If "S" is a maximal independent set in some graph, it is a maximal clique or maximal complete subgraph in the complementary graph. A maximal clique is a set of vertices that induces a complete subgraph, and that is not a subset of the vertices of any larger complete subgraph. That is, it is a set "S" such that every pair of vertices in "S" is connected by an edge and every vertex not in "S" is missing an edge to at least one vertex in "S". A graph may have many maximal cliques, of varying sizes; finding the largest of these is the maximum clique problem.