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arXiv · 1007.3807

Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices

Sang-il Oum
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Abstract

We prove that every infinite sequence of skew-symmetric or symmetric matrices M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such that M_i is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in M_j, if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymour's theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle's theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum's theorem for graphs of bounded rank-width with respect to pivot-minors.

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Sang-il Oum. 2010-07-22. Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices. https://doi.org/10.1016/j.laa.2011.09.027

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