Go to DBLP homepageComponent Order Connectivity in Directed Graphs
Abstract: 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 \(\ell \), 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 \(\ell \) vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for \(\ell =1.\) We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: \(k, \ell ,\ell +k\) and \(n-\ell \). 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 \(O^*(2^{16(n-\ell )})\) for the parameter \(n-\ell .\) We complement the latter by showing that there is no algorithm of time complexity \(O^*(2^{o({n-\ell })})\) unless ETH fails. Finally, we improve (in dependency on \(\ell \)) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter \(\ell +k\) on general digraphs from \(O^*(2^{O(k\ell \log (k\ell ))})\) to \(O^*(2^{O(k\log (k\ell ))}).\) 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 \(O^*(2^{o(k\log \ell )})\) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity \(O^*(2^{o(k\log k)}).\)
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