Obstructions for matroids of path-width at most k and graphs of linear rank-width at most k
Kanté, Mamadou Moustapha; Kim, Eun Jung; Kwon, O-joung; Oum, Sang-Il (2023), Obstructions for matroids of path-width at most k and graphs of linear rank-width at most k, Journal of Combinatorial Theory. Series B, 160, p. 15-35. 10.1016/j.jctb.2022.12.004
TypeArticle accepté pour publication ou publié
External document linkhttps://arxiv.org/abs/2109.12291
Journal nameJournal of Combinatorial Theory. Series B
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Author(s)Kanté, Mamadou Moustapha
Laboratoire d'Informatique, de Modélisation et d'Optimisation des Systèmes [LIMOS]
Kim, Eun Jung
Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Incheon National University
Department of Mathematical Sciences, KAIST
Abstract (EN)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.
Subjects / KeywordsPath-width; Matroid; Linear rank-width; Forbidden minor; Vertex-minor; Pivot-minor; Graph
Showing items related by title and author.
Kanté, Mamadou Moustapha; Kim, Eun Jung; Kwon, O-joung; Paul, Christophe (2017) Article accepté pour publication ou publié