Digraphs of bounded width

Abstract

Structural parameters for undirected graphs such as the path-width or tree-width of graphs have played a crucial role in developing a structure theory for graphs based on the minor relation and they have also found many algorithmic applications. Starting in the late 1990s, several ideas for generalizing this theory to digraphs have appeared. Broadly, for the purpose of this chapter, we distinguish these approaches into three categories: tree-width inspired, rank-width inspired and density based. The tree-width inspired approaches are based on the idea of generalizing the concept of undirected tree-width (or path-width) to digraphs. The various attempts, which we will discuss below, all have the goal of generalizing some natural property or some natural characterization of tree-width of undirected graphs to digraphs. In the same way as tree-width can be seen as a global connectivity measure for undirected graphs, the various versions of a directed analogue of tree-width measure global connectivity in digraphs. However, on digraphs, connectivity can be measured in many different natural ways. It turns out that equivalent characterizations of tree-width on undirected graphs yield different concepts on digraphs, with different properties, advantages and disadvantages.