Monday November 15th at 4.30pm

graph H is called a vertex-minor of G if H can be obtained by applying a sequence of vertex-deletions and local complementations. It will be

shown that for fixed k, if {G_1,G_2,...} is an infinite sequence of graphs of rank-width (or clique-width) at most k, then there exist i

less than j such that G_i is isomorphic to a vertex-minor of G_j. The proof is very similar to proofs of well-quasi-ordering theorem of

graphs by Robertson and Seymour and well-quasi-ordering theorem of binary matroids (or representable matroids over a fixed finite field)

by Geelen, Gerards, and Whittle. In fact, well-quasi-ordering theorem of binary matroids is implied from our theorem.