We consider the following variant of a classical pursuit-evasion problem: how many pursuers are needed to capture a single (adversarial) evader on the surface of a 3-dimensional polyhedral body? The players remain on the closed polyhedral surface, have the same maximum speed, and are always aware of each others' current positions. This generalizes the classical lion-and-the-man game, originally proposed by Rado [12], in which the players are restricted to a two-dimensional circular arena. The extension to a polyhedral surface is both theoretically interesting and practically motivated by applications in robotics where the physical environment is often approximated as a polyhedral surface. We analyze the game under the discrete-timemodel,where the players take alternate turns, however, by choosing an appropriately small time step t > 0, one can approximate the continuous time setting to an arbitrary level of accuracy. Our main result is that 4 pursuers always suffice (upper bound), and that 3 are sometimes necessary (lower bound), for catching an adversarial evader on any polyhedral surface with genus zero. Generalizing this bound to surfaces of genus g, we prove the sufficiency of (4g + 4) pursuers. Finally, we show that 4 pursuers also suffice under the "weighted region" constraints where the movement costs through different regions of the (genus zero) surface have (different) multiplicative weights.
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