Directed branch-width: A directed analogue of tree-width

TL;DR

This paper introduces directed branch-width, a new measure for directed graphs, which generalizes branch-width and relates closely to underlying undirected tree-width, enabling extended algorithmic applications.

Contribution

It defines directed branch-width, explores its properties, and demonstrates its usefulness in extending algorithmic results to a broader class of directed graphs.

Findings

Directed branch-width cannot bound underlying tree-width.

It differs from branch-width only at sources and sinks.

Enables extending algorithms from bounded tree-width to bounded directed branch-width.

Abstract

Gurski and Wanke showed that a graph class C has bounded tree-width if and only if its associated class of directed line graphs has bounded clique-width. Inevitably -- asking whether this relationship lifts to directed graphs -- we introduce a new digraph width measure: we obtain it by investigating digraphs whose directed line graphs have bounded cliquewidth. Thus, to generalize Gurski and Wanke's aforementioned result, we introduce a natural generalization of branch-width to digraphs and we name it accordingly. Directed branch-width is a genuinely directed width-measure insofar as it cannot be used to bound the value of the underlying undirected tree-width. Despite this, the two measures are still closely related: the directed branch-width of a digraph D can differ from the branch-width of its underlying undirected graph only at sources and sinks. This relationship allows us to extend a range of algorithmic results from directed graphs with bounded underlying treewidth to the strictly larger class of digraphs having bounded directed branch-width.