DAG-width is PSPACE-complete

Berwanger et al. show in [2] that for every graph G of size n and DAG-width k there is a DAG decomposition of width k and size nO(k). They also establish a polynomial time algorithm for deciding whether the DAG-width of a graph is at most a fixed number k. However, if the DAG-width of the graphs is not bounded, such algorithms 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 most other generalisations of tree width similar to DAG-width. In this paper we show that there is an infinite class of graphs such that every DAG decomposition of optimal width has size super-polynomial in n and, moreover, there is no polynomial size DAG decomposition of width at most k+k1−ε for every ε∈(0,1). 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 value is PSpace-complete.