Well-Quasi-Ordering of Matrices under Schur Complement and Applications to Directed Graphs

In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field F, any infinite sequenceM1,M2, . . . of (skew) symmetric matrices over F of bounded F-rank-width has a pair i < j, such that Mi is isomorphic to a principal submatrix of a principal pivot transform of Mj . We generalise this result to σ-symmetric matrices introduced by Rao and myself in [The Rank-Width of Edge-Coloured Graphs, arXiv:0709.1433v4]. (Skew) symmetric matrices are special cases of σ-symmetric matrices. As a byproduct, we obtain that for every infinite sequence G1, G2, . . . of directed graphs of bounded rank-width there exist a pair i < j such that Gi is a pivot-minor of Gj . Another consequence is that non-singular principal submatrices of a σ-symmetric matrix form a delta-matroid. We extend in this way the notion of representability of delta-matroids by Bouchet.