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. We also offer several interesting problems and conjectures.

翻译：本文融合警察与强盗（Cops and Robbers）和图掷石（Graph Pebbling）两个领域，引入新课题“警察与强盗掷石问题”（Cops and Robbers Pebbling）。两种范式均可描述为：沿图边移动标记物（警察）以捕获特殊标记物（强盗）。在警察与强盗问题中，所有标记物自由移动；而在图掷石问题中，部分追逐标记物移动时消失，强盗保持静止。在警察与强盗掷石问题中，部分追逐标记物（警察）随移动消失，而强盗可自由移动。我们定义图的警察掷石数为该情境下捕获强盗所需的最少警察数量，并针对多种图类给出上下界及精确值，其中涉及多种支配参数。此外，我们提出若干有趣的问题与猜想。