$($c$)$ Consider two vertices $m$ and $n$ that are simultaneously on the function$-$call stack at some point during the execution of depth$-$first search from vertex $s$ in a digraph. Which of the following must be true? $(5\%)$

$($I$)$ There is both a directed path from $s$ to $m$ and a directed path from $s$ to $n.$

$($II$)$ If there is no directed path from $m$ to $n,$ then there is a directed path from $n$ to $m.$

$($III$)$ There is both a directed path from $m$ to $n$ and a directed path from $n$ to $m.$