Minors of 3-Connected Matroids

@article{Seymour1985MinorsO3,
  title={Minors of 3-Connected Matroids},
  author={Paul D. Seymour},
  journal={Eur. J. Comb.},
  year={1985},
  volume={6},
  pages={375-382}
}
  • P. Seymour
  • Published 1 December 1985
  • Mathematics
  • Eur. J. Comb.
Triangles in 3-connected matroids
On Minors Avoiding Elements in Matroids
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.
The Smallest Rounded Sets of Binary Matroids
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.
THE EXCLUDED MINORS FOR THE MATROIDS THAT ARE BINARY OR TERNARY
We show that a matroid is binary or ternary if and only if it has no minor isomorphic to U2,5, U3,5, U2,4 ⊕ F7, U2,4 ⊕ F ∗ 7 , U2,4 ⊕2 F7, U2,4 ⊕2 F ∗ 7 , or the unique matroids obtained by relaxing
Counterexamples to conjectures on 4-connected matroids
TLDR
This paper provides counterexamples to two conjectures of Robertson is that each triple of elements in a 4-connected non-graphic matroid is in some circuit.
On Representable Matroids Having Neither U2,5– Nor U3,5–minors
Consider 3–connected matroids that are neither binary nor ternary and have neither U2,5– nor U3,5–minors: for example, AG(3, 2)′, the matroid obtained by relaxing a circuit-hyperplane of AG(3, 2).
Stabilizers of Classes of Representable Matroids
TLDR
One of the main theorems of this paper proves that if M is minor-closed and closed under duals, and N is 3- connected, then to show that N is a stabilizer it suffices to check 3-connected matroids in M that are single-element extensions or coextensions of N, or are obtained by a single- element extension followed by asingle-element coextension.
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References

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On minors of non-binary matroids
TLDR
It is proved that for every two elements of a 3-connected non-binary matroid, there is aU42 minor using them both.
Adjacency in Binary Matroids
A note on the production of matroid minors
Decomposition of regular matroids