License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2022.40
URN: urn:nbn:de:0030-drops-158507
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15850/
Kanté, Mamadou Moustapha ;
Kim, Eun Jung ;
Kwon, O-joung ;
Oum, Sang-il
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 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^{|?|^{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.
BibTeX - Entry
@InProceedings{kante_et_al:LIPIcs.STACS.2022.40,
author = {Kant\'{e}, Mamadou Moustapha and Kim, Eun Jung and Kwon, O-joung and Oum, Sang-il},
title = {{Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {40:1--40:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-222-8},
ISSN = {1868-8969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/15850},
URN = {urn:nbn:de:0030-drops-158507},
doi = {10.4230/LIPIcs.STACS.2022.40},
annote = {Keywords: path-width, matroid, linear rank-width, graph, forbidden minor, vertex-minor, pivot-minor}
}
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Keywords: |
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path-width, matroid, linear rank-width, graph, forbidden minor, vertex-minor, pivot-minor |
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Collection: |
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39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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09.03.2022 |
