  Motivated by applications in graph drawing and information visualization, we
examine the planar split thickness of a graph, that is, the smallest $k$ such
that the graph is $k$-splittable into a planar graph. A $k$-split operation
substitutes a vertex $v$ by at most $k$ new vertices such that each neighbor of
$v$ is connected to at least one of the new vertices.
  We first examine the planar split thickness of complete graphs, complete
bipartite graphs, multipartite graphs, bounded degree graphs, and genus-1
graphs. We then prove that it is NP-hard to recognize graphs that are
$2$-splittable into a planar graph, and show that one can approximate the
planar split thickness of a graph within a constant factor. If the treewidth is
bounded, then we can even verify $k$-splittability in linear time, for a
constant $k$.
