Various models to quantify the reliability of a network have been studied where certain components of the graph may fail at random and the probability that the remaining graph is connected is the proxy for reliability. We introduce a strengthening of one of these models by considering the probability that the remaining graph is cop-win. A graph is cop-win if one cop can win the game Cops and Robber. More precisely, for a graph G with nodes that are operational independently with probability p, the node cop-win reliability of G, denoted , is the probability that the operational nodes induce a cop-win subgraph of G. It is then of interest to find graphs G with n nodes and m edges such that for all p ∈ [0, 1] and all graphs H with n nodes and m edges. We show that such graphs exist among unicyclic and bicyclic graphs, respectively.

中文翻译：

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单环和双环图的节点共赢可靠性

已经研究了量化网络可靠性的各种模型，其中图的某些组件可能会随机失效，并且剩余图连接的概率是可靠性的代表。我们通过考虑剩余图是 cop-win 的概率来加强这些模型之一。如果一名警察可以赢得警察和强盗游戏，则图表是警察获胜。更准确地说，对于具有以概率p独立操作的节点的图G ， G的节点 cop-win 可靠性，表示为，是操作节点引发G的 cop-win 子图的概率。然后有趣的是找到图G具有n 个节点和m个边，使得对于所有p  ∈ [0, 1] 和所有具有n 个节点和m个边的图H。我们证明了这种图分别存在于单环和双环图中。