Consider a digraph G and a directed spanning forest F created by running DFS on G$.$

F is the vertex$-$disjoint union of directed trees. Consider one such directed tree T with root r$.$

Assume r has two different children x$,$ y in T$.$

Let X be the vertices reachable from x in T$,$ and let Y be the vertices reachable from y in T$.$

$($a$)$It is not possible that there is an arc in G from some element of X to some element of Y$.$

$($b$)$There is no path in G from some element of X to some element of Y$.$

$($c$)$It is not possible that there is an arc in G from some element of X to some element of Y$,$ AND an arc in G from some element of Y to some element of X$.$

$($d$)$Either the finishing times of all the elements of X are before the discovery times of all elements of Y$,$ or the finishing times of all the elements of Y are before the discovery times of all the elements of X$.$