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An improved neural network model for the 2page crossing number problem
 IEEE Trans. on Neural Networks
"... This item was submitted to Loughborough’s Institutional Repository by the ..."
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This item was submitted to Loughborough’s Institutional Repository by the
unknown title
, 2007
"... The crossing number of Kn is known for n � 10. We develop several simple counting properties that we shall exploit in showing by computer that cr(K11) = 100, which implies that cr(K12) = 150. We also determine the numbers of nonisomorphic optimal drawings of K9 and K10. 1 ..."
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The crossing number of Kn is known for n � 10. We develop several simple counting properties that we shall exploit in showing by computer that cr(K11) = 100, which implies that cr(K12) = 150. We also determine the numbers of nonisomorphic optimal drawings of K9 and K10. 1
PQtrees and maximal planarization  An approach to skewness
, 1998
"... This article formed the basis for our search for a new approach to a skewness heuristic. Given a non planar graph G, then a subgraph G ..."
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This article formed the basis for our search for a new approach to a skewness heuristic. Given a non planar graph G, then a subgraph G
Crossings, colorings, and cliques
, 2009
"... Albertson conjectured that if graph G has chromatic number r, then the crossing number of G is at least that of the complete graph Kr. This conjecture in the case r = 5 is equivalent to the four color theorem. It was verified for r = 6 by Oporowski and Zhao. In this paper, we prove the conjecture fo ..."
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Albertson conjectured that if graph G has chromatic number r, then the crossing number of G is at least that of the complete graph Kr. This conjecture in the case r = 5 is equivalent to the four color theorem. It was verified for r = 6 by Oporowski and Zhao. In this paper, we prove the conjecture for 7 ≤ r ≤ 12 using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on the number of edges in critical graphs, together with lower bounds by Pach et al. on the crossing number of graphs in terms of the number of edges and vertices.
Crossings, colorings, and cliques
, 2009
"... Albertson conjectured that if graph G has chromatic number r, then the crossing number of G is at least that of the complete graph Kr. This conjecture in the case r = 5 is equivalent to the four color theorem. It was verified for r = 6 by Oporowski and Zhao. In this paper, we prove the conjecture fo ..."
Abstract
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Albertson conjectured that if graph G has chromatic number r, then the crossing number of G is at least that of the complete graph Kr. This conjecture in the case r = 5 is equivalent to the four color theorem. It was verified for r = 6 by Oporowski and Zhao. In this paper, we prove the conjecture for 7 ≤ r ≤ 12 using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on the number of edges in critical graphs, together with lower bounds by Pach et al. on the crossing number of graphs in terms of the number of edges and vertices.
The Graph Crossing Number and its Variants: A Survey
"... The crossing number is a popular tool in graph drawing and visualization, but there is not really just one crossing number; there is a large family of crossing number notions of which the crossing number is the best known. We survey the rich variety of crossing number variants that have been introdu ..."
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The crossing number is a popular tool in graph drawing and visualization, but there is not really just one crossing number; there is a large family of crossing number notions of which the crossing number is the best known. We survey the rich variety of crossing number variants that have been introduced in the literature for purposes that range from studying the theoretical underpinnings of the crossing number to crossing minimization for visualization problems. 1 So, Which Crossing Number is it? The crossing number, cr(G), of a graph G is the smallest number of crossings required in any drawing of G. Or is it? According to a popular introductory textbook on combinatorics [320, page 40] the crossing number of a graph is “the minimum number of pairs of crossing edges in a depiction of G”. So, which one is it? Is there even a difference? To start with the second question, the easy answer is: yes, obviously there is a difference, the difference between counting all crossings and counting pairs of edges that cross. But maybe these different ways of counting don’t make a difference and always come out
Drawing Alternative Splicing Graphs
"... Alternative splicing of a single premRNA can give rise to different mRNA transcripts. Alternative splicing of premessenger RNA is an important layer of gene expression regulation in eukaryotic cell. Consequently, alternative splicing is an important mechanism for generating protein diversity from a ..."
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Alternative splicing of a single premRNA can give rise to different mRNA transcripts. Alternative splicing of premessenger RNA is an important layer of gene expression regulation in eukaryotic cell. Consequently, alternative splicing is an important mechanism for generating protein diversity from a single gene. Due to all these important aspects of genomic alternative splicing mechanism that produces different mRNA’s, biologists require means of analyzing and visualizations of the alternative splicing forms. In this paper, we consider ways of drawing a given alternative splicing graph on the plane. Given an alternative splicing graph G = (V, E), here in this paper we show that computing the crossing number of a given upper drawing ASG can be done in O(E  log E) time. The optimal (fewest crossing) drawing of alternative splicing graphs is related to the long known difficult problem of crossing number. Here we propose an efficient branchandbound recursive algorithm with heuristic searching strategy. Some experimental results are also discussed in the paper.