We present the first thorough theoretical analysis of the  Transitivity Editing  problem on digraphs. Herein, the task is to perform a minimum number of arc insertions or deletions in order to make a given digraph transitive. This problem has recently been identified as important for the detection of hierarchical structure in molecular characteristics of disease. Mixing up  Transitivity Editing  with the companion problems on undirected graphs, it has been erroneously claimed to be NP-hard. We correct this error by presenting a first proof of NP-hardness, which also extends to the restricted cases where the input digraph is acyclic or has maximum degree four. Moreover, we improve previous fixed-parameter algorithms, now achieving a running time of  O  (2.57  k  +  n  3) for an  n  -vertex digraph if  k  arc modifications are sufficient to make it transitive. In particular, providing an  O  ( k  2)-vertex problem kernel, we positively answer an open question from the literature. In case of digraphs with maximum degree  d  , an  O  ( k  · d  )-vertex problem kernel can be shown. We also demonstrate that if the input digraph contains no "diamond structure", then one can always find an optimal solution that exclusively performs arc deletions. Most of our results (including NP-hardness) can be transferred to the  Transitivity Deletion  problem, where only arc deletions are allowed.
['Mathias Weller', 'Christian Komusiewicz', 'Rolf Niedermeier', 'Johannes Uhlmann']
On Making Directed Graphs Transitive