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Which one of the following is TRUE?

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Consider the following two problems on undirected graphs: α: Given G(V,E), does G
have an independent set of size |v|—4? β: Given G(V,E), does G have an independent set of
size 5? Which one of the following is TRUE?
a) α is in P and β is NP-complete
b) α is NP complete and β is in P
c) Both α and β are NP-complete
d) Both α and β are in P
Answer:
Both are in P
For α : there are C(V,V-4) = C(V,4) sets of size 4, we just need to verify whether one of
these sets is independent, the time complexity for checking a set of size V-4 is independent or not
can be done in V
2
time. So for total C(V,4) = O(V
4
) sets, the time complexity will be O(V
6
),
which is polynomial.
Similarly for β : there are C(V,5) sets of size 5, we just need to verify whether one of these
sets is independent, the time complexity for checking a set of size 5 is independent or not can be
done in constant time So for total C(V,5) = O(V
5
) sets, the time complexity will be O(V
5
), which
is polynomial.

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