A Note on the Complexity of Directed Clique

TL;DR

The paper explores the complexity of the directed clique number, defined via backedge graphs $G^\ll$ under linear orders $\ll$, and proves that the decision problem DirectedClique is $\Sigma^\mathsf{P}_2$-complete when the bound $t$ is part of the input. The main result is established by a reduction from Existential-2-Level-3-CNF to DirectedClique, employing binary, copy, and clause gadgets to encode variable assignments and clause satisfaction within backedge-clique constraints. An independent contribution provides a tight polynomial upper bound for tournaments: $\overrightarrow{\omega}(T) \le \sqrt{2|V(T)|}$. Together, these results advance understanding of the complexity landscape for directed clique-like parameters beyond NP-hardness and shed light on the hardness of TournamentClique as a related problem.

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 $Σ^\mathsf{P}_{2}$-complete when $t$ is given on the input.

Theorems & Definitions (6)

• Conjecture 1
• Conjecture 2
• Theorem 3
• Theorem 3
• Theorem 6
• Corollary 7