Other authors who have studied families of binary matroids with no minors
in some subset of {F7, F7 ∗, M(K3,3), M(K5), M ∗(K3,3), M ∗(K5)} include Wal-
ton and Welsh [WW80], who examine the characteristic polynomials of ma-
troids in such classes, and Kung [Kun86], who has considered the maximum
size obtained by a simple rank-r matroid in one of these classes. Qin and
Zhou [QZ04] have characterized the internally 4-connected binary matroids with
no minor in {M(K3,3), M(K5), M ∗(K3,3), M ∗(K5)}. Moreover Zhou [Zho08]
has also characterized the internally 4-connected binary matroids that have no
M(K3,3)-minor, but which do have an M(K5)-minor. Finally we note that the
classic result of Hall [Hal43] on graphs with no K3,3-minor leads to a char-
acterization of the internally 4-connected binary matroids with no minor in
{F7, F7
M(K3,3)}, and Wagner’s [Wag37] theorem on the
graphs with no K5-minor leads to a characterization of the internally 4-connected
binary matroids with no minor in {F7, F7
The proof of Theorem 1.1 is unusual amongst results in matroid theory, in that
we rely upon a computer to check certain facts. All these checks have been carried
out using the software package Macek, developed by Petr Hlinˇ en´ y. In addition we
have written software, which does not depend upon Macek, to provide us with an
independent check of the same facts.
The Macek package is available to download, along with supporting documen-
tation. The current website is http://www.fi.muni.cz/~hlineny/MACEK. Thus
the interested reader is able to download Macek and confirm that it verifies our
assertions. Points in the proof where a computer check is required are marked by
a marginal symbol and a number. The numbers provide a reference for the web-
site of the second author (current url: http://www.maths.uwa.edu.au/~gordon),
which contains a more detailed description of the steps taken to verify each asser-
tion, and the intermediate data produced during that computer check.
We emphasize that none of the computer tests relies upon an exhaustive search
of all the matroids of a particular size or rank. The only tasks a program need
perform to verify our checks are: determine whether two binary matroids are iso-
morphic; check whether a binary matroid has a particular minor; and, generate all
the single-element extensions and coextensions of a binary matroid. Whenever the
proof requires a computer check, the text includes a complete description (inde-
pendent of any particular piece of software) of the tasks that we ask the computer
to perform. Hence a reader who is able to construct software with the capabilities
listed above could provide another verification of our assertions.
The article is organized as follows: In Chapter 2 we develop the basic definitions
and results we will need to prove Theorem 1.1. In Chapter 3 we introduce the
obius matroids and consider their properties in detail.
Chapter 4 is concerned with showing that a minimal counterexample to Theo-
rem 1.1 can be assumed to be vertically 4-connected. This chapter depends heavily
upon the Δ-Y operation and its dual; we make extensive use of results proved
by Oxley, Semple, and Vertigan [OSV00]. The central idea of Chapter 4 is that
if a binary matroid M with no M(K3,3)-minor is non-cographic and internally
4-connected but not vertically 4-connected, then by repeatedly performing Y
operations, we can produce a vertically 4-connected non-cographic binary matroid
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