Rank-width and Well-quasi-ordering of Skew-symmetric Matrices

Robertson and Seymour prove that a set of graphs of bounded tree-width is well-quasi-ordered by the graph minor relation. By extending their methods to matroids, Geelen, Gerards, and Whittle prove that a set of matroids representable over a fixed finite field are well-quasi-ordered if it has bounded branch-width. More recently, it is shown that a set of graphs of bounded rank-width (or clique-width) is well-quasi-ordered by the graph vertex-minor relation. The proof of the last one uses isotropic systems defined by A. Bouchet. We obtain a common generalization of the above three theorems in terms of skew-symmetric matrices over a fixed finite field.