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.

翻译：对于有向图$G$及其顶点上的线性序$\ll$，我们定义后向边图$G^\ll$为具有相同顶点集的无向图，其中边$\{u,w\}$属于$G^\ll$当且仅当$(u,w)$是$G$中的弧且满足$w \ll u$。有向图$G$的有向团数定义为在所有顶点线性序$\ll$下，后向边图$G^\ll$中最大团大小的最小值。一个自然的计算问题是：对于给定的有向图$G$和正整数$t$，判定$G$的有向团数是否不超过$t$。该问题在$t=1$时存在多项式算法，且已知对于每个固定的$t\ge3$（即使是锦标赛图）是\NP完全的。本文证明当$t$作为输入给定时，该问题是$Σ^\mathsf{P}_{2}$完全的。