  The splitting number of a graph $G=(V,E)$ is the minimum number of vertex
splits required to turn $G$ into a planar graph, where a vertex split removes a
vertex $v \in V$, introduces two new vertices $v_1, v_2$, and distributes the
edges formerly incident to $v$ among its two split copies $v_1, v_2$. The
splitting number problem is known to be NP-complete. In this paper we shift
focus to the splitting number of graph drawings in $\mathbb R^2$, where the new
vertices resulting from vertex splits can be re-embedded into the existing
drawing of the remaining graph. We first provide a non-uniform fixed-parameter
tractable (FPT) algorithm for the splitting number problem (without drawings).
Then we show the NP-completeness of the splitting number problem for graph
drawings, even for its two subproblems of (1) selecting a minimum subset of
vertices to split and (2) for re-embedding a minimum number of copies of a
given set of vertices. For the latter problem we present an FPT algorithm
parameterized by the number of vertex splits. This algorithm reduces to a
bounded outerplanarity case and uses an intricate dynamic program on a
sphere-cut decomposition.
