eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-03-09
40:1
40:14
10.4230/LIPIcs.STACS.2022.40
article
Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k
Kanté, Mamadou Moustapha
1
https://orcid.org/0000-0003-1838-7744
Kim, Eun Jung
2
https://orcid.org/0000-0002-6824-0516
Kwon, O-joung
3
4
https://orcid.org/0000-0003-1820-1962
Oum, Sang-il
4
5
https://orcid.org/0000-0002-6889-7286
Université Clermont Auvergne, Clermont Auvergne INP, LIMOS, CNRS, Aubière, France
Université Paris-Dauphine, PSL University, CNRS, UMR 7243, LAMSADE, Paris, France
Department of Mathematics, Incheon National University, Incheon, South Korea
Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
Department of Mathematical Sciences, KAIST, Daejeon, South Korea
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 F, the list contains only finitely many F-representable matroids, due to the well-quasi-ordering of 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 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 F, every F-representable excluded minor for the class of matroids of path-width at most k has at most 2^{|𝔽|^{O(k²)}} elements. We can therefore compute, for any integer k and a fixed finite field F, the set of 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 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(k²)}} vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol219-stacs2022/LIPIcs.STACS.2022.40/LIPIcs.STACS.2022.40.pdf
path-width
matroid
linear rank-width
graph
forbidden minor
vertex-minor
pivot-minor