In the aftermath of a large-scale disaster (such as earthquake), existing communication infrastructures are often critically impaired, preventing timely information exchange between the survivors, responders, and the coordination center. Typically, a temporary network, called Disaster Response Network (DRN), is set up using smart devices, movable base stations and easily deployable cellular antennas. However, such networks are challenged by rapid devices' energy depletion and component failures due to environmental adversities and hardware faults. State-of-the-art literature address energy challenges through intelligent routing, however robustness of DRN against component failures is largely unaddressed. In this paper, we investigate designing a novel network topology for DRNs, which is both energy-efficient and robust against component devices' failures. Specifically, the objective is to construct a sparse structure from the original DRN (termed, Sparse-DRN) while ensuring that there exists a connected tree backbone. Our performance evaluation shows that the Sparse-DRN offers a good trade-off between the energy efficiency and network robustness, while ensuring the QoS requirements i.e., packet delivery and network latency.