The Game of Cops and Eternal Robbers
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We introduce the game of Cops and Eternal Robbers played on graphs, where there are infinitely many robbers that appear sequentially over distinct plays of the game. A positive integer t is fixed, and the cops are required to capture the robber in at most t time-steps in each play. The associated optimization parameter is the eternal cop number, denoted by c_t^∞, which equals the eternal domination number in the case t=1, and the cop number for sufficiently large t. We study the complexity of Cops and Eternal Robbers, and show that game is NP-hard when t is a fixed constant and EXPTIME-complete for large values of t. We determine precise values of c_t^∞ for paths and cycles. The eternal cop number is studied for retracts, and this approach is applied to give bounds for trees, as well as for strong and Cartesian grids.
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