Computing strategies for multiple cops on infinite graphs

Abstract

Cops and robbers is a vertex pursuit game played on connected reflexive graphs. While the game is most often played with a single cop on a finite graph, we consider the variation in which n cops pursue the robber on an infinite graph. For each $n⩾2$ , we show there is a computable graph which is classically cop-win and can be won by n cops using a computable strategy, but cannot be won by $n-1$ cops using a computable strategy. Furthermore, we show that the index set of n-cop-win computable graphs is ${\mathrm{\Pi }}_1^1$ -complete.