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