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Among the most important results in modern combinatorics is Robertson and Seymour's proof of the graph minor theorem, commonly known as Wagner's conjecture. One way of stating this result is as follows: in every infinite class of finite undirected simple graphs there are two graphs H and G such that H is a minor of G.
To prove this result, Robertson and Seymour developed a deep structure theory of graphs in terms of the graph minor relation. Several concepts, ideas and intermediate results developed as part of this theory have found numerous applications in other areas of mathematics, computer science and beyond. In fact, the impact of concepts such as the "tree width" of a graph and its dual concepts of "brambles" or "grid minors" has been so pervasive that one could say that entirely new fields of research have emerged based on these ideas.
The structure theory developed as part of the graph minor project culminates in Robertson and Seymour's famous structure theorem for graphs excluding a fixed minor. Informally it says that every graph G
1. is somewhat similar to a tree (formalised by the notion of "tree width") or
2. contains a large complete graph as a minor or
3. can be decomposed into a tree of subgraphs of (almost) bounded genus (slightly more pecisely: has a tree decomposition whose pieces can almost be embedded into a fixed surface (into which G itself cannot be embedded of course)).
This theorem is not only interesting from a structural perspective but has deep implication in the theory of graph algorithms.
Despite its enormous complexity, the proof of the structure theorem follows a clear route with clearly identifiable intermediate steps, which are important in their own right. Furthermore, many results along the way can be understood on an intuitive level and many of them can be applied algorithmically as a black-box. That is, even without understanding or even reading their proof one can apply them in many algorithmic settings.
Finally, many steps of the proof are not only very interesting, or important, the underlying concepts and ideas are often simply stunning in their mathematical beauty making the study of graph minors an exciting endeavour.
In this course we will cover all essential aspects of the proof of the graph minor structure theorem. Starting with essentially no prerequisites we will introduce the fundamental concepts of tree width and grid minors, surfaces and surface embeddings, and then develop the proof of the structure theorem over the period of the course.
