A Note on the Complexity of Directed Clique

2026-02-12

AI Summary

This study addresses the problem of determining the directed clique number of a directed graph: given a digraph and an integer parameter \( t \), decide whether the minimum, over all linear orderings of the vertices, of the size of the maximum clique in the corresponding backward-edge graph is at most \( t \). By employing polynomial-time reductions and techniques for proving \( \Sigma_2^P \)-completeness, the paper establishes for the first time that this problem is \( \Sigma_2^P \)-complete when \( t \) is part of the input. This result resolves a long-standing gap in the complexity landscape between the fixed-parameter cases—where the problem lies in P or is NP-complete—and the general case with variable \( t \), thereby providing a complete characterization of the problem’s complexity across the spectrum from P and NP-completeness to higher levels of the polynomial hierarchy.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

For a directed graph $G$, and a linear order $\ll$ on the vertices of $G$, we define backedge graph $G^\ll$ to be the undirected graph on the same vertex set with edge $\{u,w\}$ in $G^\ll$ if and only if $(u,w)$ is an arc in $G$ and $w \ll u$. The directed clique number of a directed graph $G$ is defined as the minimum size of the maximum clique in the backedge graph $G^\ll$ taken over all linear orders $\ll$ 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 $t\ge3$, even for tournaments. In this note we prove that this problem is $\Sigma^\mathsf{P}_{2}$-complete when $t$ is given on the input.

Problem

• directed clique number
• computational complexity
• backedge graph
• Σ₂^P-completeness

Innovation

• directed clique number
• backedge graph
• Σ₂^P-completeness
• computational complexity
• parameterized complexity