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

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 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 Kanté, and then extended to (skew-)symmetric matrices by Kanté and Rao, we define boundaried s-labelled graphs and prove similar structure theorems for pivot-minor and linear rank-width.