All Ordinals are Cop-Robber Ordinals

Abstract

The game of cops and robbers, played on a fixed graph $G$, is a two-player game, where the cop and the robber (the players) take turns in moving to adjacent vertices. The game finishes if the cop lands on the robber's vertex. In that case we say that the cop wins. If the cop can always win, regardless of the starting positions, we say that $G$ is a cop-win graph. For a finite cop-win graph $G$ we can ask for the minimum number $n$ such that, regardless of the starting positions, the game will end in at most $n$ steps. This number is called the maximum capture time of $G$. By looking at finite paths, we see that any non-negative integer is the maximum capture time for a cop-win graph. What about infinite cop-win graphs? In this case, the notion of capture time is nicely generalised if one works with ordinals, and so the question becomes which ordinals can be the maximum capture time of a cop-win graph? These ordinals are called CR (Cop-Robber)-ordinals. In this paper we fully settle this by showing that all ordinals are CR-ordinals, answering a question of Bonato, Gordinowicz and Hahn.