On nonbinary 3-connected matroids

@article{Oxley1987OnN3,
  title={On nonbinary 3-connected matroids},
  author={James G. Oxley},
  journal={Transactions of the American Mathematical Society},
  year={1987},
  volume={300},
  pages={663-679}
}
  • J. Oxley
  • Published 1 February 1987
  • Mathematics
  • Transactions of the American Mathematical Society
It is well known that a matroid is binary if and only if it has no minor isomorphic to U2,4, the 4-point line. Extending this result, Bixby proved that every element in a nonbinary connected matroid is in a U2,4minor. The result was further extended by Seymour who showed that every pair of elements in a nonbinary 3-connected matroid is in a U2,4-minor. This paper extends Seymour's theorem by proving that if {x, y, z} is contained in a nonbinary 3-connected matroid M, then either M has a U2,4… 

Figures from this paper

The Smallest Rounded Sets of Binary Matroids
Triangles in 3-connected matroids
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.
On Minors Avoiding Elements in Matroids
Structural Results for Matroids.
This dissertation solves some problems involving the structure of matroids. In Chapter 2, the class of binary matroids with no minors isomorphic to the prism graph, its dual, and the binary affine
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 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
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 21 REFERENCES
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.
On the intersections of circuits and cocircuits in matroids
  • J. Oxley
  • Mathematics, Computer Science
    Comb.
  • 1984
TLDR
This paper determines precisely when a matroidM has a quad, a 4-element set which is the intersection of a circuit and a cocircuit, and shows that this will occur if M has a Circuit and a Cocircuit meeting in more than four elements.
A problem of P. Seymour on nonbinary matroids
TLDR
The purpose of this note is to show that fork=1, 2, 3 does not remain true and to suggest some possible alternatives.
Triples in Matroid Circuits
Connectivity in Matroids
  • W. T. Tutte
  • Mathematics
    Canadian Journal of Mathematics
  • 1966
An edge of a 3-connected graph G is called essential if the 3-connection of G is destroyed both when the edge is deleted and when it is contracted to a single vertex. It is known (1) that the only
ℓ-matrices and a Characterization of Binary Matroids
A combinatorial model for series-parallel networks
The category of pregeometries with basepoint is defined and explored. In this category two important operations are extensively characterized: the series connection S(G, H), and the parallel
Decomposition of regular matroids
On matroid connectivity
...
1
2
3
...