 
Summary: Directed TreeWidth Examples
Isolde Adler
Mathematisches Institut,
Eckerstr. 1,
D79102 Freiburg
isolde.adler@math.unifreiburg.de
5th June 2005
Abstract
In [1] Johnson, Robertson, Seymour and Thomas define the notion
of directed treewidth 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 robbermonotonely. Again a
negative answer is given. We also show that contraction of butterfly edges
can increase directed treewidth.
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 Znormal, if there is no
directed walk in D\Z with first and last vertex in S that uses a vertex from D\
