Point sets that minimize -edges, 3-decomposable drawings, and the rectilinear crossing number of

There are two properties shared by all known crossing-minimizing geometric drawings of K n , 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 + 2 2 ) ( ≤ k ) -edges, for all 0 ≤ k < n / 3 . 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 + 2 2 ) ( ≤ k ) -edges for all 0 ≤ k < n / 3 , 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 K 30 is 9726.