How many independent sets of size 4 has the Petersen graph?

Question

How many independent sets of size 4 has the Petersen graph?
(You can interpret this as Count to how many independent sets of size 4 belongs a given vertex in the Petersen graph.)
I recently come across this problem. To be honest I'm not so sure what does independent sets of size 4 mean this context. There are some answers online saying the answer to this question is 5. I would make a wild guess that it is due to the fact that we can make combinations of a set like {{1,2},{1,3},{1,4},{1,5}} in 5 ways. Effectively simplify this problem to a (5,4) combination problem, which gives us 5 as the answer. Would somebody please clarify this for me? Does Petersen graph always have 5 elements in the set? Thanks！

Answer 1

Here is the Petersen graph.

An independent set of size $4$ containing vertex $1$ must have either vertex $3$ or $4$ (otherwise it would need three of vertices $7, 8, 9, 10$, but that's clearly impossible). Suppose it contains vertex $3$. There are three remaining vertices ($8$, $9$ and $10$) not adjacent to $1$ or $3$, of which we need two that are not adjacent: the only possibility is $8$ and $10$. So the only independent set of size $4$ containing $1$ and $3$ is $\{1,3,8,10\}$. Similarly, the only independent set of size $4$ containing $1$ and $4$ is $\{1,4,7,9\}$. Thus there are two independent sets of size $4$ containing vertex $1$.

Similarly, for each of the $5$ outer vertices there are two independent sets of size $4$ containing that vertex. That makes a total of $5$ independent sets of size $4$.