Are there any good digraph width measures?

TL;DR

This paper investigates various measures of digraph width, demonstrating that any measure with desirable algorithmic and structural properties must closely relate to the underlying undirected treewidth, and introduces directed topological minors as a minimal notion of minors for digraphs.

Contribution

It proves that all reasonable digraph width measures with good algorithmic and structural properties are essentially equivalent to undirected treewidth, and introduces directed topological minors.

Findings

Any useful digraph width measure is close to the underlying undirected treewidth.

Directed topological minors are the weakest useful notion of minors for digraphs.

Most proposed digraph width measures do not differ significantly from undirected treewidth.

Abstract

Several different measures for digraph width have appeared in the last few years. However, none of them shares all the "nice" properties of treewidth: First, being \emph{algorithmically useful} i.e. admitting polynomial-time algorithms for all $\MS1$-definable problems on digraphs of bounded width. And, second, having nice \emph{structural properties} i.e. being monotone under taking subdigraphs and some form of arc contractions. As for the former, (undirected) $\MS1$ seems to be the least common denominator of all reasonably expressive logical languages on digraphs that can speak about the edge/arc relation on the vertex set.The latter property is a necessary condition for a width measure to be characterizable by some version of the cops-and-robber game characterizing the ordinary treewidth. Our main result is that \emph{any reasonable} algorithmically useful and structurally nice digraph measure cannot be substantially different from the treewidth of the underlying undirected graph. Moreover, we introduce \emph{directed topological minors} and argue that they are the weakest useful notion of minors for digraphs.