Component Order Connectivity in Directed Graphs - Bang-Jensen, Jørgen; Eiben, Eduard; Gutin, Gregory; Wahlström, Magnus; Yeo, Anders (Algorithmica, 2022)
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 16 k ) but not in time O ∗ ( 2 o ( k ) ) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O ∗ ( 2 16 k ) 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 ( k ℓ log ( k ℓ ) ) ) to O ∗ ( 2 O ( k log ( k ℓ ) ) ) . 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 ( k log ℓ ) ) 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 ( k log k ) ) .