Degreewidth on Semi-complete Digraphs

Abstract

For a digraph G and ordering of G, the degreewidth of the ordering is the maximum number of backward arcs incident to any vertex of G. The degreewidth $\varDelta (G)$ of G is defined as the minimum degreewidth of an ordering of G. A digraph G is semi-complete if every pair of vertices is connected by at least one arc, oriented if every pair of vertices is connected by at most one arc, and a tournament if every pair of vertices is connected by exactly one arc. We give a fixed parameter tractable (FPT) algorithm, with running time $\varDelta (G)^{O(\varDelta (G))}n + O(n^2)$, to compute the degreewidth of semi-complete digraphs. We then show that both the Feedback Arc Set and Cutwidth problems on semi-complete digraphs admit algorithms with running time $\varDelta (G)^{O(\varDelta (G))}n + O(n^2)$. Our algorithms resolve in the affirmative two open problems of Davot et al. [WG 2023], who asked whether there exists an FPT algorithm to compute the degreewidth of a tournament, and whether Feedback Arc Set on tournaments admits an FPT algorithm when parameterized by the degreewidth of the input digraph. Additionally, extending an argument of Davot et al. [WG 2023], we show that sorting by in-degree is a 3-approximation algorithm for degreewidth on semi-complete digraphs. Finally we prove that it is NP-hard to determine whether a given oriented digraph has degreewidth at most 2.