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Well-quasi-ordering graphs by the topological minor relation

Robertson and Seymour proved that graphs are well-quasi-ordered by the minor relation and the weak immersion relation in the prominent Graph Minors series. That is, given infinitely many graphs, one graph contains another as a minor (or a weak immersion, respectively). However, the topological minor relation does not well-quasi-order graphs. For every positive integer i, let Gi be the graph obtained from a path of length i by doubling every edge and attaching two leaves to each of the ends of the path. Then there do not exist distinct integers a, b such that Ga contains Gb as a topological minor. An old conjecture of Robertson states that for every integer k, graphs with no topological minor isomorphic to the graph obtained from a path of length k by doubling every edge are well-quasi-ordered by the topological minor relation. We recently proved this conjecture, and we will sketch our proof in this talk.