The game of Cops and Eternal Robbers

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