Abstract
In this paper we consider the digraph width measures directed feedback vertex set number, cycle rank, DAG-depth, DAG-width and Kelly-width. While the minimization problem for these width measures is generally NP-hard, we prove that it is computable in linear time for all these parameters, except for Kelly-width, when restricted to directed co-graphs. As an important combinatorial tool, we show how these measures can be computed for the disjoint union, series composition, and order composition of two directed graphs, which further leads to some similarities and a good comparison between the width measures. This generalizes and expands our former results for computing directed path-width and directed tree-width of directed co-graphs.
The paper is eligible for the best student paper award.
C. Rehs—The work of the third author was supported by the German Research Association (DFG) grant GU 970/7-1.
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- 1.
There are also further directed tree-width definitions such as allowing empty sets \(W_r\) in [23], using sets \(W_r\) of size one only for the leaves of T in [29] and using strong components within (dtw-2) in [13, Chap. 6]. Further in works of Courcelle et al. [9,10,11] the directed tree-width of a digraph G is defined by the tree-width of the underlying undirected graph. One reason for this could be the algorithmic advantages of the undirected tree-width.
- 2.
A remarkable difference to the undirected tree-width from [30] is that the sets \(W_r\) have to be disjoint and non-empty.
- 3.
XP is the class of all parameterized problems that can be solved in a certain time, see [14] for a definition.
- 4.
The proofs of the results marked with a \(\bigstar \) are omitted due to space restrictions.
- 5.
In the undirected case, reachable fragments coincide with connected components.
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Gurski, F., Komander, D., Rehs, C. (2019). Computing Digraph Width Measures on Directed Co-graphs. In: Gąsieniec, L., Jansson, J., Levcopoulos, C. (eds) Fundamentals of Computation Theory. FCT 2019. Lecture Notes in Computer Science(), vol 11651. Springer, Cham. https://doi.org/10.1007/978-3-030-25027-0_20
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