Sang-il Oum (엄상일), Obstructions for matroids of path-width at most k and graphs of linear rank-width at most k

Every minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field $\mathbb{F}$, the list contains only finitely many $\mathbb{F}$-representable matroids, due to the well-quasi-ordering of $\mathbb{F}$-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these $\mathbb{F}$-representable excluded minors in general.

We consider the class of matroids of path-width at most $k$ for fixed $k$. We prove that for a finite field $\mathbb{F}$, every $\mathbb{F}$-representable excluded minor for the class of matroids of path-width at most~$k$ has at most $2^{|\mathbb{F}{|}^{O(k^2)}}$ elements. We can therefore compute, for any integer $k$ and a fixed finite field $\mathbb{F}$, the set of $\mathbb{F}$-representable excluded minors for the class of matroids of path-width $k$, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an $\mathbb{F}$-represented matroid is at most $k$. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most $k$ has at most $2^{2^{O(k^2)}}$ vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.

This is joint work with Mamadou M. Kanté, Eun Jung Kim, and O-joung Kwon.