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 . . .xn) where xi = 1 if i ∈ A, otherwise xi = 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.