A Note on the Complexity of Directed Clique

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 $t\ge 3$, even for tournaments. In this note we prove that this problem is $\hat{I}{\pounds }_2^{\mathsf{P}}$-complete when $t$ is given on the input.