Question:
if I assume that I ran Depth First Search on a digraph $G,$ and obtained a directed spanning forest $F.$ I am interested in the set $R$ of vertices reachable from a given fixed vertex $a \in V (G).$ How do I Show in general set $R$ of vertices cannot be obtained from directed spanning forest $F$?
Answer:
Can I answer it by saying that if I have a digraph of $G:= (\{1,2,3,4\},\{(1,2),(3,4)\}$ and i perform Depth First Search $(G, 1)$ which will result in $(\{1,2\}, \{(1,2)\})$ whereby vertex $1$ is unreachable. However, how is it even a directed spanning forest if I only obtain only vertex $1$ and $2$ which is not a directed spanning forest but rather a forest. IF not how should I answer it so that the result is a spanning forest?
First you pick an initial node. As DFS progresses it will mark vertices as visited or seen. At the end, unseen vertices are those that cannot be reached by starting with the chosen initial node. The set of visited nodes corresponds to the first connected component (that to which the initial node belongs). If you want the full forest you then need to restart DFS from an unvisited node (and repeat).