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

TL;DR

This paper generalizes well-quasi-ordering results from symmetric matrices to $\sigma$-symmetric matrices, with applications to directed graphs and delta-matroids, establishing structural properties and minor relations.

Contribution

It extends the theory of well-quasi-ordering to $\sigma$-symmetric matrices and directed graphs, introducing new applications to delta-matroids and principal pivot transforms.

Findings

Infinite sequences of bounded rank-width matrices have ordered minors.

Directed graphs of bounded rank-width have pivot-minors within sequences.

Non-singular principal submatrices form delta-matroids.

Abstract

In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field $\mathbf{F}$, any infinite sequence $M_1,M_2,...$ of (skew) symmetric matrices over $\mathbf{F}$ of bounded $\mathbf{F}$-rank-width has a pair $i< j$, such that $M_i$ is isomorphic to a principal submatrix of a principal pivot transform of $M_j$. We generalise this result to $\sigma$-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 $\sigma$-symmetric matrices. As a by-product, we obtain that for every infinite sequence $G_1,G_2,...$ of directed graphs of bounded rank-width there exist a pair $i<j$ such that $G_i$ is a pivot-minor of $G_j$. Another consequence is that non-singular principal submatrices of a $\sigma$-symmetric matrix form a delta-matroid. We extend in this way the notion of representability of delta-matroids by Bouchet.