The Localization Game On Cartesian Products

The localization game is played by two players: a Cop with a team of $k$ cops, and a Robber. The game is initialised by the Robber choosing a vertex $r \in V$, unknown to the Cop. Thereafter, the game proceeds turn based. At the start of each turn, the Cop probes $k$ vertices and in return receives a distance vector. If the Cop can determine the exact location of $r$ from the vector, the Robber is located and the Cop wins. Otherwise, the Robber is allowed to either stay at $r$, or move to $r'$ in the neighbourhood of $r$. The Cop then again probes $k$ vertices. The game continues in this fashion, where the Cop wins if the Robber can be located in a finite number of turns. The localization number $\zeta(G)$, is defined as the least positive integer $k$ for which the Cop has a winning strategy irrespective of the moves of the Robber. In this paper, we focus on the game played on Cartesian products. We prove that $\zeta( G \square H) \geq \max\{\zeta(G), \zeta(H)\}$ as well as $\zeta(G \square H) \leq \zeta(G) + \psi(H) - 1$ where $\psi(H)$ is a doubly resolving set of $H$. We also show that $\zeta(C_m \square C_n)$ is mostly equal to two.