The purpose of this dissertation is to generalize some important excluded-minor theorems for graphs to binary matroids.
Chapter 3 contains joint work with Hongxun Qin, in which we show that an internally 4-connected binary matroid with no M(K5)-, M*(K5)-, M(K3,3)-, or M*(K3,3)-minor is either planar graphic, or isomorphic to F7 or F*7. As a corollary, we prove an extremal result for the class of binary matroids without these minors.
In Chapter 4, it is shown that, except for 6 'small' known matroids, every internally 4-connected non-regular binary matroid has either a K͠5- or a K͠5*-minor. Using this result, we obtain a computer-free proof of Dharmatilake's conjecture about the excluded minors for binary matroids with branch-width at most 3.
D.W. Hall proved that K5 is the only simple 3-connected graph with a K5-minor that has no K3,3-minor. In Chapter 5, we determine all the internally 4-connected binary matroids with an M(K5)-minor that have no M(K3,3)-minor.
In chapter 6, it is shown that there are only finitely many non-regular internally 4-connected matroids in the class of binary matroids with no M(K'3,3)- or M*(K'3,3})-minor, where K'3,3 is the graph obtained from K3,3 by adding an edge between a pair of non-adjacent vertices.
In Chapter 7, we summarize the results and discuss about open problems. We are particularly interested in the class of binary matroids with no M(K5)- or M*(K5)-minor. Unfortunately, we tried without success to find all the internally 4-connected members of this class. However, it is shown that the matroid J1 is the smallest splitter for the above class.