# Question

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, v), we have d[v] = d[u] + 1.

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

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

4. For each back edge (u, v), we have 0 ≤ d[v] ≤ d[u].

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

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

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

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

4. For each back edge (u, v), we have 0 ≤ d[v] ≤ d[u].

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