Algorithm to Find all Cliques in a Graph

Let V={1, 2, 3, . . . , n} be the vertex set of a graph G, P(V) the powerset of V and A∈P(V). Then A can be represented as an ordered n-tuple (x₁x₂x₃ . . .x_n) where x_i=1 if i∈A, otherwise x_i=0 (1≤i≤n). This representation is called binary count (or BC) representation of a set A and denoted as BC(A). Given a graph G of order n, it is shown that every integer m in S={0, 1, 2, . . . , 2ⁿ-1} corresponds to a subset A of V and vice versa. We introduce algorithms to find a subset A of the vertex set V={1, 2, 3, . . . , n} of a graph G that corresponds to an integer m in S={0, 1, 2, . . . , 2ⁿ-1}, verify whether A is a subset of any other subset B of V and also verify whether the sub graph <A> induced by the set A is a clique or not using BC representation. Also a general algorithm to find all the cliques in a graph G using BC representation is introduced. Moreover we have proved the correctness of the algorithms and analyzed their time complexities.