Computing the clique number of tournaments

TL;DR

The paper defines the tournament clique-number parameter $\\overrightarrow{\\omega}(T)$ as the minimum, over all vertex orderings, of the clique number of the backedge graph $T^{\\prec}$, and proves that for any $k \ge 3$, deciding whether $\\overrightarrow{\\omega}(T) \le k$ is NP-complete. It introduces gadget constructions and a lifting technique (via a $T_k$-based framework and $\\Delta$ compositions) to encode a $3$-SAT instance, establishing NP-hardness, and thereby NP-completeness, of computing the clique number of tournaments. A key methodological contribution is a tool that preserves the clique-number while enabling systematic control of subgraph occurrences, facilitating the gadget-based reduction. The paper also provides a counterexample to a Gyárfás–Sumner-type conjecture in the directed setting by constructing a forest backedge graph tournament that is not contained in any $D_k$, despite having $\\overrightarrow{\\omega}=2$ and unbounded dichromatic complexity, highlighting limits of χ-boundedness analogues for tournaments. Together, these results settle the NP-completeness of the fixed-$k$ problem, sharpen the understanding of tournament cliques, and identify new directions in directed analogues of classical graph-theoretic conjectures.

Abstract

The clique number of a tournament is the maximum clique number of a graph formed by keeping backwards arcs in an ordering of its vertices. We study the time complexity of computing the clique number of a tournament and prove that, for any integer $k \geq 3$, deciding whether a tournament has clique number at most $k$ is NP-complete. This answers an interrogation of Nguyen, Scott and Seymour. To do so, we make use of a construction which we then modify to provide a counterexample to a conjecture of Aboulker, Aubian, Charbit and Lopes.

Figures (11)

• Figure 1: The backedge graph corresponding to the ordering $1 \prec 2 \prec 3 \prec 4 \prec 5 \prec 6 \prec 7 \prec 8 \prec 9$
• Figure 2: The backedge graph of $T$ corresponding to the ordering $6 \prec 8 \prec 2 \prec 9 \prec 1 \prec 3 \prec 4 \prec 5 \prec 7$
• Figure 3: Python code to check that in every $\mathop{\mathrm{\overrightarrow{\omega}}}\nolimits$-ordering, exactly one of $uv$ or $wx$ is forward
• Figure 4: The backedge graph corresponding to the ordering $1 \prec 2 \prec 3 \prec 4 \prec 5 \prec 6 \prec 7 \prec 8$
• Figure 5: The backedge graph of $T$ corresponding to the ordering $4 \prec 7 \prec 5 \prec 8 \prec 1 \prec 2 \prec 6 \prec 3$

Theorems & Definitions (35)

• theorem 1
• theorem 2
• conjecture 3
• conjecture 4: Gyárfás, 1975 & Sumner, 1981
• conjecture 5
• theorem 6
• theorem 7
• lemma 8
• proof
• definition 9