The Game of Cops and Eternal Robbers [article]

Anthony Bonato, Melissa Huggan, Trent Marbach, Fionn Mc Inerney
2020 arXiv   pre-print
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
more » ... ps 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.
arXiv:2003.03791v2 fatcat:wfujalroezfmbbcbijhr4q5wra