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Universal obstructions for embedding extension problems
"... Let K be an induced nonseparating subgraph of a graph G, andletB be the bridge of K in G. Obstructions for extending a given 2cell 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 bran ..."
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Cited by 7 (6 self)
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Let K be an induced nonseparating subgraph of a graph G, andletB be the bridge of K in G. Obstructions for extending a given 2cell 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 2cell 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.
2Restricted Extensions Of Partial Embeddings Of Graphs
"... Let K be a subgraph of G. Suppose that we have a 2cell embedding of K in some surface and that for each Kbridge 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 ..."
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Cited by 3 (2 self)
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Let K be a subgraph of G. Suppose that we have a 2cell embedding of K in some surface and that for each Kbridge 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 Kbridges, 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 2cell embedding. 1 Introduction Let K 0 be a fixed graph together with a fixed 2cell 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 2cell em...