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
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