Complete graph minors and the graph minor structure theorem

Authors:
GwenaëL Joret;David R. Wood
Affiliations:
Département dInformatique, Université Libre de Bruxelles, Brussels, Belgium;Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia
Venue:
Journal of Combinatorial Theory Series B
Year:
2013

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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.