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Using the Graph Minor Theorem, by Robertson and Seymour, one can easily prove that there are only a finite number of excluded minors for a given surface. This immediately implies that, given a surface, the size of these excluded minors is bounded. However, no explicit bound can be derived from the Graph Minor Theorem. In 1993, Seymour proved that an excluded minor for a given surface S is of order bounded by a double exponential of the genus of S. More than thirty years later, we managed to lower this bound to a quasi-polynomial in the genus of the surface. Joint work with Ken-ichi Kawarabayashi.

[Sarah Houdaigoui] (NII, Tokyo)

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