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1/1

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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