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
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