Graph decompositions have useful applications in optimization problems that are categorized as NP-Hard. Modular Decomposition of a graph is a technique to decompose the graph into non-overlapping modules. A module M of an undirected graph G = (V, E) is commonly defined as a set of vertices such that any vertex outside of M is either adjacent or non-adjacent to all vertices in M . By the theory of modular decomposition, the modules can be categorized as parallel, series or prime modules. Series modules which are maximal and are also cliques are termed as simple series modules or critical cliques. There are modular decomposition algorithms that can be used to decompose the graph into modules and obtain critical cliques. In this current research, we present a new algorithm to decompose the graph into critical cliques without applying the process of modular decomposition. Given a simple, undirected graph G = (V, E), the runtime complexity of our proposed algorithm is O(|V| + |E|) under certain input constraints. Thus, one of our main contributions is to propose a novel algorithm for decomposing a simple, undirected graph directly into critical cliques. We apply the idea of critical cliques to propose a new way for solving the influence maximization problem in online social networks. Influence maximization in online social networks is the problem of identifying a small, initial set of influential individuals which can influence the maximum number of individuals in the network. In this research, we propose a new model of online social networks based on the notion of critical cliques. We utilize the properties of critical cliques to assign parameters for our proposed model and select an initial set of activation nodes. We then simulate the influence propagation process in the online social network using our proposed model and experimentally compare our approach to the greedy algorithm proposed by Kempe, Kleinberg and Tardos. Our main contribution in the influence maximization research is to propose a new model of online social network taking into account the structural properties of the social network graph and a new, faster algorithm for determining the initial set of influential individuals in the online social network.