Pursuit Evasion on Polyhedral Surfaces

We consider the following variant of a classical pursuit-evasion problem: how many pursuers are needed to capture a single ( adversarial ) evader on a closed polyhedral surface in three dimensions? The players move on the 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 (Littlewood in Littlewood?s miscellany, Cambridge University Press 1986 ), 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-time model, 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 four pursuers always suffice (upper bound), and that three 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 ( 4 g + 4 ) pursuers. Finally, we show that four pursuers also suffice under the “weighted region” constraints where the movement costs through different regions of the (genus zero) surface have (different) multiplicative weights.