On chains of 3-connected matroids

  title={On chains of 3-connected matroids},
  author={Robert E. Bixby and Collette R. Coullard},
  journal={Discret. Appl. Math.},
On the Structure of 3-connected Matroids and Graphs
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
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
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


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
Menger's theorem for matroids
Menge r's Theore m asse rt s t hat if x and yare ve rti ces of a graph wh ic h a re not joined b y an edge and if it takes at least k ot he r vert ices to separate x and y, the n x and y can be
A Combinatorial Decomposition Theory
Given a finite undirected graph G and A ⊆ E(G), G(A) denotes the subgraph of G having edge-set A and having no isolated vertices. For a partition {E1, E2} of E(G), W(G; E1) denotes the set V(G(E1)) ⋂
On matroid connectivity
Partial Matroid Representations
Decomposition of regular matroids