In the $(s,d)$-spy game over a graph, introduced by Cohen et al. in 2016, one spy and $k$ guards occupy vertices of a graph and, at each turn, each guard may move along one edge and the spy may move along at most $s$ edges. The guards win if, after a finite number of turns, they ensure that the spy always remains at distance at most $d$ from at least one guard. The guard number is the minimum number of guards such that the guards have a winning strategy. In this paper, we investigate the spy game variant in which the guards are placed first, before the spy. We obtain a polynomial time algorithm for every speed $s\geq 2$ and distance $d\geq 0$ when the number of guards is a constant, which leads to a fixed parameter tractable algorithm on the $P_4$-fewness of the graph. We also prove that the spy game is NP-hard even in bipartite graphs with bounded diameter, for every speed $s\geq 2$ and distance $d\geq 0$.

翻译：在图上的 $(s,d)$-间谍博弈中（由Cohen等人于2016年提出），一名间谍与$k$名守卫占据图的顶点，每回合每个守卫可沿一条边移动，而间谍可沿至多$s$条边移动。若经过有限回合后，守卫能确保间谍始终与至少一名守卫保持距离不超过$d$，则守卫获胜。守卫数定义为保证守卫存在必胜策略的最小守卫数量。本文研究守卫先于间谍部署的间谍博弈变种。当守卫数为常数时，我们针对所有速度$s\geq 2$和距离$d\geq 0$提出多项式时间算法，进而得到基于图的$P_4$-稀少性的固定参数可解算法。我们还证明，对于所有速度$s\geq 2$和距离$d\geq 0$，即便在直径有界二分图中，间谍博弈也是NP困难的。