The Maximum Clique Problem

Description Given an undirected graph G = ( V, E ), a clique S is a subset of V such that for any two elements u, v ∈ S, ( u, v ) ∈ E. Using the notation ES to represent the subset of edges which have both endpoints in clique S, the induced graph GS = ( S, ES ) is complete. Finding the largest clique in a graph is an NP-hard problem, called the maximum clique problem (MCP). Cliques are intimately related to vertex covers and independent sets. Given a graph G, and defining E* to be the complement of E, S is a maximum independent set in the complementary graph G* = ( V, E* ) if and only if S is a maximum clique in G. It follows that V – S is a minimum vertex cover in G*. There is a separate weighted form of MCP that we will not consider further here.