Problem 718 Points Given an undirected graph G = (V, E) of vertices V and edges E, such that for all v, w ∈ V, (v, w) ∈ E. In other words, there is an edge between every pair of vertices in V. Given an undirected graph G = (V, E) and a set of vertices W ⊆V, W is an independent set if for all u, v ∈ W, (u, v) ∉ E. In other words, there is no edge between any pair of vertices in W. Consider the CLIQUE and INDEPENDENT SET problems for undirected graphs that we studied in class. CLIQUE(G, k) returns true if G has a clique of size k. IS(G, k) returns true if G has an independent set of size k.
a) Define a certificate for ISCLIQUE. Show that we can verify the certificate in deterministic polynomial time.
b) Consider an undirected graph G and an integer k. Construct a new graph H from G by adding k vertices.
Show that if G has a clique of size k, then H has an independent set of size k.
Show that if H has an independent set of size k, then G has a clique of size k.
What have we shown in parts a and b? Using the fact that CLIQUE is NP-complete, what can we now conclude?