Component order connectivity in directed graphs

A directed graph D is semicomplete if for every pair x, y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D= (V, A) and a pair of natural numbers k and ℓ, we are to decide whether there is a subset X of V of size k such that the largest strongly connected component in D- X has at most ℓ vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for ℓ= 1. We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, ℓ, ℓ+ k and n- ℓ. In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O^∗(2 ^16k) but not in time O^∗(2 ^o(k)) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O^∗(2 ^16k) implies the upper bound O^∗(2 ^16(n-ℓ)) for the parameter n- ℓ. We complement the latter by showing that there is no algorithm of time complexity O^∗(2 ^o(n-ℓ)) unless ETH fails. Finally, we improve (in dependency on ℓ) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter ℓ+ k on general digraphs from O^∗(2 ^{O(kℓlog(kℓ))}) to O^∗(2 ^{O(klog(kℓ))}). Note that Drange, Dregi and van ’t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O^∗(2 ^o(klogℓ)) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O^∗(2 ^o(klogk)).