Line-of-Sight Pursuit in Monotone and Scallop Polygons

We study a turn-based game in a simply connected polygonal environment Q between a pursuer and an adversarial evader . Both players can move in a straight line to any point within unit distance during their turn. The pursuer wins by capturing the evader, meaning that their distance satisfies d(,) ≤ 1, while the evader wins by eluding capture forever. Both players have a map of the environment, but they have different sensing capabilities. The evader always knows the location of . Meanwhile, only has line-of-sight visibility: observes the evader's position only when the line segment connecting them lies entirely within the polygon. Therefore must search for when the evader is hidden from view. We provide a winning strategy for in two families of polygons: monotone polygons and scallop polygons. In both families, a straight line L can be moved continuously over Q so that (1) L n Q is a line segment and (2) every point on the boundary ?Q is swept exactly once. These are both subfamilies of strictly sweepable polygons. The sweeping motion for a monotone polygon is a single translation, and the sweeping motion for a scallop polygon is a single rotation. Our algorithms use rook's strategy during its pursuit phase, rather than the well-known lion's strategy. The rook's strategy is crucial for obtaining a capture time that is linear in the area of Q. For both monotone and scallop polygons, our algorithm has a capture time of O(n(Q) + area(Q)), where n(Q) is the number of polygon vertices.