6 2. PRELIMINARIES

(ii) A vertically 4-connected matroid with rank at least four has no triads.

(iii) Both vertically 4-connected and internally 4-connected matroids are also al-

most vertically 4-connected.

(iv) An almost vertically 4-connected matroid with no triads is vertically 4-con-

nected.

Proposition 2.2. Suppose that (X1, X2) is a k-separation of the matroid M,

and that N is a minor of M. If |E(N) ∩ Xi| ≥ k for i = 1, 2, then (E(N) ∩

X1, E(N) ∩ X2) is a k-separation of N.

Proposition 2.3. Suppose that (X, Y ) is a vertical k-separation of the matroid

M and that e ∈ Y . If e ∈ clM (X), then (X ∪ e, Y − e) is a vertical k -separation

of M, where k ∈ {k, k − 1}.

Proposition 2.4. Let e be a non-coloop element of the matroid M. Suppose

that (X, Y ) is a k-separation of M\e. Then (X ∪ e, Y ) is a k-separation of M

if and only if e ∈ clM (X) and (X, Y ∪ e) is a k-separation of M if and only if

e ∈ clM (Y ).

The following result is due to Bixby [Bix82].

Proposition 2.5. Let e be an element of the 3-connected matroid M. Then

either si(M/e) or co(M\e) is 3-connected.

Suppose that (e1, . . . , et) is an ordered sequence of at least three elements from

the matroid M. Then (e1, . . . , et) is a fan if {ei, ei+1, ei+2} is a triangle of M

whenever i ∈ {1, . . . , t−2} is odd and a triad whenever i is even. Dually, (e1, . . . , et)

is a cofan if {ei, ei+1, ei+2} is a triad of M whenever i ∈ {1, . . . , t − 2} is odd and

a triangle whenever i is even. Note that if (e1, . . . , et) is a fan and t is even, then

(et, . . . , e1) is a cofan. We shall say that the unordered set X is a fan if there is

some ordering (e1, . . . , et) of the elements of X such that (e1, . . . , et) is either a fan

or a cofan. The length of a fan X is the cardinality of X. It is straightforward to

check that if {e1, . . . , et} is a fan of M, then λM ({e1, . . . , et}) ≤ 2. The next result

is easy to confirm.

Proposition 2.6. Suppose that (X, Y ) is a 3-separation of the 3-connected

binary matroid M. If |X| ≤ 5 and rM (X) ≥ 3 then one of the following holds:

(i) X is a triad;

(ii) X is a fan with length four or five; or,

(iii) There is a four-element circuit-cocircuit

C∗

⊆ X such that X ⊆ clM

(C∗)

or

X ⊆ clM

∗

(C∗).

The next result follows directly from a theorem of Oxley [Oxl87, Theorem 3.6].

Lemma 2.7. Let T be a triangle of a 3-connected binary matroid M. If the

rank and corank of M are at least three then M has an M(K4)-minor in which T

is a triangle.

2.2. Fundamental graphs

Fundamental graphs provide a convenient way to visualize a representation of

a binary matroid with respect to a particular basis. Suppose that B is a basis of

the binary matroid M. The fundamental graph of M with respect to B, denoted

by GB(M), has E(M) as its vertex set. Every edge of GB(M) joins an element