Here we merge the two fields of Cops and Robbers and Graph Pebbling to introduce the new topic of Cops and Robbers Pebbling. Both paradigms can be described by moving tokens (the cops) along the edges of a graph to capture a special token (the robber). In Cops and Robbers, all tokens move freely, whereas, in Graph Pebbling, some of the chasing tokens disappear with movement while the robber is stationary. In Cops and Robbers Pebbling, some of the chasing tokens (cops) disappear with movement, while the robber moves freely. We define the cop pebbling number of a graph to be the minimum number of cops necessary to capture the robber in this context, and present upper and lower bounds and exact values, some involving various domination parameters, for an array of graph classes, including paths, cycles, trees, chordal graphs, high girth graphs, and cop-win graphs, as well as graph products. Furthermore we show that the analogous inequality for Graham's Pebbling Conjecture fails for cop pebbling and posit a conjecture along the lines of Meyniel's Cops and Robbers Conjecture that may hold for cop pebbling. We also offer several new problems.

翻译：本文融合了警察与强盗游戏和图卵石游戏两个领域，引入了警察与强盗卵石游戏这一新课题。两种范式均可描述为：通过沿图的边移动标记（警察）来捕获特殊标记（强盗）。在警察与强盗游戏中，所有标记均可自由移动；而在图卵石游戏中，部分追逐标记会在移动时消失，且强盗保持静止。在警察与强盗卵石游戏中，部分追逐标记（警察）会在移动时消失，而强盗可自由移动。我们定义了图的警察卵石数，即在此情境下捕获强盗所需的最少警察数量，并针对路径、环、树、弦图、高围长图、警察必胜图以及图积等一系列图类，给出了上下界和精确值，其中部分结果涉及各类支配参数。此外，我们证明了格雷厄姆卵石猜想在警察卵石情形下的类似不等式不成立，并提出了一个可能适用于警察卵石游戏的、类似梅涅尔警察与强盗猜想的猜想。本文还提出了若干新问题。