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Graph theory problem: total number of possible 'paths' through given vertices
I'm having a few difficuties with what this website calls a "graph theory question".
Write the number of possible routes through given vertices. Each vertex can be visited in either an odd or even number of turns.
This is what I came up with
The blue arrow points to the centre.
The red arrow points to the centre again, and the other red arrow points to the centre once.
Is this the right answer?
A:
You correctly answered the question, but have not answered a second question:
Given that there are $n$ nodes to visit in $n$ turns, how many combinations are possible for the first turn?
Consider the $n$ blue arrows, and the $n$ red arrows. If the first arrow is taken in an even number of turns, then for each arrow there are two options: take the next turn first, or keep going. If the first arrow is taken in an odd number of turns, there are only one option: take the next turn first.
If the first arrow is taken in an even number of turns, there are two options: take the next turn first, or keep going. If the first arrow is taken in an odd number of turns, there is only one option: take the next turn first.
If the first arrow is taken in an odd number of turns, then for each of the $n$ arrows there are $2$ options: take the next turn first, or keep going. If the first arrow is taken in an even number of turns, there are $1$ options: take the next turn first.
This means there are $2n$ different possibilities for the first turn.
Now there are $
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