Rank-Width and Well-Quasi-Ordering

Robertson and Seymour (1990) proved that graphs of bounded tree-width are well-quasi-ordered by the graph minor relation. By extending their arguments, Geelen, Gerards, and Whittle (2002) proved that binary matroids of bounded branch-width are well-quasi-ordered by the matroid minor relation. We prove another theorem of this kind in terms of rank-width and vertex-minors. For a graph G = (V,E) and a vertex v of G, a local complementation at v is an operation that replaces the graph induced on neighbors of v by its complement graph. A graph H is called a vertex-minor of G if H can be obtained by applying a sequence of vertex-deletions and local complementations. Rank-width was defined by Oum and Seymour to investigate clique-width; they showed that graphs have bounded rank-width if and only if they have bounded clique-width. We prove that graphs of bounded rank-width are well-quasi-ordered by the vertex-minor relation; in other words, for every infinite sequence G1, G2, . . . of graphs of rank-width (or clique-width) at most k, there exist i < j such that Gi is isomorphic to a vertex-minor of Gj . This implies that there is a finite list of graphs such that a graph has rank-width at most k if and only if it contains no one in the list as a vertex-minor. The proof uses the notion of isotropic systems defined by Bouchet.