Computing the clique number of tournaments
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 ≥ 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.
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