Excluded minor theorems for graphs are some of the most celebrated results in graph structure theory. In this talk I will give an overview of excluded minor results that play a useful role in matroid theory.
For a given class of matroids with a specified structure, a minimal excluded minor is a matroid that is not in the class, but of which every single-element deletion and contraction is in the class. Such matroids are minimal obstructions for membership in the class. For example, the complete graph on five vertices, K5 , and the complete bipartite graph with three vertices in each class, K3,3 , are the minimal obstructions to planarity in graphs. Similarly, several important classes of matroids such as binary, ternary, and quaternary matroids can be characterized in terms of their minimal obstructions. Using software we computed several previously unknown rank-3 minimal excluded minors for matroids representable over the field of order 5.
The opposite approach to excluded minor problems is to select a certain set of matroids for exclusion and describe the structure of the resulting class. Such results are called decomposition results. Results of this type were initiated by Seymour in his paper on the decomposition of regular matroids. I will describe how the problem of almost-graphic matroids (this is the class of matroids for which either M\e or M/e is a graphic matroid, for every element e) was solved by giving decomposition results for several classes of matroids.
The computer-generated minimal excluded minors were joint work with Robert Kingan and the result on almost-graphic matroids was joint work with Manoel Lemos.