Finding a subdivision of a prescribed digraph of order 4

The 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 algorithmic pointof view. In this paper we give further support to several open conjectures and speculations about algorithmiccomplexity of finding F-subdivisions. In particular, up to 5 exceptions, we completely classify forwhich 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 problemin digraphs, some of the polynomial-time solvable cases involve relatively complicated algorithms

Finding a subdivision of a prescribed digraph of order 4

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

On disjoint directed cycles with prescribed minimum lengths

In this paper, we show that the k-Linkage problem is polynomial-time solvable for digraphs with circumference at most 2. We also show that the directed cycles of length at least 3 have the Erdős-Pósa Property : for every n, there exists an integer t_n such that for every digraph D, either D contains n disjoint directed cycles of length at least 3, or there is a set T of t_n vertices that meets every directed cycle of length at least 3. From these two results, we deduce that if F is the disjoint union of directed cycles of length at most 3, then one can decide in polynomial time if a digraph contains a subdivision of F