For an arbitrary fixed surface S, a linear time algorithm is presented that for a given graph G either finds an embedding of G in S or identifies a subgraph of G that is homeomorphic to a minimal forbidden subgraph for embeddability in S. A side result of the proof of the algorithm is that minimal forbidden subgraphs for embeddability in S cannot be arbitrarily large. This yields a constructive proof of the result of Robertson and Seymour that for each closed surface there are only finitely many minimal forbidden subgraphs. The results and methods of this paper can be used to solve more general embedding extension problems.

Let K be an induced non-separating subgraph of a graph G, andletB be the bridge of K in G. Obstructions for extending a given 2-cell embedding of K to an embedding of G in the same surface are considered. It is shown that it is possible to find a nice obstruction which means that it has bounded branch size up to a bounded number of “almost disjoint ” millipedes. Moreover, B contains a nice subgraph ˜ B with the following properties. If K is 2-cell embedded in some surface and F is a face of K, then ˜ B admits exactly the same types of embeddings in F as B. A linear time algorithm to construct such a universal obstruction ˜ B is presented. At the same time, for every type of embeddings of ˜ B, an embedding of B ofthesametypeis determined. 1

A linear time algorithm is presented that, for a given graph G, finds an embedding of G in the torus whenever such an embedding exists, or exhibits a subgraph\Omega of G of small branch size that cannot be embedded in the torus. 1 Introduction Let K be a subgraph of G, and suppose that we are given an embedding of K in some surface. The embedding extension problem asks whether it is embedding extension problem possible to extend the embedding of K to an embedding of G in the same surface, and any such embedding is an embedding extension of K to G. An embedding extension obstruction for embedding extensions is a subgraph\Omega of G \Gamma E(K) such that obstruction the embedding of K cannot be extended to K [ \Omega\Gamma The obstruction is small small if K [\Omega is homeomorphic to a graph with a small number of edges. If\Omega is small, then it is easy to verify (for example, by checking all the possibilities Supported in part by the Ministry of Science and Technolo...