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Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k

Authors Mamadou Moustapha Kanté , Eun Jung Kim , O-joung Kwon , Sang-il Oum



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Author Details

Mamadou Moustapha Kanté
  • Université Clermont Auvergne, Clermont Auvergne INP, LIMOS, CNRS, Aubière, France
Eun Jung Kim
  • Université Paris-Dauphine, PSL University, CNRS, UMR 7243, LAMSADE, Paris, France
O-joung Kwon
  • Department of Mathematics, Incheon National University, Incheon, South Korea
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
Sang-il Oum
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
  • Department of Mathematical Sciences, KAIST, Daejeon, South Korea

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Mamadou Moustapha Kanté, Eun Jung Kim, O-joung Kwon, and Sang-il Oum. Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 40:1-40:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.STACS.2022.40

Abstract

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • path-width
  • matroid
  • linear rank-width
  • graph
  • forbidden minor
  • vertex-minor
  • pivot-minor

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References

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