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Embedding graphs containing K5subdivisions
 Ars Combinatoria
"... Given a nonplanar graph G with a subdivision of K5 as a subgraph, we can either transform the K5subdivision into a K3,3subdivision if it is possible, or else we obtain a partition of the vertices of G\K5 into equivalence classes. As a result, we can reduce a projective planarity or toroidality al ..."
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Cited by 8 (2 self)
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Given a nonplanar graph G with a subdivision of K5 as a subgraph, we can either transform the K5subdivision into a K3,3subdivision if it is possible, or else we obtain a partition of the vertices of G\K5 into equivalence classes. As a result, we can reduce a projective planarity or toroidality algorithm to a small constant number of simple planarity checks [6] or to a K3,3subdivision in the graph G. It significantly simplifies algorithms presented in [7], [10] and [12]. We then need to consider only the embeddings on the given surface of a K3,3subdivision, which are much less numerous than those of K5. 1.
Simpler Projective Plane Embedding
, 2000
"... A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n ..."
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Cited by 3 (0 self)
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A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V . Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. Archdeacon's survey [2] provides an excellent introduction to topologica...
Forbidden Minors to Graphs with Small Feedback Sets
 DISCRETE MATHEMATICS
, 1996
"... Finite obstruction set characterizations for lower ideals in the minor order are guaranteed to exist by the Graph Minor Theorem. In this paper wecharacterize several families of graphs with small feedback sets, namely k 1 Feedback Vertex Set, k 2 Feedback Edge Set and #k 1 ,k 2 ##Feedback Ver ..."
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Cited by 2 (1 self)
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Finite obstruction set characterizations for lower ideals in the minor order are guaranteed to exist by the Graph Minor Theorem. In this paper wecharacterize several families of graphs with small feedback sets, namely k 1 Feedback Vertex Set, k 2 Feedback Edge Set and #k 1 ,k 2 ##Feedback Vertex#Edge Set, for small integer parameters k 1 and k 2 . Our constructive methods can compute obstruction sets for any minorclosed family of graphs, provided the pathwidth #or treewidth# of the largest obstruction is known.
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 2 (2 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
CLASSIFICATION OF RINGS WITH GENUS ONE ZERODIVISOR GRAPHS
"... Abstract. This paper investigates properties of the zerodivisor graph of a commutative ring and its genus. In particular, we determine all isomorphism classes of finite commutative rings with identity whose zerodivisor graph has genus 1. 1. ..."
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Abstract. This paper investigates properties of the zerodivisor graph of a commutative ring and its genus. In particular, we determine all isomorphism classes of finite commutative rings with identity whose zerodivisor graph has genus 1. 1.
The obstructions for toroidal graphs with no K3,3’s
, 2005
"... Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no K3,3subdivisions that coincide with the toroidal graphs with no K3,3minors. These graphs admit a unique decomposition into planar components and have short lists of obstructions. We provide ..."
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Cited by 1 (0 self)
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Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no K3,3subdivisions that coincide with the toroidal graphs with no K3,3minors. These graphs admit a unique decomposition into planar components and have short lists of obstructions. We provide the complete lists of four forbidden minors and eleven forbidden subdivisions for the toroidal graphs with no K3,3’s and prove that the lists are sufficient.
A Simpler and Faster Torus Embedding Algorithm
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A Faster Algorithm for Torus Embedding
, 2006
"... Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and di ..."
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Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and discuss how it was inspired by the quadratic time planar embedding algorithm of Demoucron, Malgrange and Pertuiset. We show that it is faster in practice than the only fully implemented alternative (also exponential) and explain how both the algorithm itself and the knowledge gained during its development might be used to solve the wellstudied problem of finding the complete set of torus obstructions.
THE KURATOWSKI COVERING CONJECTURE FOR GRAPHS OF ORDER < 10 FOR THE NONORIENTABLE SURFACES OF Genus 3 and 4
, 2008
"... Kuratowski proved that a finite graph embeds in the plane if it does not contain a subdivision of either K5 or K3,3, called Kuratowski subgraphs. A conjectured generalization of this result to all nonorientable surfaces says that a finite minimal forbidden subgraph for the nonorientable surface of ..."
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Kuratowski proved that a finite graph embeds in the plane if it does not contain a subdivision of either K5 or K3,3, called Kuratowski subgraphs. A conjectured generalization of this result to all nonorientable surfaces says that a finite minimal forbidden subgraph for the nonorientable surface of genus ˜g can be written as the union of ˜g + 1 Kuratowski subgraphs such that the union of each pair of these fails to embed in the projective plane, the union of each triple of these fails to embed in the Klein bottle if ˜g ≥ 2, and the union of each triple of these fails to embed in the torus if ˜g ≥ 3. We show that this conjecture is true for all minimal forbidden subgraphs of order < 10 for the nonorientable surfaces of genus 3 and 4.