Question

How many nonisomorphic caterpillars are there with six vertices?

Short answer

Therefore, 6 nonisomorphic caterpillars are there with six vertices.

Step-by-step solution

General form 

Definition of tree: A tree is a connected undirected graph with no simple circuits.

Definition of Circuit: It is a path that begins and ends in the same vertex.

Definition of Caterpillar: A caterpillar is a tree that contains a simple path such that every vertex not contained in this path is adjacent to a vertex in the path.

Evaluate the nonisomorphic caterpillars

Given that, a caterpillar with 6 vertices.

Case (1): Path of length 5

A path is a caterpillar, so the path of length 5 will be a caterpillar. Then, there is 1 nonisomorphic caterpillar which consists of a longest path of length 5.

Case (2): Path of length 4

Let us consider a path of length 4. We can attach the remaining vertex at one of the three vertices in the middle. So, the 6^th vertex could be attached to the 2^nd, 3^rd and 4^th vertex.

However, attaching the 6^th vertex to the 2^nd or 4^th vertex will lead to isomorphic graphs.

That is a total of 2 non-isomorphic caterpillars consisting of a longest track of length 4.

Case (3): Path of length 3

Let us consider a path of length 3. We can the attach the remaining two vertices at the two vertices in the middle of the path.

So, the total of 2 nonisomorphic caterpillar which consists of a longest path of length 3.

Case (4): Path of length 2

Let us consider a path of length 2. The only possible option is that the other tree vertices are all connected to the middle vertex in the path.

So, the total of 1 nonisomorphic caterpillar which consists of a longest path of length 2.

That is, \({\bf{1 + 2 + 2 + 1 = 6}}\)

Conclusion: There are 6 nonisomorphic caterpillars with 6 vertices.