A depth-first forest classifies the edges of a graph into tree, back, forward, and cross edges. A breadth-first tree can also be used to classify the edges reachable from the source of the search into the same four categories.

a. Prove that in a breadth-first search of an undirected graph, the following properties hold:

1. There are no back edges and no forward edges.

2. For each tree edge (u, ν), we have ν.d = u.d + 1.

3. For each cross edge (u, ν), we have ν.d = u.d or ν.d = u.d + 1.

b. Prove that in a breadth-first search of a directed graph, the following properties hold:

1. There are no forward edges.

2. For each tree edge (u, ν), we have ν.d = u.d + 1.

3. For each cross edge (u, ν), we have ν.d ≤ u.d + 1.

4. For each back edge (u, ν), we have 0 ≤ ν.d ≤ u.d.