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Containment: A Variation of Cops and Robber
Graphs and Combinatorics ( IF 0.7 ) Pub Date : 2020-02-19 , DOI: 10.1007/s00373-020-02140-5
Danny Crytser , Natasha Komarov , John Mackey

We consider “Containment”: a variation of the graph pursuit game of Cops and Robber in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop), and the cops win by “containing” the robber—that is, by occupying all \(\deg (v)\) of the edges incident with a vertex v while the robber is at v. We develop bounds that relate the minimal number of cops, \(\xi (G)\), required to contain a robber to the well-known “cop-number” c(G) in the original game: in particular, \(c(G) {\le } \xi (G) {\le } \gamma (G) \Delta (G)\). We note that \(\xi (G) {\ge } \Delta (G)\) for all graphs G, and analyze several families of graphs in which equality holds, as well as several in which the inequality is strict. We also give examples of graphs which require an unbounded number of cops in order to contain a robber, and show that there exist cubic graphs on n vertices with \(\xi (G) = \Omega (n^{1/6})\).



中文翻译:

遏制:警察和强盗的变化

我们考虑“包容”:警察和强盗的图追求游戏的一种变体,其中警察从边移动到相邻边,强盗从顶点移动到相邻顶点(但不能沿着被警察占据的边移动),并且警察通过“遏制”强盗而获胜,也就是说,当强盗位于v时,占领占据顶点v的边的所有\(\ deg(v)\)。我们开发了一个边界,该边界与包含抢劫犯的最小警察数量\(\ xi(G)\)有关,而后者是原始游戏中众所周知的“警察编号” cG):特别是\( c(G){\ le} \ xi(G){\ le} \ gamma(G)\ Delta(G)\)。我们注意到\(\ xi(G){\ ge} \ Delta(G)\)对于所有图G,分析几个拥有相等性的图族,以及几个不等式严格的图族。我们还给出了一些图的示例,这些图需要无限数量的警察才能容纳强盗,并显示在\(\ xi(G)= \ Omega(n ^ {1/6})的n个顶点上存在立方图。\)

更新日期:2020-02-19
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