Independent Set- a set of vertices in a graph no two of which are adjacent.

Maximal Independent set problem - Given a graph $G$, find the size of the maximal independent set in it. This problem in NP-hard.

Independent set decision problem - Given a graph $G$ and a number $k$, does $G$ have an independent set of size $k$. This problem is NP-complete (NP-hard but in NP).

Now, in the given problem $\beta$ corresponds to the Independent set decision problem. But there is a difference there. We have $5$ instead of $k$. And this drastically changes the problem statement. We can now give a polynomial time **deterministic **algorithm for $\beta$.

- Find all vertex sets of size $5$. We get ${}^{|V|}C_5$ such vertex sets
- For each of them check if there is any adjacent vertices. This check can be done in constant time if we use an Adjacency matrix representation

Thus the whole time complexity reduces to ${}^{|V|}C_5$ which is $O\left({|V|}^5\right)$ and hence polynomial. $({}^{|V|}C_k$ is not polynomial but ${}^{|V|}C_5$ is $)$.

Problem $\alpha$ is asking for an independent set of size $|V| - 4$. This is equivalent to asking if $G$ has a vertex cover* of size $4$. Following a similar approach as done for $\beta$ this problem also can be done in polynomial time.

So, both $\alpha$ and $\beta$ are in $P$.

**D** choice.

Vertex cover of a graph $G$ is the set of vertices such that each edge of the graph is incident on atleast one of those vertices.

Independent Set and Vertex cover Reduction: https://www.cs.cmu.edu/~ckingsf/bioinfo-lectures/npcomplete.pdf