20.500.12556/DiRROS-25076 The game of Cops and Robber on geodesic spaces The game of Cops and Robber is traditionally played on a finite graph. The purpose of this article is to introduce and analyze the game that is played on an arbitrary geodesic space (a compact, path-connected space endowed with intrinsic metric). It is shown that the game played on metric graphs is essentially the same as the discrete game played on abstract graphs and that for every compact geodesic surface there is an integer $c$ such that $c$ cops can win the game against one robber, and $c$ only depends on the genus $g$ of the surface. It is shown that $c=3$ for orientable surfaces of genus $0$ or $1$ and nonorientable surfaces of crosscap number $1$ or $2$ (with any number of boundary components) and that $c=O(g)$ and that $c=\Omega(\sqrt{g})$ when the genus $g$ is larger. The main motivation for discussing this game is to view the cop number (the minimum number of cops needed to catch the robber) as a new geometric invariant describing how complex is the geodesic space. pursuit-evasion game games on graphs Cops and Robber game geodesic space metric surface true false true Angleški jezik Ni določen Neznano 2026-01-09 08:42:56 2026-01-09 08:42:56 2026-01-10 03:48:47 0000-00-00 00:00:00 2025 0 0 str. 1827-1860 iss. 6 Vol. 77 Dec. 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2025-12-01 519.17 0008-414X 10.4153/S0008414X24000543 264039171 the-game-of-cops-and-robber-on-geodesic-spaces.pdf the-game-of-cops-and-robber-on-geodesic-spaces.pdf 1 71D987CD39ABBCEF97ACAF826962DFB1 e7f38a5769a1786fa2030bbe7f2902d89b7d5a3f6bec13cad7e1cede36a103d1 2f27fe25-ed2f-11f0-94a7-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=37691 https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/game-of-cops-and-robber-on-geodesic-spaces/E33876051132DB114E090A9340774B49 0 3bcddd68-ed2f-11f0-94a7-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=37692 Inštitut za matematiko, fiziko in mehaniko 0 0 0