Topics in matroid union

Abstract

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.