Analysis of the interdependencies between interconnected critical infrastructures can help enhance the robustness of the individual infrastructures as well as the overall interconnected infrastructures. One of the most studied interdependent critical infrastructure network scenarios is a power grid connected to a backbone telecommunications network. In this interdependent infrastructure scenario, the robustness of the entire system is usually analyzed in the context of cascading failure models in the power grid. However, this paper focuses on targeted attacks, where an attack on a telecommunications network node directly affects a connected power grid node, and vice versa. Cascading failures are outside the scope of this paper because the objective is to enhance the robustness of the interconnections between the infrastructures. In order to mitigate the impacts of targeted attacks on the interdependent infrastructures, three interdependency matrices for connecting the infrastructures are specified and analyzed. The analysis identifies the interdependency matrix that best reduces the impacts of targeted attacks and the propagation of failures between the infrastructures. Additionally, the impacts of interconnecting a power grid to different telecommunications networks, each with different susceptibilities to targeted attacks, is evaluated