We generalise the popular cops and robbers game to multi-layer graphs, where each cop and the robber are restricted to a single layer (or set of edges). We show that initial intuition about the best way to allocate cops to layers is not always correct, and prove that the multi-layer cop number is neither bounded from above nor below by any increasing function of the cop numbers of the individual layers. We determine that it is NP-hard to decide if $k$ cops are sufficient to catch the robber, even if every cop layer is a tree and a set of isolated vertices. However, we give a polynomial time algorithm to determine if $k$ cops can win when the robber layer is a tree. Additionally, we investigate a question of worst-case divisions of a simple graph into layers: given a simple graph $G$, what is the maximum number of cops required to catch a robber over all multi-layer graphs where each edge of $G$ is in at least one layer and all layers are connected? For cliques, suitably dense random graphs, and graphs of bounded treewidth, we determine this parameter up to multiplicative constants. Lastly we consider a multi-layer variant of Meyniel's conjecture, and show the existence of an infinite family of graphs whose multi-layer cop number is bounded from below by a constant times $n / \log n$, where $n$ is the number of vertices in the graph.

翻译：我们将流行的警察与小偷博弈推广至多层图，其中每个警察和小偷被限制在单一层（或边集）中。我们发现，关于如何最优分配警察到各层的直观判断并不总是正确的，并证明多层警察数量既不受各层警察数量任何递增函数的上界限制，也不受其下界限制。我们确定，即使每个警察层是树与孤立顶点集的组合，判断$k$个警察是否足以抓获小偷仍然是NP难的。然而，当小偷层为树时，我们给出了一个确定$k$个警察能否获胜的多项式时间算法。此外，我们研究了简单图最坏情况分层问题：给定简单图$G$，在所有满足$G$的每条边至少属于一个层且所有层均连通的多层图中，抓获小偷所需的最大警察数量是多少？对于团图、适当稠密的随机图以及有界树宽图，我们给出了该参数在乘法常数意义下的精确值。最后，我们考虑了Meyniel猜想的多层图变体，并证明存在无限图族，其多层警察数量的下界为常数乘以$n / \log n$，其中$n$为图的顶点数。