Decision problem - vertex with path to all other vertcies
Suppose we are given a directed graph $G$ and two vertices $s,t$. Add a new vertex $a$ and an edge from $a$ to $s$. Also, add an edge from $t$ to all vertices in the graph apart from $a$.

Since $a$ has no incoming edges, if there is any vertex from which all other vertices are reachable, it must be $a$. For this to happen, in particular $t$ must be reachable from $a$. Since the only outgoing edge from $a$ points at $s$, this can only happen if $t$ is reachable from $s$. In that case, all other vertices are also reachable from $a$, via $t$, since we added outgoing edges from $t$ to all vertices apart from $a$.