All Ordinals are Cop-Robber Ordinals

TL;DR

The paper addresses which ordinals can appear as the maximum capture time $\rho(G)$ in cop-win graphs on infinite networks. It introduces CR-ordinals and provides explicit two-part constructions: $\mathcal{G}_{\gamma}$ for infinite limit ordinals with $\rho(\mathcal{G}_{\gamma})=\gamma$, and $\mathcal{G}_{\gamma+n}$ for successor ordinals with $\rho(\mathcal{G}_{\gamma+n})=\gamma+n$. These constructions, together with the prior result that $\omega$ is a CR-ordinal, establish that every ordinal is a CR-ordinal. The work thereby completes the CR-ordinal landscape and sheds light on how pursuit-evasion dynamics realize any ordinal capture time in infinite graphs.

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.

Theorems & Definitions (26)

• Proposition 1
• proof
• Lemma 2
• proof
• Lemma 3
• Lemma 4
• proof
• Corollary 5
• proof
• Lemma 6