Directed tree-width examples

In [T. Johnson, N. Robertson, P.D. Seymour, R. Thomas, Directed tree-width, J. Combin. Theory Ser. B 82 (2001) 138–154] 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 [P.D. Seymour, R. Thomas, Graph searching and a min–max theorem for tree-width, J. Combin. Theory Ser. B 58 (1993) 22–33] 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.