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
Oxley, J. (2019). A matroid extension result. SIAM Journal on Discrete Mathematics, 33 (1), 138-152. https://doi.org/10.1137/18M1187155