# In class, we covered the Clique problem, in which we were given a graph G and had to find the…

In class, we covered the Clique problem, in which we were given a graph G and had to find the largest clique in G. Recall that a clique is a set of nodes such that every node is connected to every other node in the set (i.e., everybody in the set knows everybody else in the set). The Clique-3 problem is the same problem, except that we are guaranteed that every node has degree at most 3.

Suppose that I want to prove that Clique-3 is NP-Complete. To do this, I make the following argument: We know that the Clique problem is NPComplete. Clique-3 reduces to Clique, because if we can solve Clique for graphs in general, then clearly we can also solve it for graphs where the nodes have degree at most 3. Thus, Clique-3 is also NP-Complete.

Part a: What is wrong with the above argument?

Part b: Show that Clique-3 is not NP-Complete (assuming that P 6= NP). (Hint: what is the largest possible clique in a graph where the nodes have degree at most 3?)

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