Some Excluded-Minor Theorems for Binary Matroids

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(K_3,3)-, or M^*(K_3,3)-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 K_3,3-minor. In Chapter 5, we determine all the internally 4-connected binary matroids with an M(K₅)-minor that have no M(K_3,3)-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 K_3,3 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.