On chains of 3-connected matroids

@article{Bixby1986OnCO,
  title={On chains of 3-connected matroids},
  author={Robert E. Bixby and Collette R. Coullard},
  journal={Discret. Appl. Math.},
  year={1986},
  volume={15},
  pages={155-166}
}
On the Structure of 3-connected Matroids and Graphs
TLDR
It is proved that the essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan, a maximal partial wheel, containing both and if an essential element e of M is in more than one fan, then that fan has three or five elements.
Vertically N-contractible elements in 3-connected matroids
In this paper we establish a variation of the Splitter Theorem. Let $M$ and $N$ be simple 3-connected matroids. We say that $x\in E(M)$ is vertically $N$-contractible if $si(M/x)$ is a 3-connected
On fixing elements in matroid minors
TLDR
The aim of this note is to prove that, for all sufficiently largen, the collection of n-element 3-connected matroids having some minor in F is also (3, 1)-rounded.
Finding a small 3-connected minor maintaining a fixed minor and a fixed element
TLDR
This result generalizes a theorem of Truemper and can be used to prove Seymour’s 2-roundedness theorem, as well as a result of Oxley on triples in nonbinary matroids.
Linearly Synthesizing 2-Connected Simplicial Graphs
It is proved that for any two 2-connected, smooth, and simplicial graphs G and H such that H is homeomorphic to a subgraph of G, there is a sequence of 2-connected subgraphs G0 I G1 I. . . I Gr = G
Matroid decomposition
Extensions of Tutte's wheels-and-whirls theorem

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Partial Matroid Representations
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