Document Type

Article

Publication Date

1-1-2019

Abstract

Adding elements to matroids can be fraught with difficulty. In the Vámos matroid V 8 , there are four pairs X 1 , X 2 , X 3 , and X 4 that partition E(V 8 ) such that (X 1 ∪ X 2 , X 3 ∪ X 4 ) is a 3-separation while exactly three of the local connectivities (X 1 , X 3 ), (X 1 , X 4 ), (X 2 , X 3 ), and (X 2 , X 4 ) are one, with the fourth being zero. As is well known, there is no extension of V 8 by a nonloop element p such that X j ∪ p is a circuit for all j. This paper proves that a matroid can be extended by a fixed element in the guts of a 3-separation provided no Vámos-like structure is present.

Publication Source (Journal or Book title)

SIAM Journal on Discrete Mathematics

First Page

138

Last Page

152

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