Graph searching - structural properties

图搜索-结构特性

• 批准号: RGPIN-2017-05065
• 负责人: Hahn, Gena
• 金额: $ 2.91万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=757422
• 关键词: Graph; searching; structural; properties

Cops-and-robbersgames have been studied since the 1980's, especially after thediscovery of connections with various graph widths. Algorithmic and complexity questionsseem to dominate the field because of applications in optimisation, among other things, but some basic questions remain unanswered: the cop-number of a graph, a characterisation of k-cop-win graphs, the length of an optimal game, etc.We wish to study these basic questions for the game as defined by Nowakowski andWinkler, and by Quillot, and to consider new models. Our graph theorygroup (two researchers, up to five students) has been studying theinfluence of loops on the number of cops needed (there are graphs that alternate between cop-winand not with the addition of loops one by one starting with a loopless graph). In the summer 2016we have worked on a version of the game in which the cops have to catchthe robber at a specified vertex (this is no longer a total knowledge game, the robber does not know the vertex). We wish to continuestudying these variants where we have some preliminary results. Our algorithm to decide whether k cops can catch r robbers on a given(finite) graph is used in robotics for robot motion planning and we think that specifying a capture vertex and an algorithm to describe a strategy would also be useful to roboticists.Further, we would like to generalise by considering not only the number of cops needed to catch therobber, but also the cost of doing so. Perhaps having a few more copswould cost less? The main problem here is defining the cost. We have considered severalpossibilities and will continue in this direction.Since finite regular graphs are not cop-win unless complete, Cayley graphs have not been much considered. We think theyare worth looking at again, especially in connection with some of the models (vaguely) describedabove.Last, we wish to study cop-win infinite graphs. We believe that understanding the infinite brings an understanding of the finite. Cops-and-robbers games on infinite graphs are different (there are many cop-win vertex transitive infinite graphs but only complete finite ones). Arecent paper by Lehner suggests that we have been looking the problem backwards, overlooking the fact that the reverse of a well-order is a well-order for finite graphs but not for infinite ones. Thus any attempt atcharacterising infinite cop-win graphs through an ordering like that for finite ones is doomed tofailure. This indicates that even forfinite graphs we should reconsider our approach. We propose to do just that.A part of the study of infinite cop-win graphs are the implications to structural properties of graphs. Generalising aconstruction from our 2009 paper we have examples of universal countable graphs that aredifferent from the unique (ultra)homogeneous countable graph (the random, or Rado, graph).We wish to continue studying such structures.

警察和抢劫犯的游戏自20世纪80年代以来一直在研究，特别是在发现各种图宽度的连接之后。数学和复杂性questionsseem占主导地位的领域，因为在优化中的应用，除其他外，但一些基本的问题仍然没有答案：一个图的COP数，一个表征的K-COP-WIN图，长度的最佳游戏，等等。我们希望研究这些基本问题的游戏所定义的Nowakowski和Winkler，和Quillot，并考虑新的模型。我们的图论小组（两名研究人员，最多五名学生）一直在研究循环对所需的cops数量的影响（有一些图在cop-win之间交替，而不是从无环图开始一个接一个地添加循环）。在2016年夏天，我们开发了一个版本的游戏，其中警察必须在指定的顶点抓住强盗（这不再是一个完全的知识游戏，强盗不知道顶点）。我们希望继续研究这些变种，我们有一些初步的结果。我们的算法，以决定是否k警察可以赶上r强盗在一个给定的（有限）图是用于机器人机器人运动规划，我们认为，指定一个捕获顶点和算法来描述一个战略也将是有用的roboticists.此外，我们想推广，不仅考虑警察需要抓住therobber的数量，但这样做的成本。或许多几个警察能省点钱？这里的主要问题是确定成本。我们已经考虑了几种可能性，并将继续在这个方向上。由于有限的正则图不是cop-win，除非完全，凯莱图没有得到太多的考虑。我们认为它们值得再看一遍，特别是与上面（模糊地）描述的一些模型有关。我们相信，理解无限带来了对有限的理解。无限图上的Cops-and-Robbers博弈是不同的（有许多Cops-win顶点传递无限图，但只有完全有限图）。Lehner最近的一篇论文表明，我们一直在向后看这个问题，忽略了一个事实，即一个良序的逆是有限图的良序，但不是无限图的良序。因此，任何试图通过类似于有限图的排序来刻画无限cop-win图的尝试都注定要失败。这表明即使对于有限图，我们也应该重新考虑我们的方法。无限cop-win图的研究的一部分是图的结构性质的影响。从我们2009年的论文中，我们有一些不同于唯一（超）齐次可数图（随机或Rado图）的泛可数图的例子。我们希望继续研究这样的结构。