Abstract : 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 NPcomplete.
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.