D-width: A more natural measure for directed tree width

Abstract

Due to extensive research on tree-width for undirected graphs and due to its many applications in various fields it has been a natural desire for many years to generalize the idea of tree decomposition to directed graphs, but since many parameters in tree-width behave very differently in the world of digraphs, the generalization is still in its preliminary steps. In this paper, after surveying the main work that has been done on this topic, we propose a new simple definition for directed tree-width and prove a special case of the min-max theorem (duality theorem) relating haven order, bramble number, and tree-width on digraphs. We also compare our definition with previous definitions and study the behavior of some tree-width like parameters such as brambles and havens on digraphs. © Springer-Verlag Berlin Heidelberg 2005.