The $st$-Planar Edge Completion Problem is Fixed-Parameter Tractable

The problem of deciding whether a biconnected planar digraph $G=(V,E)$ can be augmented to become an $st$-planar graph by adding a set of oriented edges $E^{\mathrm{\prime }}\subseteq V\times V$ is known to be NP-complete. We show that the problem is fixed-parameter tractable when parameterized by the size of the set $E^{\mathrm{\prime }}$.