A simpler proof of the excluded minor theorem for higher surfaces

We give a simple proof of the fact (which follows From the Robertson-Seymour theory) that a graph which is minimal of genus g cannot contain a subdivision of a large grid. Combining this with the tree-width theorem and the quasi-well-ordering of graphs of bounded tree-width in the Robertson-Seymour theory, we obtain a simpler proof of the generalized Kuratowski theorem for each fixed surface. The proof requires no previous knowledge of graph embeddings. (C) 1997 Academic Press.