A *quasi-kernel* in a directed graph D is an independent subset of vertices $S$ so that for every $v \in V(D)-S$ either $v->s$ for some $s \in S$ or $v->w->s$ for some $w \in V(D)-S, s \in S$.

**Equivalently**, a set $S$ is a quasi-kernel if $d(s,t) \geq 2$ for every $s,t \in S$ and $d(v,s) \leq 2$ for every $v \in V(D)-S,s \in s$.

Chvatal & Lovasz had proved that every digraph has a quasi-kernel (very easy proof by induction) and Jacob & Meyniel had proved that a graph without a kernel has at least three quasi-kernels.

My question is:

Is counting the number of quasi-kernels NP-hard?