2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210746In Graph Minor III, Robertson and Seymour conjecture that the tree-width of a planar graph and that of its dual differ by at most one. We prove that given a hypergraph H on a surface of Euler genus k, the tree-width of H^* is at most the maximum of tw(H) + 1 + k and the maximum size of a hyperedge of H^*.Discrete Mathematicstext