8 2. PRELIMINARIES

Proposition 2.9. Let B be a basis of a matroid M. Let (X, Y ) be an exact

k-separation of M[X ∪ Y, B] and let e1, . . . , et be a blocking sequence of (X, Y ).

Suppose that |X| k and that e1 is either in parallel or in series to x ∈ X in M[X∪

Y ∪ e1, B] where x / ∈ clM[X∪Y,

B]

(Y ) and x / ∈ clM[X∪Y,

∗

B]

(Y ). If {e1, x} ⊆ B or if

{e1, x} ∩ B = ∅ then let B = B, and otherwise let B be the symmetric difference

of B and {x, e1}. Then M[(X − x) ∪ Y ∪ e1, B ]

∼

= M[X ∪ Y, B] and e2, . . . , et is a

blocking sequence of the k-separation ((X − x) ∪ e1, Y ) in M[(X − x) ∪ Y ∪ e1, B ].

Proposition 2.10. Suppose that N is a 3-connected matroid such that

|E(N)| ≥ 8 and N contains a four-element circuit-cocircuit X. If M is an inter-

nally 4-connected matroid with an N-minor, then there exists a 3-connected single-

element extension or coextension N of N, such that X is not a circuit-cocircuit of

N and N is a minor of M.

Proof. Let B be a basis of M and let X and Y be disjoint subsets of E(M)

such that M[X ∪ Y, B]

∼

= N, where X is a four-element circuit-cocircuit of M[X ∪

Y, B]. Thus (X, Y ) is an exact 3-separation of M[X ∪Y, B]. Since |X|, |Y | ≥ 4 and

M is internally 4-connected it cannot be the case that (X, Y ) induces a 3-separation

of M. Lemma 2.8 implies that there is a blocking sequence e1, . . . , et of (X, Y ).

Let us suppose that B, X, and Y have been chosen so that t is as small as possible.

Since (X, Y ∪ e1) is not a 3-separation of M[X ∪ Y ∪ e1, B] it follows that X is

not a circuit-cocircuit in M[X ∪Y ∪e1, B]. Thus if M[X ∪Y ∪e1, B] is 3-connected

there is nothing left to prove. Therefore we will assume that M[X ∪Y ∪e1, B] is not

3-connected. Hence e1 is in series or parallel to some element in M[X ∪ Y ∪ e1, B],

and in fact e1 is in parallel or series to an element x ∈ X, since X is not a circuit-

cocircuit of M[X ∪Y ∪e1, B]. But Proposition 2.9 now implies that our assumption

on the minimality of t is contradicted. This completes the proof.

2.4. Splitters

Suppose that M is a minor-closed class of matroids. A splitter of M is a

matroid M ∈ M such that any 3-connected member of M having an M-minor

is isomorphic to M. We present here two different forms of Seymour’s Splitter

Theorem.

Theorem 2.11. [Sey80, (7.3)] Suppose M is a minor-closed class of matroids.

Let M be a 3-connected member of M such that |E(M)| ≥ 4 and M is neither a

wheel nor a whirl. If no 3-connected single-element extension or coextension of M

belongs to M, then M is a splitter for M.

Theorem 2.12. [Oxl92, Corollary 11.2.1] Let N be a 3-connected matroid with

|E(N)| ≥ 4. If N is not a wheel or a whirl, and M is a 3-connected matroid with a

proper N-minor, then M has a 3-connected single-element deletion or contraction

with an N-minor.

Recall that for a binary matroid M we use EX (M) to denote the set of binary

matroids with no M-minor.

Proposition 2.13. The only 3-connected matroid in EX (M(K3,3)) that is reg-

ular but non-cographic is M(K5).

Proof. Walton and Welsh [WW80] note (and it is easy to confirm) that

M(K5) is a splitter for EX (F7, F7 ∗, M(K3,3)). This implies the result, since the