Directed tree-width examples

In [1] Johnson, Robertson, Seymour and Thomas define the notion of directed tree-width dtw(D) of a directed graph D. They ask whether dtw(D) ≥ k− 1 implies that D has a haven of order k. A negative answer is given. Furthermore they define a generalisation of the robber and cops game of [3] to digraphs. They ask whether it is true that if k cops can catch the robber on a digraph, then they can do so robber-monotonely. Again a negative answer is given. We also show that contraction of butterfly edges can increase directed tree-width. 1 Definitions All graphs and digraphs are finite. We recall the most important definitions from [1]1. Let D be a digraph. A set S ⊆ D \ Z is Z-normal, if there is no directed walk in D\Z with first and last vertex in S that uses a vertex from D\ (Z∪S). Consider the digraph D depicted in Figure 1, where V (D) = {0, 1, 2, 3} and E(D) = {(0, 1), (1, 0), (0, 2), (2, 0), (0, 3), (3, 0), (1, 3), (3, 1)}. Then {1, 2} is {0}-normal, but {2, 3} is not {0}-normal. Note that there is an addendum [2] to [1]. The addendum concerns the algorithmic aspects of diredted tree-width which are not investigateed in this paper.