Directed Tree-Width Examples Isolde Adler Summary: Directed Tree-Width Examples Isolde Adler Mathematisches Institut, Eckerstr. 1, D-79102 Freiburg isolde.adler@math.uni-freiburg.de 5th June 2005 Abstract 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\ Collections: Computer Technologies and Information Sciences