# Graphs, Digraphs, Trees and Forests

I am posting one problem from Exercise 2.2, I need answer for 2.9.

I am posting another question from Exercise 3.1:

Problem 3.2.) Prove that a graph G is a forest if and only if every induced subgraphs of G contains a vertex of degree at most 1.

Problem 3.1) Draw all forests of order 6.

See attached file for full problem description.

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#### Solution Preview

3.1 )) Attached figures of Forests of order 6.

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3.2 ))

Proof:

Need to prove in both directions

==>)

Here we need to prove

Every induced subgraphs of G contains a vertex of degree at most 1 => G is a forest -- (1)

This is same as saying that

"If G is not a forest ==> Not all induced subgraphs have a vertex of degree at most 1." -- (2)

( The above fact follows from logic and this method of proof is known as proof by contrapositive.)

So let us assume that G is not a forest. Therefore there is a cycle in G. Consider the vertices of the cycle. Now consider the induced subgraph on the vertices. This graph ...

#### Solution Summary

Graphs, Digraphs, Trees and Forests are investigated in the following posting.