We study a variation of the cops and robber game characterising treewidth, where in each play at most q cops can be placed in order to catch the robber, where q is a parameter of the game. We prove that if k cops have a winning strategy in this game, then k cops have a monotone winning strategy. As a corollary we obtain a new characterisation of bounded depth treewidth, and we give a positive answer to an open question by Fluck, Seppelt and Spitzer (2024), thus showing that graph classes of bounded depth treewidth are homomorphism distinguishing closed. Our proof of monotonicity substantially reorganises a winning strategy by first transforming it into a pre-decomposition, which is inspired by decompositions of matroids, and then applying an intricate breadth-first "cleaning up" procedure along the pre-decomposition (which may temporarily lose the property of representing a strategy), in order to achieve monotonicity while controlling the number of cop placements simultaneously across all branches of the decomposition via a vertex exchange argument. We believe this can be useful in future research.

翻译：我们研究了刻画树宽的一种警察与小偷博弈变体，其中每局游戏最多可放置q个警察来抓捕小偷，q为游戏参数。我们证明：若k个警察在此游戏中存在必胜策略，则k个警察存在单调必胜策略。作为推论，我们得到了有界深度树宽的新刻画，并对Fluck、Seppelt与Spitzer（2024）提出的开放问题给出肯定回答，从而表明有界深度树宽的图类是同态区分封闭的。我们的单调性证明通过以下方式对必胜策略进行实质重组：首先将其转化为受拟阵分解启发的预分解，随后沿该预分解执行精密的广度优先"清理"流程（该流程可能暂时失去表征策略的性质），并通过顶点交换论证同时控制分解所有分支上的警察放置次数。我们相信该方法对未来研究具有价值。