Question: A graph G = (V, E) has a vertex cover of size k if there is a subset V 0 V of size k such

A graph G = (V, E) has a vertex cover of size k if there is a subset V 0 V of size k such that for every edge (u, v) E at least one of u and v is in V 0 .

(a) The vertex cover decision problem is to determine if there is a vertex cover of size m in a graph G = (V, E) where m |V |. Show this problem is NP by writing a polynomial-time nondeterministic algorithm to solve it.

(b) What is the worst-case time complexity of the verification stage of this algorithm?

(c) Show that the vertex cover decision problem is NP-complete by reducing the clique decision problem to it. (Hint: consider the complement of the graph, that is the graph with the same vertices but with edges only between all pairs of vertices that are not adjacent in the original graph.)

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