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Generically Dependent Continuant
RDF description of Point sets that minimize (≤k)-edges, 3-decomposable drawings, and the rectilinear crossing number of K30 - http://orbis.uaslp.mx/vivo/individual/scopus2835
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There are two properties shared by all known crossing-minimizing geometric drawings of Kn, for n a multiple of 3. First, the underlying n-point set of these drawings minimizes the number of (≤k)-edges, that means, has exactly 3(k 22)(≤k)-edges, for all 0≤k<n3. Second, all such drawings have the n points divided into three groups of equal size; this last property is captured under the concept of 3-decomposability. In this paper we show that these properties are closely related: every n-point set with exactly 3(k 22)(≤k)-edges for all 0≤k<n3, is 3-decomposable. The converse, however, is easy to see that it is false. As an application, we prove that the rectilinear crossing number of K30 is 9726. © 2011 Elsevier B.V. All rights reserved.
3-decomposability; k-edges; Rectilinear crossing number
There are two properties shared by all known crossing-minimizing geometric drawings of Kn, for n a multiple of 3. First, the underlying n-point set of these drawings minimizes the number of (≤k)-edges, that means, has exactly 3(k%2b22)(≤k)-edges, for all 0≤k<n3. Second, all such drawings have the n points divided into three groups of equal size; this last property is captured under the concept of 3-decomposability. In this paper we show that these properties are closely related: every n-point set with exactly 3(k%2b22)(≤k)-edges for all 0≤k<n3, is 3-decomposable. The converse, however, is easy to see that it is false. As an application, we prove that the rectilinear crossing number of K30 is 9726. © 2011 Elsevier B.V. All rights reserved.
Point sets that minimize (≤k)-edges, 3-decomposable drawings, and the rectilinear crossing number of K30
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