Document # Component Order Connectivity in Directed Graphs

### Authors Jørgen Bang-Jensen , Eduard Eiben, Gregory Gutin, Magnus Wahlström, Anders Yeo ## File

LIPIcs.IPEC.2020.2.pdf
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## Cite As

Jørgen Bang-Jensen, Eduard Eiben, Gregory Gutin, Magnus Wahlström, and Anders Yeo. Component Order Connectivity in Directed Graphs. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 2:1-2:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.IPEC.2020.2

## 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 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 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(k𝓁 log (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(klog 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(klog k)).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Parameterized complexity and exact algorithms
##### Keywords
• Parameterized Algorithms
• component order connectivity
• directed graphs
• semicomplete digraphs

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## References

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