Classifying edges by breadth-first search 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; /, we have :d D u:d C 1. 3. For each cross edge .u; /, we have :d D u:d or :d D u:d C 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; /, we have :d D u:d C 1. 3. For each cross edge .u; /, we have :d u:d C 1. 4. For each back edge .u; /, we have 0 :d u:d.

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