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Communication networks are vulnerable to natural disasters, such as earthquakes or floods, as well as to physical attacks, such as an electromagnetic pulse (EMP) attack. Such real-world events happen in specific geographical locations and disrupt specific parts of the network. Therefore, the geographical layout of the network determines the impact of such events on the network's connectivity. In this paper, we focus on assessing the vulnerability of (geographical) networks to such disasters. In particular, we aim to identify the most vulnerable parts of the network. That is, the locations of disasters that would have the maximum disruptive effect on the network in terms of capacity and connectivity. We consider graph models in which nodes and links are geographically located on a plane. First, we consider a simplistic bipartite graph model and present a polynomial-time algorithm for finding a worst-case vertical line segment cut. We then generalize the network model to graphs with nodes at arbitrary locations. We model the disaster event as a line segment or a disk and develop polynomial-time algorithms that find a worst-case line segment cut and a worst-case circular cut. Finally, we obtain numerical results for a specific backbone network, thereby demonstrating the applicability of our algorithms to real-world networks. Our novel approach provides a promising new direction for network design to avert geographical disasters or attacks.