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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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Graph and map isomorphism and all polyhedral embeddings in linear time
 IN PROCEEDINGS OF THE 40TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC
, 2008
"... For every surface S (orientable or nonorientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of facewidth at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g), where g is the genus of S. Th ..."
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Cited by 15 (5 self)
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For every surface S (orientable or nonorientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of facewidth at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g), where g is the genus of S. This is the first algorithm for which the degree of polynomial in the time complexity does not depend on g. The above result is based on two linear time algorithms, each of which solves a problem that is of independent interest. The first of these problems is the following one. Let S be a fixed surface. Given a graph G and an integer k≥3, we want to find an embedding of G in S of face width at least k, or conclude that such an embedding does not exist. It is known that this problem is NPhard when the surface is not fixed. Moreover, if there is an embedding, the algorithm can give all embeddings of facewidth at least k, up to Whitney equivalence. Here, the facewidth of an embedded graph G is the minimum number of points of G in which some noncontractible closed curve in the surface intersects the graph. In the proof of the above algorithm, we give a simpler proof and a better bound for the theorem by Mohar and Robertson concerning the number of polyhedral embeddings of 3connected graphs.
Crossing and weighted crossing number of nearplanar graphs
 ALGORITHMICA
, 2009
"... A nonplanar graph G is nearplanar if it contains an edge e such that G − e is planar. The problem of determining the crossing number of a nearplanar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop minmax formulas involving efficiently computable lower and u ..."
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Cited by 10 (2 self)
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A nonplanar graph G is nearplanar if it contains an edge e such that G − e is planar. The problem of determining the crossing number of a nearplanar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop minmax formulas involving efficiently computable lower and upper bounds. These minmax results are the first of their kind in the study of crossing numbers and improve the approximation factor for the approximation algorithm given by Hliněny and Salazar (Graph Drawing GD’06). On the other hand, we show that it is NPhard to compute a weighted version of the crossing number for nearplanar graphs.
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.
An algorithm for embedding graphs in the torus
"... An efficient algorithm for embedding graphs in the torus is presented. Given a graph G, the algorithm either returns an embedding of G in the torus or a subgraph of G which is a subdivision of a minimal nontoroidal graph. The algorithm based on [13] avoids the most complicated step of [13] by applyi ..."
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Cited by 1 (1 self)
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An efficient algorithm for embedding graphs in the torus is presented. Given a graph G, the algorithm either returns an embedding of G in the torus or a subgraph of G which is a subdivision of a minimal nontoroidal graph. The algorithm based on [13] avoids the most complicated step of [13] by applying a recent result of Fiedler, Huneke, Richter, and Robertson [5] about the genus of graphs in the projective plane, and simplifies other steps on the expense of losing linear time complexity. 1
Linear Connectivity Forces Large Bipartite Minors
, 2004
"... Let a be an integer. It is proved that for any s and k, there exists a constant N = N(s, k, a) such that every 31 2 (a+1)connected graph with at least N vertices either contains a subdivision of Ka,sk or a minor isomorphic to s disjoint copies of Ka,k. In fact, we prove that connectivity 3a + 2 and ..."
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Let a be an integer. It is proved that for any s and k, there exists a constant N = N(s, k, a) such that every 31 2 (a+1)connected graph with at least N vertices either contains a subdivision of Ka,sk or a minor isomorphic to s disjoint copies of Ka,k. In fact, we prove that connectivity 3a + 2 and minimum degree at least 31 2 (a + 1) − 3 are enough. The condition “a subdivision of Ka,sk ” is necessary since G could be a complete bipartite graph K31 where m could be arbitrarily 2 (a+1),m, large. The requirement on N(s, k, a) vertices is necessary since there exist graphs without Kaminor whose connectivity is Θ(a √ log a). When s = 1 and k = a, this implies that every 31 2 (a+1)connected graph with at least N(a) vertices has a Kaminor. This is the first result where a linear lower bound on the connectivity in terms of a forces a Kaminor. This was also conjectured in [68, 47, 69, 39]. Our result generalizes a recent result of Böhme and Kostochka [4] and resolves a conjecture of FonDerFlaass [16]. Our result together with a recent result in [25] also implies that there exists an absolute constant c such that there are only finitely many ckcontractioncritical graphs without Kk as a minor and there are only finitely many ckconnected ckcolorcritical graphs without Kkminors. These results are related to the wellknown conjecture of Hadwiger [17]. Our result was also motivated by the wellknown result of Erdős and Pósa [15]. Suppose that G is 31
K a,k minors in graphs of bounded treewidth
 J. COMBIN. THEORY SER. B
, 2000
"... It is shown that for any positive integers k and w there exists a ..."