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The embeddability of a graph in a given surface is determined entirely by the polygon matroid of the graph. That is also true for cellular embeddability in nonorientable surfaces but not in orientable surfaces. An embedding of a finite graph T in a surface S is a homeomorphism of Y, regarded as a topological space, with a closed subset of S. In order to know in which surfaces T embeds it suffices to consider only the compact surfaces: the orientable ones T of genus g (Euler characteristic 2-2g)doi:10.2307/2047303 fatcat:qyzwvrn7wbd6pghlzuw24mgrdm