Exercise 7.

We define a forest to be a graph with no cycles.

1. Explain why this is a good name. That is, explain why a forest is a union of trees.

2. Suppose \(F\) is a forest consisting of \(m\) trees and \(v\) vertices. How many edges does \(F\) have? Explain.

3. Prove that any graph \(G\) with \(v\) vertices and \(e\) edges that satisfies \(v \lt e+1\) must contain a cycle (i.e., not be a forest).