  A \emph{complete geometric graph} consists of a set $P$ of $n$ points in the
plane, in general position, and all segments (edges) connecting them. It is a
well known question of Bose, Hurtado, Rivera-Campo, and Wood, whether there
exists a positive constant $c<1$, such that every complete geometric graph on
$n$ points can be partitioned into at most $cn$ plane graphs (that is,
noncrossing subgraphs). We answer this question in the affirmative in the
special case where the underlying point set $P$ is \emph{dense}, which means
that the ratio between the maximum and the minimum distances in $P$ is of the
order of $\Theta(\sqrt{n})$.
