Backedge Graphs of Tournaments: Algorithms and Complexity

Abstract

A tournament $T=(V,A)$ on $n$ vertices is an orientation of the complete graph $K_n$. The backedge graph of $T$ with respect to an ordering of $V$ is the undirected graph on vertex set $V$ whose edge set corresponds to the arcs directed from a later vertex to an earlier vertex in the ordering. Tournaments are dense digraphs, and an appropriate choice among the $n!$ orderings of $V$ can yield a backedge graph that provides a concise representation of the tournament. However, the algorithmic problem of determining whether a tournament admits a backedge graph in a given class of undirected graphs varies in complexity. Moreover, this problem is often closely tied to computing existing parameters of tournaments, such as the dichromatic number, clique number \cite{Aboulker et al., 2023}, and degreewidth \cite{Davot et al., 2023} of a tournament. We extend the notion of degreewidth by introducing directional degreewidth, which separately bounds the left-degrees and right-degrees of vertices in addition to bounding the total degrees in the backedge graph. We investigate the algorithmic complexity of verifying bounds on directional degreewidth and obtain an algorithm which runs in polynomial time when the total degree is unbounded, or in fixed-parameter tractable time parameterized by the total degree bound otherwise. We also provide a polynomial-time algorithm for computing a $P_3$-free backedge graph of a tournament, if it exists. Together with existing results, this settles the complexity of determining whether a tournament admits an $H$-free backedge graph for every graph $H$ on three vertices.