Computing the degreewidth of a digraph is hard

TL;DR

This paper proves that computing the degreewidth of a digraph is NP-hard, settling an open question and analyzing related parameters in graph theory.

Contribution

It establishes NP-hardness for determining degreewidth at most 1 in oriented graphs, answering an open problem and exploring parameter relationships.

Findings

Proves NP-hardness of computing degreewidth for oriented graphs.

Complements the hardness result with a discussion on backedge graph parameters.

Settles the last open case for oriented graphs regarding degreewidth.

Abstract

Given a digraph, an ordering of its vertices defines a backedge graph, namely the undirected graph whose edges correspond to the arcs pointing backwards with respect to the order. The degreewidth of a digraph is the minimum over all ordering of the maximum degree of the backedge graph. We answer an open question by Keeney and Lokshtanov [WG 2024], proving that it is \NP-hard to determine whether an oriented graph has degreewidth at most $1$, which settles the last open case for oriented graphs. We complement this result with a general discussion on parameters defined using backedge graphs and their relations to classical parameters.