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Topics in matroid union

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thesis
posted on 2022-03-07, 21:26 authored by Ali M. Hameed
The operation of matroid union was introduced by Nash-Williams in 1966. A matroid is indecomposable if it cannot be written in the form M = M1 V M2, where r(M1),r(M2) > 0. In 1971 Welsh posed the problem of characterizing indecomposable matroids, this problem has turned out to be extremely difficult. As a partial solution towards its progress, Cunningham characterized binary indecomposable matroids in 1977. In this thesis we present numerous results in topics of matroid union. Those include a link between matroid connectivity and matroid union, such as the implication of having a 2-separation in the matroid union, and under what conditions is the union 3-connected. We also identify which elements in binary and ternary matroids are non-fixed. Then we create a link between having non-fixed elements in binary and ternary matroids and the decomposability of such matroids, and the effect of removing non-fixed elements from binary and ternary matroids. Moreover, we show results concerning decomposable 3-connected ternary matroids, such as what essential property every decomposable 3-connected ternary matroid must have, how to compose a ternary matroid, and what a 3-connected ternary matroid decomposes into. We also give an alternative statement and an alternative proof of Cunningham's theorem from the perspective of fixed and non-fixed elements.

History

Copyright Date

2008-01-01

Date of Award

2008-01-01

Publisher

Te Herenga Waka—Victoria University of Wellington

Rights License

Author Retains All Rights

Degree Discipline

Mathematics

Degree Grantor

Te Herenga Waka—Victoria University of Wellington

Degree Level

Masters

Degree Name

Master of Science

Victoria University of Wellington Item Type

Awarded Research Masters Thesis

Language

en_NZ

Victoria University of Wellington School

School of Mathematics, Statistics and Operations Research

Advisors

Whittle, Geoff