Question $6$ : Graphs $-$ Foraging for Rocks $..........................................10$ points

Suppose that a node of our undirected graph $G$ represents a clearing in a forest and an edge of our graph represents a trail between two clearings. For each node $($i$.$e$.$ clearing$)$ we have a count of how many rocks are in that particular clearing.

In this problem, you're given an undirected graph $G$ and a node $S$ and you want to identify how many rocks can be reached by following paths from $S$ to other clearings. For example suppose the graph below was your input graph $G$ :

Examples: If your $S$ was the node $S_1$ above, you would find a total of $9$ rocks $($see the blue colored nodes$).$ Similarly, if your $S$ was the node $S_2$ above, you would find that there were a total of $5$ rocks. Finally, if your $S$ was the node $S_3$ above, you would find a total of $15$ rocks.

We define a graph as follows: