International audienceThe problem of when a given digraph contains a subdivision of a fixed digraph F is considered. Bang-Jensen et al. [2] laid out foundations for approaching this problem from the algorith-mic 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
