Complete Graph Minors and the Graph Minor Structure Theorem
arXiv:1105.3549 · doi:10.1016/j.jctb.2012.09.001
Abstract
The graph minor structure theorem by Robertson and Seymour shows that every graph that excludes a fixed minor can be constructed by a combination of four ingredients: graphs embedded in a surface of bounded genus, a bounded number of vortices of bounded width, a bounded number of apex vertices, and the clique-sum operation. This paper studies the converse question: What is the maximum order of a complete graph minor in a graph constructed using these four ingredients? Our main result answers this question up to a constant factor.