Finding subdigraphs in digraphs of bounded directed treewidth

TL;DR

This paper investigates the complexity of finding specific subdigraphs in directed graphs with bounded directed treewidth, revealing that only directed paths and stars can be efficiently identified.

Contribution

It demonstrates that, beyond directed paths, only stars can be efficiently found in digraphs of bounded directed treewidth, providing new positive and negative results.

Findings

Directed paths and stars are the only subdigraphs efficiently found in bounded directed treewidth digraphs.

The paper generalizes existing results on subdigraph detection.

It offers new insights and directions for future research in digraph algorithms.

Abstract

It is well known that directed treewidth does not enjoy the nice algorithmic properties of its undirected counterpart. There exist, however, some positive results that, essentially, present XP algorithms for the problem of finding, in a given digraph $D$, a subdigraph isomorphic to a digraph $H$ that can be formed by the union of $k$ directed paths (with some extra properties), parameterized by $k$ and the directed treewidth of $D$. Our motivation is to tackle the following question: Are there subdigraphs, other than the directed paths, that can be found efficiently in digraphs of bounded directed treewidth? In a nutshell, the main message of this article is that, other than the directed paths, the only digraphs that seem to behave well with respect to directed treewidth are the stars. For this, we present a number of positive and negative results, generalizing several results in the literature, as well as some directions for further research.