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 initialized by the Robber choosing a vertex r is an element of 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) >= max{zeta(G), zeta(H)} as well as - zeta(G square H) <= zeta (G) + psi(H) - 1, where psi(H) is the doubly resolving number of H. We also show that zeta(C-m square C-n)is mostly equal to two. (C) 2021 Elsevier B.V. All rights reserved.