We consider a pursuit-evasion game that describes the process of extinguishing a fire burning on the nodes of an undirected graph. We denote the minimum number of firefighters required by ffn(G) and provide almost sharp bounds to this graph parameter for complete binary trees. We show that deciding whether ffn(G) <= m for given G and m is NP-hard. Furthermore, we show that shortest strategies can have superpolynomial length, leaving open whether the problem is in NP. We provide a construction that allows for transferring these results to a well-established Cops and Robbers variant called the "Hunter and Rabbit game".

翻译：我们考虑一种追逃博弈，该博弈描述了在无向图节点上扑灭火焰的过程。我们用ffn(G)表示所需消防员的最小数量，并为完全二叉树提供该图参数的几乎严格界。我们证明，对于给定的图G和整数m，判定ffn(G) ≤ m是NP困难的。此外，我们证明最短策略的长度可以是超多项式的，这使该问题是否属于NP类保持开放。我们提出一种构造方法，可将这些结果迁移到一个成熟的“警察与小偷”变体——即“猎人与兔子博弈”中。