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A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded treewidth
, 2008
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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.
Embedding a Graph Into the Torus in Linear Time
, 1994
"... 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 ..."
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Cited by 4 (0 self)
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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...
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...
Obstructions for 2Möbius band embedding extension problem
 SIAM J. Discrete Math
, 1997
"... 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 embe ..."
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Cited by 3 (3 self)
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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 Kbridge (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. EachedgeofaKbridge B having an endpoint in K is a foot of B. The vertices
k3,kminors in large 7connected graphs
, 2006
"... It is shown that for any positive integer k there exists a constant N = N(k) such that every 7connected graph of order at least N contains K3,k as a minor. 1 ..."
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It is shown that for any positive integer k there exists a constant N = N(k) such that every 7connected graph of order at least N contains K3,k as a minor. 1
Projective plane and Möbius band obstructions
, 1997
"... Let S be a compact surface with possibly nonempty boundary ∂S and let G be a graph. Let K be a subgraph of G embedded in S such that ∂S ⊆ K. An embedding extension of K to G is an embedding of G in S which coincides on K with the given embedding of K. Minimal obstructions for the existence of embed ..."
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Let S be a compact surface with possibly nonempty boundary ∂S and let G be a graph. Let K be a subgraph of G embedded in S such that ∂S ⊆ K. An embedding extension of K to G is an embedding of G in S which coincides on K with the given embedding of K. Minimal obstructions for the existence of embedding extensions are classified in cases when S is the projective plane or the Möbius band (for several “canonical” choices of K). Linear time algorithms are presented that either find an embedding extension, or return a “nice” obstruction for the existence of extensions.
Obstructions for simple embeddings
"... Suppose that K ⊆ G is a graph embedded in some surface and F is a face of K with singular branches e and f such that F ∪ ∂F is homeomorphic to the torus minus an open disk. An embedding extension of K to G is a simple embedding if each Kbridge embedded in F is attached to at most one appearance of ..."
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Suppose that K ⊆ G is a graph embedded in some surface and F is a face of K with singular branches e and f such that F ∪ ∂F is homeomorphic to the torus minus an open disk. An embedding extension of K to G is a simple embedding if each Kbridge embedded in F is attached to at most one appearance of e and at most one appearance of f on ∂F. Combinatorial structure of minimal obstructions for existence of simple embedding extensions is described. Moreover, a linear time algorithm is presented that either finds a simple embedding, or returns an obstruction for existence of such embeddings. 1