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Towards Unavoidable Minors of Binary 4-connected Matroids

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posted on 08.12.2021, 02:02 by Susan JowettSusan Jowett

We show that for every n ≥ 3 there is some number m such that every 4-connected binary matroid with an M (K3,m)-minor or an M* (K3,m)-minor and no rank-n minor isomorphic to M* (K3,n) blocked in a path-like way, has a minor isomorphic to one of the following: M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, or a rank-n matroid closely related to the cycle matroid of a double wheel, which we call a non graphic double wheel. We also show that for all n there exists m such that the following holds. If M is a 4-connected binary matroid with a sufficiently large spanning restriction that has a certain structure of order m that generalises a swirl-like flower, then M has one of the following as a minor: a rank-n spike, M (K4,n), M* (K4,n), the cycle matroid of an n-spoke double wheel, the cycle matroid of a rank-n circular ladder, the cycle matroid of a rank-n Möbius ladder, a matroid obtained by adding an element in the span of the petals of M (K3,n) but not in the span of any subset of these petals and contracting this element, a rank-n non graphic double wheel, M* (K3,n) blocked in a path-like way or a highly structured 3-connected matroid of rank n that we call a clam.

History

Copyright Date

01/01/2019

Date of Award

01/01/2019

Publisher

Te Herenga Waka—Victoria University of Wellington

Rights License

Author Retains Copyright

Degree Discipline

Matroid Theory

Degree Grantor

Te Herenga Waka—Victoria University of Wellington

Degree Level

Doctoral

Degree Name

Doctor of Philosophy

ANZSRC Type Of Activity code

1 PURE BASIC RESEARCH

Victoria University of Wellington Item Type

Awarded Doctoral Thesis

Language

en_NZ

Victoria University of Wellington School

School of Mathematics and Statistics

Advisors

Whittle, Geoff