Let G = (V(G), E(G)) be an undirected simple connected graph. A network is usually represented by an undirected simple graph where vertices represent processors and edges represent links between processors. Finding the vulnerability values of communication networks modeled by graphs is important for network designers. The vulnerability value of a communication network shows the resistance of the network after the disruption of some centers or connection lines until a communication breakdown. The domination number and its variations are the most important vulnerability parameters for network vulnerability. Some variations of domination numbers are the 2-domination number, the bondage number, the reinforcement number, the average lower domination number, the average lower 2-domination number, and so forth. In this paper, we study the vulnerability of cycles and related graphs, namely, fans, k-pyramids, and n-gon books, via domination parameters. Then, exact solutions of the domination parameters are obtained for the above-mentioned graphs.