Abstract
The map of relations between the different directed width measures in general still has some blank spots. In this work we fill in many of these open relations for the restricted class of semicomplete digraphs. To do this we show the parametrical equivalence between directed path-width, directed tree-width, DAG-width, and Kelly-width. Moreover, we show that directed (linear) clique-width is upper bounded in a function of directed tree-width on semicomplete digraphs. In general the algorithmic use of many of these parameters is fairly restricted. On semicomplete digraphs our results allow to combine the nice computability of directed tree-width with the algorithmic power of directed clique-width. Moreover, our results have the effect that on semicomplete digraphs every digraph problem, which is describable in monadic second-order logic on quantification over vertices and vertex sets, is fixed parameter tractable for all of the shown width measures if a decomposition of bounded width is given.
This work is partly funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 388221852.
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A remarkable difference to the undirected tree-width [23] is that the sets \(W_r\) have to be disjoint and non-empty.
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Gurski, F., Komander, D., Rehs, C., Wiederrecht, S. (2021). Directed Width Parameters on Semicomplete Digraphs. In: Du, DZ., Du, D., Wu, C., Xu, D. (eds) Combinatorial Optimization and Applications. COCOA 2021. Lecture Notes in Computer Science(), vol 13135. Springer, Cham. https://doi.org/10.1007/978-3-030-92681-6_48
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