Customer networks go through constant changes. They may expand or shrink once they are formed. In dynamic environments, it is a critical corporate challenge to identify and manage influential customer groups in a cost effective way. In this context, we apply inverse optimization theory to suggest an efficient method to manage customer networks. In this paper, we assume that there exists a subset of nodes that might have a large effect on the network and that the network can be modified via some strategic actions. Rather than making efforts to find influential nodes whenever the network changes, we focus on a subset of selective nodes and perturb as little as possible the interaction between nodes in order to make the selected nodes influential in the given network. We define the following problem based on the inverse optimization. Given a graph and a prescribed node subset, the objective is to modify the structure of the given graph so that the fixed subset of nodes becomes a minimum dominating set in the modified graph and the cost for modification is minimum under a fixed norm. We call this problem the inverse dominating set problem and investigate its computational complexity.