Obstructions for matroids of path-width at most k and graphs of linear rank-width at most k
Abstract
Every minor-closed class of matroids of bounded branch-width can be characterized by a list of excluded minors, but unlike graphs, this list may need to be infinite in general. However, for each fixed finite field , the list needs to contain only finitely many -representable matroids, due to the well-quasi-ordering of -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
-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
, every -representable excluded minor for the class of matroids of path-width at most k has at most elements. We can therefore compute, for any integer k and a fixed finite field , the set of -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 -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 vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.