Directed Graphs and Network Flows · Challenge problem

A digraph has vertices $1,\dots,7$ with arcs $1\to2$, $2\to3$, $3\to1$ (a $3$-cycle), $4\to5$, $5\to4$ (a $2$-cycle), $6\to7$, $7\to6$ (a $2$-cycle), and $1\to4$, $4\to6$, $1\to6$. Among all $\binom{7}{2}=21$ pairs of vertices, how many pairs are NOT mutually reachable (there fails to be a directed path in at least one of the two directions)?