  The problem of when a given digraph contains a subdivision of a fixed digraph
$F$ is considered. Bang-Jensen et al. laid out foundations for approaching this
problem from the algorithmic point of view. In this paper we give further
support to several open conjectures and speculations about algorithmic
complexity of finding $F$-subdivisions. In particular, up to 5 exceptions, we
completely classify for which 4-vertex digraphs $F$, the $F$-subdivision
problem is polynomial-time solvable and for which it is NP-complete. While all
NP-hardness proofs are made by reduction from some version of the 2-linkage
problem in digraphs, some of the polynomial-time solvable cases involve
relatively complicated algorithms.
