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Computing the degreewidth of a digraph is hard

Pierre Aboulker, Nacim Oijid, Robin Petit, Mathis Rocton, Christopher-Lloyd Simon

TL;DR

It is proved 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.

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.

Computing the degreewidth of a digraph is hard

TL;DR

It is proved 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.

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 , 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.
Paper Structure (12 sections, 8 theorems, 10 equations, 2 figures)

This paper contains 12 sections, 8 theorems, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Computing degreewidth is -complete
  3. Preliminary lemmas
  4. Proof of \ref{['thm:k-Max-Degree is NP-complete for k >= 1']}
  5. A Unifying Viewpoint
  6. Generalisation from graphs to digraphs via symmetric digraphs
  7. Digraph parameters
  8. Dichromatic Number and Directed Clique Number
  9. Directed Vertex Cover is Feedback Vertex Set
  10. Back to the Degreewidth
  11. The decision problems associated with a parameter
  12. Optimal Linear Arrangement, Cutwidth and Bandwidth

Key Result

theorem 2

For every integer $k \geq 1$, the problem $k$-Degreewidth is -complete.

Figures (2)

  • Figure 1: Transfer digraph.
  • Figure 2: Representation of a clause gadget.

Theorems & Definitions (17)

  • definition 1: $k$-Degreewidth
  • theorem 2: $1$- is -complete
  • lemma 3
  • proof
  • remark 4
  • theorem 4: $1$- is -complete
  • claim 5
  • lemma 6
  • proof
  • corollary 7
  • ...and 7 more