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...

Abstract. Let K = C ∪ e1 ∪ e2 be a subgraph of G, consisting of a cycle C and disjoint paths e1 and e2, connecting two interlacing pairs of vertices in C. Suppose that K is embedded in the MöbiusbandinsuchawaythatC lies on its boundary. An algorithm is presented which in linear time extends the embedding of K to an embedding of G, if such an extension is possible, or finds a “nice ” obstruction for such embedding extensions. The structure of obtained obstructions is also analysed in details. Key words. surface embedding, obstruction, Möbius band, algorithm AMS subject classifications. 05C10, 05C85, 68Q20 1. Introduction. Let K be a subgraph of a graph G. A K-bridge (or a Kcomponent)inG is a subgraph of G which is either an edge e ∈ E(G)\E(K) (together with its endpoints) which has both endpoints in K, or it is a connected component of G − V (K) together with all edges (and their endpoints) between this component and K. EachedgeofaK-bridge B having an endpoint in K is a foot of B. The vertices

Let K be a subgraph of G. Suppose that we have a 2-cell embedding of K in some surface and that for each K-bridge in G one or two simple embeddings in faces of K are prescribed. A linear time algorithm is presented that either finds an embedding of G extending the embedding of K in the same surface using only prescribed embeddings of K-bridges, or finds an obstruction which certifies that such an extension does not exist. It is described how the obtained obstructions can be transformed into minimal obstructions in linear time. The geometric and combinatorial structure of minimal obstructions is also analyzed. At the end we apply the above algorithm to solve general embedding extension problems where the embedding of K is a closed 2-cell embedding. 1 Introduction Let K 0 be a fixed graph together with a fixed 2-cell embedding in some (closed) surface. Let G be a graph containing a subgraph K homeomorphic to K 0 . The embedding of K 0 and the homeomorphism K ! K 0 determine a 2-cell em...