How to compute digraph width measures on directed co-graphs

Abstract

In this paper we consider the digraph width measures directed path-width, directed tree-width, directed feedback vertex set number, directed feedback arc set number, cycle rank, DAG-depth, DAG-width and Kelly-width of recursively defined digraphs. While the minimization problem for these width measures is generally NP-hard, we prove that it is computable in linear time for all these parameters, except for Kelly-width, when restricted to directed co-graphs. As an important combinatorial tool, we show how these measures can be computed for the disjoint union, order composition, directed union, and series composition of two directed graphs, which further leads to some similarities. Although it is often not possible to compare them in general, we achieved a good comparison between the width measures within this framework. The equality of directed path-width and directed tree-width on directed co-graphs generalizes the known results for undirected co-graphs of Bodlaender and Möhring.