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This paper introduces and solves a visibility-based escort planning problem. This novel problem, which is closely related to the well-researched family of visibility-based pursuit-evasion problems in robotics, entails an escort agent tasked with escorting a vulnerable agent, called the VIP, in a 2-dimensional environment. The escort protects the VIP from adversaries that pose line-of-sight threats. We describe a correct and complete planning algorithm whose inputs are a simply-connected polygonal map of the environment, starting locations for the escort and the VIP, along with a goal location to which the VIP agent should be safely moved. The algorithm computes trajectories for the escort and VIP which allow the VIP to reach its goal without coming into the line-of-sight of the adversary at any time. During the execution of these trajectories, the adversary is allowed to move along any continuous path that does not enter into the line-of-sight of the escort. The algorithm proceeds by dividing the environment into a collection of conservative regions and planning the escort's movements as a sequence of these regions via breadth-first search over an information graph. The trajectory of the VIP can then be constructed by tracing the 'safe zones' swept out by the escort's trajectory. We describe an implementation of this algorithm and present computed examples of escort agent strategies in diverse environments.