Computing the degreewidth of a digraph is hard

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.

💡 Research Summary

The paper introduces and studies the computational complexity of a newly defined directed‑graph parameter called degree‑width. For a digraph D and a total ordering ≺ of its vertices, the back‑edge graph D≺ is obtained by turning every arc that points “backwards” with respect to ≺ into an undirected edge. The degree‑width →Δ(D) is the minimum possible maximum degree of D≺ over all vertex orderings. This definition naturally extends the classical maximum‑degree parameter from undirected graphs to directed graphs via back‑edge graphs.

The main technical contribution is a proof that, for every integer k ≥ 1, the decision problem k‑Degree‑Width (i.e., “does a given digraph have degree‑width at most k?”) is NP‑complete. The case k = 1 had been the only remaining open case from earlier work, so this result settles the full complexity landscape for degree‑width.

To establish NP‑hardness, the authors reduce from 3‑SAT. For each clause of a 3‑SAT formula they construct a gadget D_j that contains three literal vertices ℓ_i^j and their “exit” counterparts eℓ_i^j, a collection of transfer digraphs T_{2k} (p‑disjoint length‑2 paths from a source to a sink) and T_{2k+1}, and a directed cycle C_j that links the eℓ_i^j vertices with auxiliary vertices c_i^j. Transfer digraphs of size 2k enforce that any feedback‑arc set (FAS) that disconnects the source from the sink must contain at least k arcs incident to each of the source and sink; transfer digraphs of size 2k+1 force a stricter bound of k+1.

The reduction works as follows. If the formula φ is satisfiable, one can select, for each clause, a literal set to true and place the corresponding arc (eℓ_i^j, c_i^j) into the feedback‑arc set F. All other transfer digraphs are then “cut” by adding exactly 2k arcs, guaranteeing that the underlying undirected graph D