5 - Graph minors: generalizing Kuratowski's theorem

Summary

In their Graph Minors Project, Robertson and Seymour proved that, in any infinite set of graphs, one is a minor of another. In particular, if S is a surface, the set of minor-minimal graphs that are not embeddable in S is finite.

Two central results of the Graph Minors Project are:

• if the graphs in the infinite set have bounded tree-width, then one is a minor of the other;

• graphs with large tree-width have large grids as minors.

We present the ‘simple’ proofs of these two facts, and adapt an argument of Thomassen that shows how to apply them to prove the finiteness of the set of minor-minimal non-S-embeddable graphs.

Introduction

This chapter is a self-contained introduction to graph minors. It contains a complete proof of the generalization of Kuratowski's theorem to higher surfaces; more importantly, it is a major step in understanding the whole Graph Minors Project of Robertson and Seymour. The only background needed is some familiarity with connectivity issues (essentially variations of Menger's theorem and a willingness to view cutsets from different perspectives). Our experience with the arguments presented here is that we need to be able to focus on both the big picture and on the details. There are many small points that require their own little arguments and we have attempted to provide these in sufficient detail to make it easier not to lose sight of the big picture.