A Note on the Complexity of Directed Clique

Gutowski, Grzegorz; Rams, Mikołaj

doi:10.4230/LIPIcs.WG.2026.22

Abstract

For a directed graph G, and a linear order ≪ on the vertices of G, we define the backedge graph G^≪ to be the undirected graph on the same vertex set with edge {u,w} in G^≪ if and only if (u,w) is an arc in G and w ≪ u. The directed clique number of a directed graph G is defined as the minimum size of the maximum clique in the backedge graph G^≪ taken over all linear orders ≪ on the vertices of G. A natural computational problem is to decide for a given directed graph G and a positive integer t, if the directed clique number of G is at most t. This problem has polynomial algorithm for t = 1 and is known to be NP-complete for every fixed t ≥ 3, even for tournaments. In this note we prove that this problem is Σ^𝖯₂-complete when t is given on the input.

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Grzegorz Gutowski and Mikołaj Rams. A Note on the Complexity of Directed Clique. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 22:1-22:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.22

Author Details

Grzegorz Gutowski

Institute of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

Mikołaj Rams

Institute of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

Funding

Gutowski, Grzegorz: Partially supported by grant no. 2023/49/B/ST6/01738 from National Science Centre, Poland.

References

Pierre Aboulker, Guillaume Aubian, Pierre Charbit, and Raul Lopes. Clique number of tournaments, 2023. URL: https://doi.org/10.48550/arXiv.2310.04265.
Pierre Aboulker, Nacim Oijid, Robin Petit, Mathis Rocton, and Christopher-Lloyd Simon. Computing the degreewidth of a digraph is hard, 2024. URL: https://arxiv.org/abs/2407.19270.
Sanjeev Arora and Boaz Barak. Computational Complexity: A Modern Approach. Cambridge University Press, 2009.
Guillaume Aubian. Computing the clique number of tournaments, 2024. URL: https://doi.org/10.48550/arXiv.2401.07776.
Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979.
Ilhee Kim. On Containment Relations in Directed Graphs. PhD thesis, Princeton University, 2013. URL: http://arks.princeton.edu/ark:/88435/dsp012801pg46x.
Víctor Neumann-Lara. The dichromatic number of a digraph. Journal of Combinatorial Theory, Series B, 33(3):265-270, 1982. URL: https://doi.org/10.1016/0095-8956(82)90046-6.
Tung Nguyen, Alex Scott, and Paul Seymour. Some results and problems on tournament structure. Journal of Combinatorial Theory, Series B, 173:146-183, 2025. URL: https://doi.org/10.1016/j.jctb.2025.02.002.
Marcus Schaefer and Christopher Umans. Completeness in the polynomial-time hierarchy: Part I: A compendium. SIGACT News, 33(3):32-49, 2002. URL: https://doi.org/10.1145/582475.582484.
Marcus Schaefer and Christopher Umans. Completeness in the polynomial-time hierarchy: Part II. SIGACT News, 33(4):22-36, 2002. URL: https://doi.org/10.1145/601819.601829.
Marcus Schaefer and Christopher Umans. Completeness in the polynomial-time hierarchy: A compendium, 2008. URL: https://ovid.cs.depaul.edu/documents/phcom.pdf.
Larry J. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3(1):1-22, 1976. URL: https://doi.org/10.1016/0304-3975(76)90061-X.
Celia Wrathall. Complete sets and the polynomial-time hierarchy. Theoretical Computer Science, 3(1):23-33, 1976. URL: https://doi.org/10.1016/0304-3975(76)90062-1.

A Note on the Complexity of Directed Clique

Authors Grzegorz Gutowski , Mikołaj Rams

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