Defining clusters, part two: cliques

In an undirected graph G=(V,E) a clique is a subgraph induced by node set S⊆V (i.e., G_S = (S,E_S)) such that the density of G_S is 1. Put another way, in a clique, every pair of nodes is adjacent.

Example: Consider undirected graph G=(V,E) drawn below.

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The following are cliques:

• The subgraph induced by {c,d,e,f}
• The subgraph induced by {b,f}

The following are not cliques:

• The subgraph induced by {a,b,c}
• The subgraph induced by {b,c,d,e}, which is drawn below

Image may be NSFW. Clik here to view. This subgraph is not a clique: Its density is less than 1.

The subgraph induced by {c,d,e,f} is the largest clique in G: the clique with the most nodes. Usually the largest clique in a graph is the most interesting clique in that graph.