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

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

2011-02-10

arxiv.org/abs/1102.2134v2

Mathematics

Combinatorics

35 pages. Revised version with a section for directed graphs

Scientific paper

In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field $\bF$, any infinite sequence $M_1,M_2,...$ of (skew) symmetric matrices over $\bF$ of bounded \emph{$\bF$-rank-width} has a pair $i< j$, such that $M_i$ is isomorphic to a principal submatrix of a \emph{principal pivot transform} of $M_j$. We generalise this result to \emph{$\sigma$-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

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