An Upper Bound on the Size of Obstructions for Bounded Linear Rank-Width

TL;DR

This paper establishes a doubly exponential upper bound on the size of obstructions for graphs, matrices, and matroids with bounded linear rank-width, solving a significant open problem in graph theory and matroid theory.

Contribution

It introduces a new bound on the size of forbidden minors for linear rank-width, extending the pseudo-minor order technique to this context and providing structure theorems for pivot-minors.

Findings

Doubly exponential upper bound on forbidden vertex-minors for graphs of bounded linear rank-width.

Extension of pseudo-minor order to linear rank-width and pivot-minors.

Resolution of an open problem by Jeong, Kwon, and Oum.

Abstract

We provide a doubly exponential upper bound in $p$ on the size of forbidden pivot-minors for symmetric or skew-symmetric matrices over a fixed finite field $\mathbb{F}$ of linear rank-width at most $p$. As a corollary, we obtain a doubly exponential upper bound in $p$ on the size of forbidden vertex-minors for graphs of linear rank-width at most $p$. This solves an open question raised by Jeong, Kwon, and Oum [Excluded vertex-minors for graphs of linear rank-width at most $k$. European J. Combin., 41:242--257, 2014]. We also give a doubly exponential upper bound in $p$ on the size of forbidden minors for matroids representable over a fixed finite field of path-width at most $p$. Our basic tool is the pseudo-minor order used by Lagergren [Upper Bounds on the Size of Obstructions and Interwines, Journal of Combinatorial Theory Series B, 73:7--40, 1998] to bound the size of forbidden graph minors for bounded path-width. To adapt this notion into linear rank-width, it is necessary to well define partial pieces of graphs and merging operations that fit to pivot-minors. Using the algebraic operations introduced by Courcelle and Kant\'e, and then extended to (skew-)symmetric matrices by Kant\'e and Rao, we define boundaried $s$-labelled graphs and prove similar structure theorems for pivot-minor and linear rank-width.