Question

A forest is a graph where each of the connected components is a tree. In particular, a tree is a forest. Prove that a graph is a forest if and only if it does not have any cycles.

a) The statement is true; provide a proof.
b) The statement is false; counterexample required.
c) The statement is partially true; additional conditions needed.
d) The statement is ambiguous; further clarification needed.

Answer 1

A graph is a forest if and only if it does not have any cycles.

Step-by-step explanation:

To prove that a graph is a forest if and only if it does not have any cycles, we need to prove both directions.

First, let's prove that if a graph is a forest, then it does not have any cycles.

A forest is a graph where each of the connected components is a tree. In a tree, there is a unique path between any two vertices, and there are no cycles.

If a graph is a forest, each connected component is a tree, which means there are no cycles in any connected component. Therefore, the graph as a whole does not have any cycles.

Now, let's prove the other direction.

If a graph does not have any cycles, we need to show that it is a forest.

If the graph does not have any cycles, it means that there are no closed paths.

Each connected component in the graph can be seen as a separate tree, as there are no cycles within each connected component. Hence, the graph is a forest.

Therefore, both directions have been proven, and we can conclude that a graph is a forest if and only if it does not have any cycles.