# Comparing Linear Width Parameters for Directed Graphs

## TL;DR

This paper introduces and compares several linear width parameters for directed graphs, establishing their relationships, equivalences under certain restrictions, and analyzing their computational complexity.

## Contribution

It defines new linear width parameters for directed graphs and compares them with existing parameters, establishing equivalences and complexity results.

## Key findings

- Several directed linear width parameters are equivalent.
- Computing these parameters is hard in general.
- Characterizations for graphs with small width are provided.

## Abstract

In this paper we introduce the linear clique-width, linear NLC-width, neighbourhood-width, and linear rank-width for directed graphs. We compare these parameters with each other as well as with the previously defined parameters directed path-width and directed cut-width. It turns out that the parameters directed linear clique-width, directed linear NLC-width, directed neighbourhood-width, and directed linear rank-width are equivalent in that sense, that all of these parameters can be upper bounded by each of the others. For the restriction to digraphs of bounded vertex degree directed path-width and directed cut-width are equivalent. Further for the restriction to semicomplete digraphs of bounded vertex degree all six mentioned width parameters are equivalent. We also show close relations of the measures to their undirected versions of the underlying undirected graphs, which allow us to show the hardness of computing the considered linear width parameters for directed graphs. Further we give first characterizations for directed graphs defined by parameters of small width.