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**1 - 2**of**2**### DAG-width is PSPACE-complete

, 2014

"... Berwanger et al. show in [BDH+12] that for every graph G of size n and DAG-width k there is a DAG decomposition of width k of size nO(k). This gives a poly-nomial time algorithm for determining the DAG-width of a graph for any fixed k. However, if the DAG-width of the graphs from a class is not boun ..."

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Berwanger et al. show in [BDH+12] that for every graph G of size n and DAG-width k there is a DAG decomposition of width k of size nO(k). This gives a poly-nomial time algorithm for determining the DAG-width of a graph for any fixed k. However, if the DAG-width of the graphs from a class is not bounded, such algo-rithms become exponential. This raises the question whether we can always find a DAG decomposition of size polynomial in n as it is the case for tree width and all generalisations of tree width similar to DAG-width. We show that there is an infinite class of graphs such that every DAG decompo-sition of optimal width has size super-polynomial in n and, moreover, there is no polynomial size DAG decomposition which would approximate an optimal decom-position up to an additive constant. In the second part we use our construction to prove that deciding whether the DAG-width of a given graph is at most a given constant is PSpace-complete. 1

### Graph Searching Games and Width Measures for Directed Graphs

"... In cops and robber games a number of cops tries to capture a robber in a graph. A variant of these games on undirected graphs characterises tree width by the least number of cops needed to win. We consider cops and robber games on digraphs and width measures (such as DAG-width, directed tree width o ..."

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In cops and robber games a number of cops tries to capture a robber in a graph. A variant of these games on undirected graphs characterises tree width by the least number of cops needed to win. We consider cops and robber games on digraphs and width measures (such as DAG-width, directed tree width or D-width) corresponding to them. All of them generalise tree width and the game characterising it. For the DAG-width game we prove that the problem to decide the minimal number of cops required to capture the robber (which is the same as deciding DAG-width), is PSPACE-complete, in contrast to most other similar games. We also show that the cop-monotonicity cost for directed tree width games cannot be bounded by any function. As a consequence, D-width is not bounded in directed tree width, refuting a conjecture by Safari. A large number of directed width measures generalising tree width has been proposed in the literature. However, only very little was known about the relation between them, in particular about whether classes of digraphs of bounded width in one measure have bounded width in another. In this paper we establish an almost complete order among the most prominent width measures with respect to mutual boundedness.