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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 ∈ 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 ζ(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 thatwhere ψ(H) is a doubly resolving set of H. We also show that ζ(Cm Cn) is mostly equal to two.
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