  A topological graph is a graph drawn in the plane. A topological graph is
$k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the
problem of partitioning the edges of a $k$-plane graph such that each partite
set forms a graph with a simpler structure. While this problem has been studied
for $k=1$, we focus on optimal $2$-plane and $3$-plane graphs, which are
$2$-plane and $3$-plane graphs with maximum density. We prove the following
results. (i) It is not possible to partition the edges of a simple optimal
$2$-plane graph into a $1$-plane graph and a forest, while (ii) an edge
partition formed by a $1$-plane graph and two plane forests always exists and
can be computed in linear time. (iii) We describe efficient algorithms to
partition the edges of a simple optimal $2$-plane graph into a $1$-plane graph
and a plane graph with maximum vertex degree $12$, or with maximum vertex
degree $8$ if the optimal $2$-plane graph is such that its crossing-free edges
form a graph with no separating triangles. (iv) We exhibit an infinite family
of simple optimal $2$-plane graphs such that in any edge partition composed of
a $1$-plane graph and a plane graph, the plane graph has maximum vertex degree
at least $6$ and the $1$-plane graph has maximum vertex degree at least $12$.
(v) We show that every optimal $3$-plane graph whose crossing-free edges form a
biconnected graph can be decomposed, in linear time, into a $2$-plane graph and
two plane forests.
