Component Order Connectivity in Directed Graphs
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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 strong connectivity 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 parametered 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(klog k)).
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