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 $c_t^{\infty},$ 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 $c_t^{\infty}$ 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.

中文翻译：

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警察与永恒强盗的游戏

我们介绍了在图表上玩的警察和永恒强盗的游戏，其中有无限多的强盗在不同的游戏玩法中依次出现。正整数 $t$ 是固定的，并且要求警察在每场比赛中最多在 $t$ 时间步长内抓捕强盗。相关的优化参数是永恒的 cop 数，用 $c_t^{\infty} 表示，它等于 $t=1,$ 情况下的永恒支配数和足够大的 $t 的 cop 数。 Cops 和 Eternal Robbers 的复杂性，并证明当 $t$ 是一个固定常数时游戏是 NP-hard 并且对于大 $t$ 值的 EXPTIME-complete。我们确定路径和循环的 $c_t^{\infty}$ 的精确值。研究了用于收回的永恒 cop 数，这种方法适用于为树以及强网格和笛卡尔网格提供边界。