Excluded Minors of Rank 3 for Orientability and Representability

Hidefumi HIRAISHI
Sonoko MORIYAMA

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E101-A    No.9    pp.1355-1362
Publication Date: 2018/09/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E101.A.1355
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
Category: 
Keyword: 
matroid,  representable matroid,  matroid minor,  

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Summary: 
We investigate excluded minor characterizations of two fundamental classes of matroids: orientable matroids and representable matroids. We prove (i) for any fixed field F, there exist infinitely many excluded minors of rank 3 for the union of the class of orientable matroids and the class of F-representable matroids, and (ii) for any fixed field F with characteristic 0, there exist infinitely many orientable excluded minors of rank 3 for intersection of the class of orientable matroids and the class of F-representable matroids. We show these statements by explicitly constructing infinite families of excluded minors.