The Power of the Score Sequence of a Tournament

2026-07-16

AI Summary

This work investigates the solvability of classical directed graph problems using only the score sequence—that is, the in-degree sequence—of a tournament. The authors propose a unified framework grounded in the theory of cycle-reversal invariants to characterize all graph properties determined solely by in-degrees. They introduce a joint representation combining skeleton graphs and in-degree sequences, extending their results to general directed graphs. Leveraging this approach, they design efficient algorithms for acyclicity testing, topological sorting, reachability, strongly connected component decomposition, and vertex ordering across multiple computational models, including the streaming model, two-party communication model, and cut-query model, achieving performance that matches or surpasses the current state of the art in several settings.

Abstract

What problems can one solve on a tournament if only its score sequence is known? Tournaments are oriented complete graphs that form an extensively-studied class of directed graphs (digraphs), both from combinatorial and algorithmic perspectives. Over the years, researchers have identified multiple classical digraph problems that can be solved on a tournament from only its score sequence (indegree sequence). These problems include acyclicity testing and topological sorting [Chakrabarti, Ghosh, McGregor, and Vorotnikova; SODA'20], $s,t$-reachability, strong connectivity, and decomposition into strongly connected components (SCC) [Ghosh and Kuchlous; ESA'24], and vertex-ordering problems such as cutwidth and optimal linear arrangement [Barbero, Paul, and Pilipczuk; ICALP'17]. These prior works showed the sufficiency of the score sequence by designing distinct algorithms for the individual problems. In this work, we give a simple unified framework that solves all these problems using only indegrees and, in fact, completely characterises the class of problems that is determined by the indegree information: problems whose answers are invariant under cycle reversals. This characterisation is a special case of a much more general result that we establish: for any arbitrary digraph, the knowledge of its skeleton (underlying undirected graph) and the vertex indegrees completely determines its properties that are invariant under cycle reversal. As a byproduct of our results, we obtain algorithms for a variety of connectivity-based, cut-based, and vertex-ordering problems on tournaments and ``almost tournaments'' in the streaming, the two-player communication, and the cut-query models of computation. Some of these algorithms match existing optimal bounds and others provide bounds improving the state of the art.

Problem

• tournament
• score sequence
• cycle reversal
• indegree sequence
• digraph properties

Innovation

• score sequence
• cycle reversal invariance
• tournament
• unified framework
• streaming algorithms