Obstructions for matroids of path-width at most k and graphs of linear rank-width at most k

Every minor-closed class of matroids bounded branch-width can be characterized by a list excluded minors, but unlike graphs, this may need to infinite in general. However, for each fixed finite field F, the needs contain only finitely many F-representable matroids, due well-quasi-ordering under taking matroid minors [J.F. Geelen, A.M.H. Gerards, and G. Whittle (2002)]. But proof is non-constructive does not provide any algorithm computing these We consider path-width at most k k. prove that every minor has 2|F|O(k2) elements. therefore compute, integer set k, gives as corollary polynomial-time checking whether an F-represented also pivot-minor graphs having linear rank-width 22O(k2) vertices, which results similar algorithmic consequence graphs.