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Practical Toroidality Testing
 Proc. of the Eighth Annual ACMSIAM Symposium on Discrete Algorithms
, 1996
"... We describe an algorithm for embedding graphs on the torus (doughnut) which we implemented first in C, and then in C++. Although the algorithm is exponential in the worst case, it was very effective for indicating the small graphs which are torus obstructions. We have completed examination of the gr ..."
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Cited by 12 (2 self)
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We describe an algorithm for embedding graphs on the torus (doughnut) which we implemented first in C, and then in C++. Although the algorithm is exponential in the worst case, it was very effective for indicating the small graphs which are torus obstructions. We have completed examination of the graphs on up to 10 vertices and the 11 vertex ones up to 24 edges, and of these 3884 are topological obstructions, and 2249 are also minor order obstructions. A cursory search of 12 and 13 vertex graphs resulted in several more. We purport that this approach has proved practical as it has permitted us to compile what we believe to be the biggest collection of torus obstructions in the world to date. 1 Introduction A graph is said to be embedded on a surface if it is drawn there with no crossing edges. A graph is planar if it can be drawn on the sphere, and is toroidal if it can be drawn on the torus (a sphere with one handle). The genus of a planar graph is zero, and a nonplanar graph which ...
KuratowskiType Theorems for Average Genus
 J. Combinatorial Theory B
, 1992
"... Graphs of small average genus are characterized. In particular, a Kuratowskitype theorem is obtained: except for finitely many graphs, a cutedgefree graph has average genus less than or equal to 1 if and only if it is a necklace. We provide a complete list of those exceptions. A Kuratowskitype th ..."
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Cited by 6 (3 self)
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Graphs of small average genus are characterized. In particular, a Kuratowskitype theorem is obtained: except for finitely many graphs, a cutedgefree graph has average genus less than or equal to 1 if and only if it is a necklace. We provide a complete list of those exceptions. A Kuratowskitype theorem for graphs of maximum genus 1 is also given. Some of the methods used in obtaining these results involve variations of a classical result of Whitney. April 27, 1992 1 Supported by Engineering Excellence Award from Texas A&M University, and by the National Science Foundation under Grant CCR9110824. 2 Supported by ONR Contract N00014850768. CUCS01992 1 Introduction By the average genus of a graph G, we mean the average value of the genus of the imbedding surface, taken over all orientable imbeddings of G. This value is evidently a rational number, and it is clearly an invariant of the homeomorphism type of a graph. The average genus of individual graphs is in the GrossFurst ...
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...
A Faster Algorithm for Torus Embedding
, 2004
"... 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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Cited by 1 (0 self)
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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.
Simplified O(n) Planarity Algorithms
, 2001
"... A graph is planar if it can be drawn on the plane with vertices at unique locations and no edge intersections except at the vertex endpoints. Due to the wealth of interest from the computer science community, there are a number of remarkable but complex O(n) planar embedding algorithms. This pape ..."
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A graph is planar if it can be drawn on the plane with vertices at unique locations and no edge intersections except at the vertex endpoints. Due to the wealth of interest from the computer science community, there are a number of remarkable but complex O(n) planar embedding algorithms. This paper presents an O(n) planar embedding algorithm that avoids a number of the complexities of prior approaches (an early version of this work was presented at the January 1999 Symposium on Discrete Algorithms). In July 1999
Order, Chaos and Algorithms
, 2006
"... In these notes I will look at some very exciting parts of mathematics related to order in apparent chaos, and surprising connections with applications in algorithm design. I assume that the reader has a basic background with the ..."
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In these notes I will look at some very exciting parts of mathematics related to order in apparent chaos, and surprising connections with applications in algorithm design. I assume that the reader has a basic background with the