Obstructions For 2-Möbius Band Embedding Extension Problem
SIAM Journal on Discrete Mathematics
A Linear Time Algorithm for Embedding Graphs in an Arbitrary Surface
SIAM Journal on Discrete Mathematics
Journal of the ACM (JACM)
Embedding Graphs in the Torus in Linear Time
Proceedings of the 4th International IPCO Conference on Integer Programming and Combinatorial Optimization
Testing planarity of partially embedded graphs
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
A kuratowski-type theorem for planarity of partially embedded graphs
Proceedings of the twenty-seventh annual symposium on Computational geometry
A Kuratowski-type theorem for planarity of partially embedded graphs
Computational Geometry: Theory and Applications
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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. Obstructions for existence of extensions of the embedding of K to an embedding of G are studied. It is shown that minimal obstructions possess certain combinatorial structure that can be described in an algebraic way by means of forcing chains of K-bridges. The geometric structure of minimal obstructions is also described. It is shown that they have "millipede" structure that was observed earlier in some special cases (disc, Möbius band). As a consequence it is proved that if one is allowed to reroute the branches of K, one can obtain a subgraph K' of G homeomorphic to K for which an obstruction of bounded branch size exists. The precise combinatorial and geometric structure of corresponding obstructions can be used to get a linear time algorithm for either finding an embedding extension or discovering minimal obstructions.