A split is a
partition of the vertex set into two sets (X,Y) such that the cut-rank of X
is less than 2 and both X and Y have at least two vertices. A graph is
prime (with respect to the split decomposition) if it is connected and has no
splits. A graph G is k+β-rank-connected if for every set X of
vertices with the cut-rank less than k, β£Xβ£ or β£V(G)βXβ£ is less than k+β. We prove that every prime
3+2-rank-connected graph G with at least 10 vertices has a prime
3+3-rank-connected pivot-minor H such that β£V(H)β£=β£V(G)β£β1. As a corollary, we show that every excluded pivot-minor for the
class of graphs of rank-width at most k has at most (3.5β 6kβ1)/5
vertices for kβ₯2. We also show that the excluded pivot-minors for the
class of graphs of rank-width at most 2 have at most 16 vertices.Comment: 19 pages; Lemma 5.3 is now fixe