In a question I had asked earlier, I was interested in knowing whether we could decide in polynomial time whether, for a directed graph $G$ with every one of its vertices belonging to an edge, a size-$w$ watch set, with $w$ at least half the number of vertices in $G$, exists (see the linked question). Several users showed that, in fact, such a polynomial time decider does not exist.
I'm interested in whether a polynomial time decider exists for a variant of this problem. Now, suppose that $G = (V,E)$ is an $n$-vertex directed graph with the property that every vertex in $G$ has an edge entering it. That is, for each vertex $v \in V$, there is an edge $(u,v)$ in $E$. With this in mind, consider the problem of deciding whether, for a given $w \geq 2n/3$, $G$ has a size-$w$ watch set. With the constraint that every vertex has an edge entering it, I'm not sure that we are able to reduce from arbitrary directed graphs, as Discrete lizard showed we could do in my earlier question. What I suspect is that any graph of this type must have a size-$w$ watch set, with $w \geq 2n/3$.