Recently defined exponential domination number is reported as a new measure to graph vulnerability. It is a methodology, emerged in graph theory, for vulnerability analysis of networks. Also, it gives more sensitive results than other available measures. Exponential domination number has great significance both theoretically and practically for designing and optimizing networks. In this paper, it is studied how some of the graph types perform when they suffer a vertex failure. When its vertices are corrupted, the vulnerability of a graph can be calculated by the exponential domination number which gives more information about the characterization of the network.