  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 has exactly $3\binom{k+2}{2}$ $(\le k)$-edges, for all $0\le
k < n/3$. 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 tightly
related: every $n$-point set with exactly $3\binom{k+2}{2}$ $(\le k)$-edges for
all $0\le k < n/3$, is 3-decomposable. As an application, we prove that the
rectilinear crossing number of $K_{30}$ is 9726.
