## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 307-351 |

Number of pages | 45 |

Journal | International Journal of Computational Geometry and Applications |

Volume | 29 |

Issue number | 4 |

DOIs | |

State | Published - 2019 |

### Bibliographical note

Funding Information:This work was supported in part by the Institute for Mathematics and its Applications and in part by NSF Grant DMS-1156701. Volkan Isler was supported in part by NSF Grant IIS-0917676. We thank Narges Noori for helpful conversations and feedback, and we are grateful to the anonymous referees for their suggestions on improving this exposition.

## Keywords

- Pursuit-evasion
- monotone polygon
- sweepable polygon