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Q: Undirected vs. directed connectivity.

(a) Prove that in any connected undirected graph G = (V, E) there is a vertex v ? V whose removal leaves G connected. (Hint: Consider the DFS search tree for G.)

(b) Give an example of a strongly connected directed graph G = (V, E) such that, for every v ? V , removing v from G leaves a directed graph that is not strongly connected.

(c) In an undirected graph with 2 connected components it is always possible to make the graph connected by adding only one edge. Give an example of a directed graph with two strongly connected components such that no addition of one edge can make the graph strongly connected.

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